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~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 1 : §Ó˚° ˆòy°à!ï˛§Ó˚° ˆòy°à!ï˛§Ó˚° ˆòy°à!ï˛§Ó˚° ˆòy°à!ï˛§Ó˚° ˆòy°à!ï˛
àë˛öàë˛öàë˛öàë˛öàë˛ö
1.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
1.2 §Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#
1.2.2 §Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛
1.2.2 §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î
1.2.3 xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
1.2.4 §Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey (parameter)
1.3 §Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´
1.4 §Ó˚° ˆòy°à!ï˛§¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î
1.4.1 !fl±Ç û˛Ó˚
1.4.2 !fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚
1.4.3 §Ó˚° ˆòy°Ü˛
1.4.4 ˆÎÔ!àܲ ˆòy°Ü˛
1.4.5 ÓƒyÓï≈˛ ˆòy°Ü˛
1.4.6 ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ ôyÓ˚ܲ Óï≈˛ö#
1.4.7 !fl±Ç myÓ˚y Î%_´ û˛Ó˚ Î%@¿
1.5 §Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y
1.6 §yÓ˚yÇ¢
1.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1.8 í˛z_Ó˚üy°y
1.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
•z!ï˛˛õ)ˆÏÓ≈ xy˛õ!ö öyöy ôÓ˚ˆÏöÓ˚ à!ï˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö– xy˛õ!ö °«˛ƒ ܲˆÏÓ˚ ÌyܲˆÏÓö ˆÎñ ~Ó˚ üˆÏôƒ
ˆÜ˛yö ˆÜ˛yö!ê˛Ó˚ ˆ«˛ˆÏe à!ï˛¢#° Ó›!ê˛ !ö!ò≈‹T §üÎ˚ xhsˇÓ˚ ˛õ)ˆÏÓ≈Ó˚ xÓfliyö Óy xyÜ,˛!ï˛ˆÏï˛ !ú˛ˆÏÓ˚ xyˆÏ§– í˛zòy•Ó˚î
!•§yˆÏÓ â%Ó˚hsˇ ˛õyáyÓ˚ ˆÓœí˛ñ ˆòy°öy Óy â!í˛¸Ó˚ ˆ˛õu%˛°yˆÏüÓ˚ ܲÌy Ó°y ÎyÎ˚– çˆÏ°Ó˚ í˛z˛õÓ˚ !òˆÏÎ˚ Îáö ï˛Ó˚Düy°y
x@ˇÃ§Ó˚ •Î˚ ï˛áö ç°ï˛ˆÏ° û˛y§üyö ˆÜ˛yöÄ Ó›ˆÏܲ xy˛õ!ö ~û˛yˆÏÓ à!ï˛¢#° •ˆÏï˛ ˆòáˆÏÓö– ~•z ôÓ˚ˆÏöÓ˚
à!ï˛ˆÏܲ xyüÓ˚y Ó!° ‘˛õÎ≈yÓ,_ à!ï˛’ (periodic motion)– §Ó˚° ˆòy°à!ï˛Ä ~ܲ ôÓ˚ˆÏöÓ˚ ˛õÎ≈yÓ,_ à!ï˛ñ ÎyÓ˚
ày!î!ï˛Ü˛ !ӈϟ’£Ïî á%Ó §•ˆÏç•z ܲÓ˚y §Ω˛Ó– ~•z ~ܲˆÏܲ xyüÓ˚y §Ó˚° ˆòy°à!ï˛Ó˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï≈˛ ~ÓÇ
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~•z à!ï˛Ó˚ ôü≈=!° xyˆÏ°yã˛öy ܲÓ˚Ó– ~ôÓ˚ˆÏöÓ˚ à!ï˛§¡õߨ Ó›Ó˚ Îy!sfܲ ¢!_´Ó˚ §ÇÓ˚«˛îÄ ˛õÓ˚#«˛y ܲÓ˚Ó– §Ó˚°
ˆòy°à!ï˛Ó˚ í˛zòy•Ó˚î !•§yˆÏÓ Ü˛ˆÏÎ˚ܲ!ê˛ Îy!sfܲ ˆòy°ˆÏܲÓ˚ í˛zòy•Ó˚î åÈyí˛¸yÄ ~•z ~ܲˆÏܲ xyüÓ˚y ï˛!í˛¸Í xyˆÏÓ¢
Ä ôyÓ˚ܲ myÓ˚y à!ë˛ï˛ ï˛!í˛¸ÍÓï≈˛ö#Ó˚ üˆÏôƒ ï˛!í˛¸ˆÏï˛Ó˚ ≤ÃÓy• !ÓˆÏÓã˛öy ܲÓ˚Ó– xy˛õ!ö °«˛ƒ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ
~•z Óï≈˛ö#ˆÏï˛ ï˛!í˛¸Í ≤ÃÓy• ày!î!ï˛Ü˛ !òܲ !òˆÏÎ˚ §Ó˚° ˆòy°à!ï˛ §¡õߨ ˆòy°ˆÏܲÓ˚ xö%Ó˚*˛õ–
≤ÃÜ,˛!ï˛ˆÏï˛ Ó‡ ôÓ˚ˆÏöÓ˚ à!ï˛ !ÓhflÏyÓ˚ ΈÏÌ‹T x“ ÌyܲˆÏ° §Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛ ˛õy°ö ܲˆÏÓ˚– ~•z ܲyÓ˚ˆÏî
˛õòyÌ≈!ÓòƒyÎ˚ §Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c x˛õ!Ó˚§#ü– ~•z ~ܲˆÏܲÓ˚ üyôƒˆÏü xy˛õ!ö §Ó˚° ˆòy°à!ï˛Ó˚ ˆû˛Ôï˛ Ä
ày!î!ï˛Ü˛ ôü≈ §¡∫ˆÏ¶˛ !Ó¢ò ˛õ!Ó˚ã˛Î˚ °yû˛ ܲÓ˚ˆÏÓö
í˛ zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒ
~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö
§Ó˚° ˆòy°à!ï˛Ó˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï≈˛=!° !ÓÓ,ï˛ Ü˛Ó˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
§Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î!ê˛ fliy˛õö Ä ˆ§!ê˛Ó˚ §üyôyö ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
§Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°ö¢#° Ó›Ó˚ à!ï˛¢!_´Ó˚ !•§yÓ Ü˛Ó˚ˆÏï˛ ~ÓÇ ¢!_´Ó˚ §ÇÓ˚«˛î ≤Ã!ï˛˛õyòö ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
!Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛¡õö¢#° Ó›ï˛ˆÏsfÓ˚ à!ï˛Ó˚ üˆÏôƒ §yò,¢ƒ ˆòáyˆÏï˛ ˛õyÓ˚ˆÏÓö–
1.2 §Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#§Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#§Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#§Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#§Ó˚° ˆòy°à!ï˛Ó˚ §ÇK˛y Ä ˜Ó!¢‹TƒyÓ°#
ˆÎ ˆÜ˛yöÄ ˛õÎ≈yÓ,_ à!ï˛ˆÏܲ•z §Ó˚° ˆòy°à!ï˛ Ó°y ÎyÎ˚ öy– ˆÜ˛yöÄ Ó›ï˛sf ܲˆÏÎ˚ܲ!ê˛ ¢ï≈˛ ˛õy°ö ܲÓ˚ˆÏ°ñ
ï˛ˆÏÓ•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛zqÓ •ˆÏï˛ ˛õyˆÏÓ˚– xyüyÓ˚ ~•z ¢ï≈˛=!°ˆÏܲ ày!î!ï˛Ü˛û˛yˆÏÓ ≤Ãܲy¢ ܲˆÏÓ˚ §Ó˚°
ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î àë˛ö ܲÓ˚Ó– ~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö ˆÌˆÏܲ xyüÓ˚y §Ó˚° ˆòy°à!ï˛Ó˚ ôü≈=!°ˆÏï˛
í˛z˛õö#ï˛ •ˆÏï˛ ˛õyÓ˚Ó– ≤Ã̈Ïü §Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛=!° xyˆÏ°yã˛öy ܲÓ˚y Îyܲ–
1.2.1 §Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛§Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛§Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛§Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛§Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛
ˆÜ˛yöÄ Ó›Ü˛îyÓ˚ à!ï˛ˆÏܲ ï˛áö•z §Ó˚° ˆòy°à!ï˛ Ó°y ÎyÎ˚ñ Îáö !öˆÏ¡¨Ó˚ ¢ï≈˛=!° ˛õy!°ï˛ •Î˚:
(i) ӛܲîy!ê˛Ó˚ à!ï˛ ~ܲ §Ó˚°ˆÏÓ˚áyÎ˚ xyÓm ÌyˆÏܲ ~ÓÇ
(ii) ӛܲîy!ê˛Ó˚ í˛z˛õÓ˚ ˆ§!ê˛Ó˚ à!ï˛˛õˆÏÌÓ˚ í˛z˛õÓ˚ xÓ!fliï˛ ˆÜ˛yöÄ !ö!ò≈‹T !Ó®% x!û˛ü%ˆÏáñ ˙ !Ó®% ˆÌˆÏܲ
ӛܲîy!ê˛Ó˚ ò)Ó˚ˆÏcÓ˚ §üyö%˛õyï˛# ~ܲ!ê˛ ≤Ãï˛ƒyöÎ˚ܲ Ó° ܲyç ܲˆÏÓ˚–
~ˆÏ«˛ˆÏe ܲˆÏÎ˚ܲ!ê˛ !Ó£ÏÎ˚ üˆÏö Ó˚yáˆÏï˛ •ˆÏÓ– ˛≤ÃÌüï˛ñ ˆÜ˛yöÄ ò,ì˛¸ Ó›Ó˚ ã˛°ö (translational) à!ï˛Ó˚ ˆ«˛ˆÏe
Î!ò í˛z˛õˆÏÓ˚Ó˚ ¢ï≈˛=!° ≤ÈÏÎyçƒ •Î˚ñ ï˛ˆÏÓ ï˛yÓ˚ û˛Ó˚ˆÏܲw!ê˛ Ó›Ü˛îyÓ˚ üï˛ xyã˛Ó˚î ܲÓ˚ˆÏÓ ~ÓÇ Ó›!ê˛ §Ó˚°
ˆòy°à!ï˛ˆÏï˛ ã˛°ˆÏï˛ ÌyܲˆÏÓ– !mï˛#Î˚ï˛ñ ˆÎ ˆÜ˛yöÄ Ó›ï˛sf Î!ò ˆÜ˛yˆÏöy ~ܲ!ê˛ Ó˚y!¢Ó˚ SÎÌy ~ܲ!ê˛ ˆÜ˛yîñ
ï˛!í˛¸ï˛yôyöñ ï˛!í˛¸Í≤ÃÓy•V ˆ«˛ˆÏe í˛z˛õˆÏÓ˚Ó˚ ¢ï≈˛=!°Ó˚ xö%Ó˚*˛õ ¢ï≈˛ ≤ÈÏÎyçƒ •Î˚ñ ï˛ˆÏÓ ˆ§•z Ó˚y!¢!ê˛Ó˚ §Ó˚°
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ˆòy°à!ï˛Ó˚ xö%Ó˚*˛õ ˆòy°ö °«˛ƒ ܲÓ˚y ÎyˆÏÓ– ˛õÓ˚Óï≈˛# 1.4 xLjϢ í˛zòy•Ó˚î=!°Ó˚ üˆÏôƒ xy˛õ!ö ~ çyï˛#Î˚
ˆòy°ö ˆòáˆÏï˛ ˛õyˆÏÓö– ~=!°ˆÏܲ xyüÓ˚y §Ó˚° ˆòy°à!ï˛ öyˆÏü x!û˛!•ï˛ ܲÓ˚Ó–
1.2.2. §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î§Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î§Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î§Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î§Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î
ôÓ˚y Îyܲ m û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ Ó›Ü˛îyÓ˚ à!ï˛ xyüyˆÏòÓ˚ !öˆÏò≈¢ï˛ˆÏsfÓ˚ x xˆÏ«˛ xyÓk˛ ~ÓÇ à!ï˛˛õˆÏÌÓ˚ í˛z˛õÓ˚
!ö!ò≈‹T !Ó®%!ê˛ !öˆÏò≈¢ï˛ˆÏsfÓ˚ ü)°!Ó®% (origin), 0– ӛܲîy!ê˛ 0
!Ó®%ˆÏï˛ ÌyܲˆÏ° ï˛yÓ˚ í˛z˛õÓ˚ ˆÜ˛yö Ó°•z ܲyç ܲˆÏÓ˚ öy– ܲyˆÏç•z
~ܲüye O !Ó®%ˆÏï˛•z ˆ§!ê˛ !fliÓ˚ ÌyܲˆÏï˛ ˛õyˆÏÓ˚– ~çöƒ 0 !Ó®% Ïܲ
ӛܲîyÓ˚ §yüƒ!Ó®% Ó°y •ˆÏÓ– ӛܲîy!ê˛ Îáö ü)°!Ó®% ˆÌˆÏܲ x
ò)Ó˚ˆÏc P !Ó®%ˆÏï˛ ÌyˆÏܲñ ï˛áö ï˛yÓ˚ í˛z˛õÓ˚ x-~Ó˚ §üyö%˛õyï˛# -kx
≤Ãï˛ƒyöÎ˚ܲ Ó° 0 !Ó®% x!û˛ü%ˆÏá ܲyç ܲˆÏÓ˚– ˆÎáyˆÏö k ~ܲ!ê˛ §üyö%˛õyï˛ ô Óܲ– °«˛î#Î˚ ˆÎñ x Ó˚y!¢ Ä
≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ !ã˛•´ §Ó≈òy•z !Ó˛õÓ˚#ï˛ ~ÓÇ !ÓˆÏÎ˚yà !ã˛•´!ê˛ ~çöƒ•z ÓƒÓ•*ï˛ •ˆÏÎ˚ˆÏåÈ– ~áö !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚
§)e xö%§yˆÏÓ˚ û˛Ó˚ Ä cÓ˚ˆÏîÓ˚ =îú˛° ~ÓÇ Ó° F -~Ó˚ §üï˛y ˆÌˆÏܲ xy˛õ!ö !°áˆÏï˛ ˛õyˆÏÓ˚ö –
= = −mx F kx ............. 1.1
xÌÓyñ
ω= − = − 2kx x x
m
, ˆÎáyˆÏö ày!î!ï˛Ü˛ §%!ÓôyÓ˚ çöƒ km
Ó˚y!¢!ê˛Ó˚ fliyˆÏö + ω 2
ˆ°áy •ˆÏÎ˚ˆÏåÈ˛– 1.1 §ü#ܲÓ˚î!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î– §ü#ܲÓ˚î!ê˛ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ
x -~Ó˚ üyö Îáö ˛õ!ç!ê˛û˛ ï˛áö ӛܲîyÓ˚ cÓ˚î ˆöˆÏà!ê˛û˛ xÌ≈yÍ ï˛áö ӛܲîyÓ˚ ˆÓà x !òˆÏܲ ü)°!Ó®%Ó˚
!Ó˛õÓ˚#ï˛ü%ˆÏá ÌyܲˆÏ° ï˛yÓ˚ üyö ܲüˆÏï˛ ÌyܲˆÏÓ ~ÓÇ x -~Ó˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ ü)°!Ó®% x!û˛ü%ˆÏá ÌyܲˆÏ° ï˛yÓ˚
üyö Óyí˛¸ˆÏï˛ ÌyܲˆÏÓ–
!Ó˛õÓ˚#ï˛e´ˆÏüñ x -~Ó˚ üyö Îáö ˆöˆÏà!ê˛û˛ñ ï˛áö ӛܲîyÓ˚ cÓ˚î ˛õ!ç!ê˛û˛ñ xÌ≈yÍ ˆÓà x !òˆÏܲ ü)°!Ó®%
x!û˛ü%ˆÏá ÌyܲˆÏ° ï˛yÓ˚ ˆÓà Óyí˛¸ˆÏï˛ ÌyܲˆÏÓ ~ÓÇ ˆÓà x ~Ó˚ ˛!Ó˛õÓ˚#ï˛ !òˆÏܲ ü)°!Ó®%Ó˚ !Ó˛õÓ˚#ï˛ü%ˆÏá ÌyܲˆÏ°
ï˛yÓ˚ üyö ܲüˆÏï˛ ÌyܲˆÏÓ–
1.1 §ü#ܲÓ˚î!ê˛ ~ܲ!ê˛ ˜Ó˚!áܲñ §ü§_¥ñ !mï˛#Î˚ e´ˆÏüÓ˚ (linear homogeneous second order) xÓܲ°
§ü#ܲÓ˚î– xyüÓ˚y çy!ö ~Ó˚ §üyôyˆÏö ò%•z!ê˛ x!ö!ò≈‹T ô Óܲ ÌyܲˆÏÓ– ~ÓyÓ˚ ~Ó˚ §üyôyö!ê˛ ˆòáy Îyܲ–
1.2.3 xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
§ü#ܲÓ˚î 1.1 •Î˚ï˛ xy˛õöyÓ˚ ܲyˆÏåÈ á%Ó•z ˛õ!Ó˚!ã˛ï˛– ~áyˆÏö x Ó˚y!¢!ê˛ §üÎ˚ t-~Ó˚ ~üö ~ܲ!ê˛ xˆÏ˛õ«˛Ü˛
ÎyˆÏܲ ò%•zÓyÓ˚ xÓܲ°ö ܲÓ˚ˆÏ° – ω 2 §•à §ˆÏüï˛ ˙ ~ܲ•z xˆÏ˛õ«˛Ü˛ ˛õyÄÎ˚y ÎyÎ˚– xyüÓ˚y çy!ö sinω t ~ÓÇ
cosω t ~•z ôÓ˚ˆÏöÓ˚ xˆÏ˛õ«˛Ü˛– ~ ò%!ê˛ˆÏܲ !ü!°ˆÏÎ˚ xyüÓ˚y 1.1 xÓܲ°ö §ü#ܲÓ˚î!ê˛Ó˚ §yôyÓ˚î §üyôyö !°áˆÏï˛
˛õy!Ó˚ :
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x = A sin ωt + B cost ωt ....... 1.2
ˆÎáy Ïö A Ä B x!ö!ò≈‹T ô Óܲ– ~áyˆÏö §yôyÓ˚î §üyôyö Ó°ˆÏï˛ ~ê˛y•z ˆÓyé˛yÎ˚ ˆÎñ ˙ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚
x§Çრ!ӈϢ£Ï §üyôyö ÌyܲˆÏ°Äñ ˆÎ=!° §Ó•z x!ö!ò≈‹T ô Óܲ=!°Ó˚ !ӈϢ£Ï üyö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyˆÏÓ–
ˆÎüöñ 1.2 ~Ó˚ ˆ«˛ˆÏe
B = 0 ôÓ˚ˆÏ° x = A sinωt, xyÓyÓ˚
A = 0 ôÓ˚ˆÏ° x = Bcosωt
xy˛õ!ö û˛yÓˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ
j te ω
Ó˚y!¢!ê˛Ä ~ܲ!ê˛ §Ω˛yÓƒ §üyôyö– !ܲv ~!ê˛
( )ω ω+cos sint j t
~Ó˚
§üyö– xÌ≈yÍñ ~!ê˛
ωcos t
Ä
ωsin t
~Ó˚ ~ܲ!ê˛ ˜Ó˚!áܲ §Ç!ü◊î– §%ï˛Ó˚yÇ 1.2 §üyôyˆÏö ~!ê˛ˆÏܲ
xy°yòyû˛yˆÏÓ xhsˇû%≈˛_´ ܲÓ˚yÓ˚ ≤ÈÏÎ˚yçö ˆö•z–
~áö Î!ò xyüÓ˚y ôˆÏÓ˚ !ö•z
θ θ= − =sin , cosA a B a
.............1.3
ï˛ˆÏÓ 1.2 §ü#ܲÓ˚î ˆÌˆÏܲ θ ω θ ω= − +sin sin cos cosx a t a t
Óyñ x =
( )ω θ+cosa t
............1.4
~áyˆÏö xyüÓ˚y A Ä B ~Ó˚ fliyˆÏö xöƒ ò%•z x!ö!ò≈‹T ô Óܲ a Ä θ ÓƒÓ•yÓ˚ ܲˆÏÓ˚!åÈ– x!ö!ò≈‹T ô ÓܲmˆÏÎ˚Ó˚
üyö t = 0 §üˆÏÎ˚ §Ó˚î x ~Ó˚ üyö x0 ~ÓÇ ˆÓà
x ~Ó˚ üyö x0 ˆÌˆÏܲ §•ˆÏç•z !öî≈Î˚ ܲÓ˚y ÎyÎ˚– 1.4
§ü#ܲÓ˚î Ä ~Ó˚ ˆÌˆÏܲ °∏˛ ˆÓˆÏàÓ˚ Ó˚y!¢üy°y x = –aωsin(ωt + θ) ˆÌ Ïܲ
x0 = a cosθ
~ÓÇ 0 sin= −x aω θ ˛õyÄÎ˚y ÎyÎ˚–
§%ï˛Ó˚yÇñ
22 00 2
⎛ ⎞= +⎜ ⎟
⎝ ⎠x
a xω
~ÓÇ
1 0
0
tan−= − xx
θ ω
............1.5
~ÓyÓ˚ xy˛õ!ö !öˆÏç•z ~ܲ!ê˛ §•ç xö%¢#°ö# ܲˆÏÓ˚ ˆòá%ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# : 1
SܲV 1.2 §ü#ܲÓ˚ˆÏîÓ˚ A Ä B ô Óܲ ò%•z!ê˛Ó˚ üyö x0 ~ÓÇ
0x
~Ó˚ !•§yˆÏÓ !öî≈Î˚ ܲÓ˚&ö–
SáV üˆÏö ܲÓ˚&ö t = 0 ~ÓÇ t =
πω2
§üˆÏÎ˚ §Ó˚ˆÏîÓ˚ üyö ÎÌye´ˆÏü
0x
~ÓÇ x– ï˛y•ˆÏ° 1.4 §ü#ܲÓ˚ˆÏî
a ~ÓÇ θ ~Ó˚ üyö ܲ#⁄
13
SàV üˆÏö ܲÓ˚&ö t = 0 §üˆÏÎ˚ §Ó˚î x0 ~ÓÇ
πω=
2t
§üˆÏÎ˚ ˆÓà v– ˆ§ˆÏ«˛ˆÏe ˙ §ü#ܲÓ˚ˆÏî a ~ÓÇ
θ ~Ó˚ üyö !ܲ⁄
1.2.4 §Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey§Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey§Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey§Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey§Ó˚° ˆòy°à!ï˛Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ!Ó˚üyey (parameter)
!ÓhflÏyÓ˚ !ÓhflÏyÓ˚ !ÓhflÏyÓ˚ !ÓhflÏyÓ˚ !ÓhflÏyÓ˚ : 1.4 §ü#ܲÓ˚î ˆÌˆÏܲ §•ˆÏç•z ˆÓyé˛y ÎyÎ˚ ˆÎñ x ~Ó˚ üyö §üˆÏÎ˚Ó˚ §ˆÏD ‘+a’ Ä ‘–a’, ~•z ò%•z
§#üyÓ˚ üˆÏôƒ Äë˛yöyüy ܲˆÏÓ˚– §yüƒ!Ó®% ˆÌˆÏܲ ӛܲîyÓ˚ §ˆÏÓ≈yFã˛ ò)Ó˚c ‘a’ ˆÜ˛ §Ó˚° ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ Ó°y
•Î˚– ~!ê˛ x ~Ó˚ ò%•z §#üyfli üyˆÏöÓ˚ ÓƒÓôyˆÏöÓ˚ xˆÏô≈ܲ–
xy˛õ!ö ˛õ)ˆÏÓ≈•z ˆòˆÏáˆÏåÈö ˆÎñ ӛܲîyÓ˚ ˆÓà ( )sin= = − +v x a tω ω θ – xö%Ó˚*˛õû˛yˆÏÓñ ˆÓˆÏàÓ˚ ~•z
Ó˚y!¢!ê˛Ó˚ xÓܲ°ö ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ cÓ˚î
( )2 cos= = − +f x a tω ω θ
flõfiê˛•z ˆÓyé˛y ÎyÎ˚ ˆÎñ ӛܲîyÓ˚ ˆÓà ~ÓÇ cÓ˚î Ä §üˆÏÎ˚Ó˚ §ˆÏD §Ó˚° ˆòy°à!ï˛Ó˚ xö%Ó˚&˛õû˛yˆÏÓ
˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– 1.4 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˆÓyé˛y ÎyÎ˚
ˆÓˆÏàÓ˚ !ÓhflÏyÓ˚ =
ωa
~ÓÇ cÓ˚ˆÏîÓ˚ !ÓhflÏyÓ˚ =
ω2aò¢y : 1.4 §ü#ܲÓ˚ˆÏî ( )ω θ+t Ó˚y!¢!ê˛ˆÏܲ t §üˆÏÎ˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ §Ó˚ˆÏîÓ˚ ò¢y (phase) Ó°y •Î˚–
t = 0 §üˆÏÎ˚ ~•z Ó˚y!¢Ó˚ üyö 'θ' - ˆÜ˛ ≤ÃyÓ˚!Ω˛Ü˛ ò¢y (initial phase) Óy ò¢y ô Óܲ (phase constant)
Ó°y •Î˚– ò¢yÓ˚ IJõÓ˚•z ӛܲîyÓ˚ ï˛yÍ«˛!îܲ xÓfliyö !öû≈˛Ó˚ ܲˆÏÓ˚–
~áö xy˛õ!ö §Ó˚ˆÏîÓ˚ ò¢yÓ˚ §ˆÏD ˆÓà Ä cÓ˚ˆÏîÓ˚ ò¢yÓ˚ ï%˛°öy ܲÓ˚ˆÏï˛ xy@ˇÃ•# •ˆÏï˛ ˛õyˆÏÓ˚ö– °«˛ƒ ܲˆÏÓ˚
ˆòá%ö
( ) πω ω θ ω ω θ⎛ ⎞= − + = + +⎜ ⎟⎝ ⎠sin cos
2v a t a t
~ÓÇ ( ) ( )ω ω θ ω ω θ π= − + = + +2 2cos cosf a t a t
xÌ≈yÍñ §Ó˚ˆÏîÓ˚ ï%˛°öyÓ˚ ˆÓˆÏàÓ˚ ò¢y
π2
ˆÜ˛yˆÏî ~ÓÇ cÓ˚ˆÏîÓ˚ ò¢y π ܲyˆÏî
x@ˇÃÓï≈˛# ÌyˆÏܲ– !ã˛e 1.2 ˆï˛ §Ó˚îñ ˆÓà
Ä cÓ˚ ÏîÓ˚ ò¢yÓ˚ ≤à Ïû˛ò òáy Ïöy • ÏÎ˚ ÏåÈ–
!ã˛e 1.2 -§Ó˚° ˆòy°à!ï˛ˆÏï˛ §Ó˚îñ
ˆÓà Ä cÓ˚î
§Óî
oÓî
ˆÓà
ωt+θ
14
ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ : ω Ó˚y!¢!ê˛ ~ܲܲ §üˆÏÎ˚ ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö §)!ã˛ï˛ ܲˆÏÓ˚– ~•z Ó˚y!¢!ê˛ˆÏܲ ˆÜ˛Ô!îܲ ܲ¡õyB˛
(angular frequency) Ó°y •Î˚–
˛õÎ≈yÎ˚ܲy° ˛õÎ≈yÎ˚ܲy° ˛õÎ≈yÎ˚ܲy° ˛õÎ≈yÎ˚ܲy° ˛õÎ≈yÎ˚ܲy° : ˆÎ §üÎ˚ xhsˇÓ˚ §Ó˚° ˆòy°à!ï˛§¡õߨ ܲîyÓ˚ xÓfliyöñ ˆÓà •zï˛ƒy!ò ~ܲ•z xÓfliyÎ˚ !ú˛ˆÏÓ˚
xyˆÏ§ñ ï˛yˆÏܲ ˙ ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy° ӈϰ– ~•z §üˆÏÎ˚ ˆÎˆÏ•ï%˛ ò¢y
π2
˛õ!Ó˚üyˆÏî Ó,!k˛ ˛õyÎ˚ñ xï˛~Ó
˛õÎ≈yÎ˚ܲy°ˆÏܲ T myÓ˚y !öˆÏò≈¢ ܲˆÏÓ˚
( ) 2t T t+ + = + +ω θ ω θ π
xÌ≈yÍñ
πω= 2
T
Óy π2 mk
...........1.6 (a)
ˆÎˆÏ•ï%˛ §yüƒ!Ó®% ˆÌˆÏܲ ≤Ã!ï˛ ~ܲܲ §Ó˚ˆÏî
km
˛õ!Ó˚üyî cÓ˚î âˆÏê˛ñ ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy° ~û˛yˆÏÓÄ ˆ°áy
ÎyÎ˚ :
π= 2T
ܲ¡õyB˛ ܲ¡õyB˛ ܲ¡õyB˛ ܲ¡õyB˛ ܲ¡õyB˛ : ~ܲܲ §üˆÏÎ˚ §Ó˚° ˆòy°à!ï˛Ó˚ Îï˛=!° ˛õ)î≈ ˛õÎ≈yÎ˚ (cycle) x!ï˛e´yhsˇ •Î˚ñ ï˛yˆÏܲ•z ˙ ˆòy°à!ï˛Ó˚
ܲ¡õyB˛ ӈϰ– ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ üyö ˆÌˆÏܲ ܲ¡õyˆÏB˛Ó˚ üyö ˛õyÄÎ˚y ÎyÎ˚ :
1 12 2
kT m
= = =ωυ π π ........... 1.6(b)
1.4 §ü#ܲÓ˚ˆÏî •zFåÈy ܲÓ˚ˆÏ° xyüÓ˚y ω ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ T Óy υ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚– ÎÌyñ( ) ( )2cos cos cos 2tx a t a a t
T⎛ ⎞= + = + = +⎝ ⎠
πω θ θ πυ θ
1.3 §Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´§Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´§Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´§Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´§Ó˚° ˆòy°à!ï˛§¡õߨ ӛܲîyÓ˚ Îy!sfܲ ¢!_´
ӛܲîy!ê˛Ó˚ Îy!sfܲ ¢!_´ !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ ˆÎyàú˛°– ˆÎ ˆÜ˛yöÄ xÓfliyˆÏö ӛܲîyÓ˚ !fli!ï˛¢!_´
≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ ˆ§!ê˛ˆÏܲ §yüƒ!Ó®% ˆÌˆÏܲ ˆ§•z xÓfliyˆÏö xyöˆÏï˛ ˆÎ ܲyÎ≈ ܲÓ˚y •Î˚ ï˛yÓ˚ §üyö–
~•z ܲyˆÏÎ≈Ó˚ ˛õ!Ó˚üyî
2 2 2
0 0
1 12 2
x x
U Fdx kxdx kx m x= = = =∫ ∫ ω
.......1.7
x˛õÓ˚ ˛õˆÏ«˛ñ à!ï˛¢!_´Ó˚ ˛õ!Ó˚üyî ( )2 2 2 21 1 sin2 2
K mv ma t= = +ω ω θ
=
( )2 2 2 21 cos2
m a a t⎡ ⎤− +⎣ ⎦ω ω θ
~ܲܲ §Ó˚ˆÏî í˛zq(ï˛ cÓ˚î
15
=
( )2 2 212
m a x−ω
= ( )−2 212
k a x .......1.8
∴ ˆüyê˛ Îy!sfܲ ¢!_´ñ E = U + K
=
( )2 2 2 2 21 12 2
m x m a x+ −ω ω
= ω 2 212
m a =
212
ka
......1.9
°«˛î#Î˚ ˆÎñ ˆüyê˛ Îy!sfܲ ¢!_´ x Óy t ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#° öÎ˚ñ Îy Îy!sfܲ ¢!_´Ó˚ §ÇÓ˚«˛î §)!ã˛ï˛ ܲˆÏÓ˚–
!ã˛e 1.3 (a) ˆï˛ x ~Ó˚ §ˆÏD !fli!ï˛ ¢!_´ñ à!ï˛¢!_´
Ä ˆüyê˛ Îy!sfܲ ¢!_´Ó˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •°–
ˆòy°ˆÏöÓ˚ ~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yˆÏÎ˚ x ~Ó˚ üyö ˆÎüö
+ a Ä –a ~Ó˚ üˆÏôƒ ã˛°yˆÏú˛Ó˚y ܲˆÏÓ˚ñ ˆï˛üö•z
Îy!sfܲ ¢!_´ §¡õ)î≈Ó˚*ˆÏ˛õ à!ï˛¢!_´ ˆÌˆÏܲ
§¡õ)î≈Ó˚*ˆÏ˛õ !fli!ï˛¢!_´ ~•z ò%•z ã˛Ó˚ü xÓfliyÓ˚ üˆÏôƒ
xyˆÏ®y!°ï˛ •Î˚– 1.3 (b) !ã˛ˆÏe ~•z ˛õ!Ó˚Óï≈˛ö!ê˛
!ã˛ey!Î˚ï˛ •ˆÏÎ˚ˆÏåÈ– 1.3(a) Ä (b) !ã˛e ò%•z!ê˛ °«˛ƒ
ܲÓ˚ˆÏ° ˆÓyé˛y ÎyˆÏÓ ˆÎñ x = 0 xÌ≈yÍñ §yüƒ !Ó®%ˆÏï˛
!fli!ï˛ ¢!_´ ¢)öƒ Ä §Ó≈!ö¡¨– x˛õÓ˚˛õˆÏ«˛ñ
à!ï˛¢!_´ ~•z !Ó®%ˆÏï˛ §Ó≈y!ôܲ ~ÓÇ 21
2ka
~Ó˚ §üyö– Îáö
= ±x a
xÌ≈yÍñ à!ï˛Ó˚ ò%•z
≤Ãyhsˇ !Ó®%ˆÏï˛ à!ï˛¢!_´ ¢)öƒñ !ܲv !fli!ï˛¢!_´
§Ó≈y!ôܲ Ä
212
ka
~Ó˚ §üyö– Îáö
=2
ax
,
ï˛áö
⎛ ⎞= =⎜ ⎟⎝ ⎠
221 1
2 42a
U K ka
~ÓÇ
⎛ ⎞= − =⎜ ⎟⎝ ⎠
22 21 1
2 2 4a
K K a ka
xÌ≈yÍñ !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üüyö–
!fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üÎ˚ §yˆÏ˛õ«˛ àí˛¸!fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üÎ˚ §yˆÏ˛õ«˛ àí˛¸!fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üÎ˚ §yˆÏ˛õ«˛ àí˛¸!fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üÎ˚ §yˆÏ˛õ«˛ àí˛¸!fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ §üÎ˚ §yˆÏ˛õ«˛ àí˛¸
üyö üyö üyö üyö üyö (time average)
§Ó˚° ˆòy°à!ï˛Ó˚ §ˆÏD §¡õ!Ü≈˛ï˛ ˆÜ˛yöÄ
ˆû˛Ôï˛ Ó˚y!¢Ó˚ (z) §üÎ˚ §yˆÏ˛õ«˛ àí˛¸ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ xyüÓ˚y ~•z §ÇK˛y!ê˛ ÓƒÓ•yÓ˚ ܲÓ˚Ó :
§Ó˚ˆÏîÓ˚ §ˆÏD à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ ˛õ!Ó˚Óï≈˛ö
= + = 212
E u k ka= 212
U kx= −2 21( )
2K k a k
x = –a x = 0 x = 0
16
àí˛¸
( )= = =∫ ∫ ∫0 0 0
1T T T
Z z t zdt dt zdtT
~áyˆÏö §üyܲ°ö=!° ~ܲ!ê˛ §¡õ)î≈ ˛õÎ≈yˆÏÎ˚Ó˚ í˛z˛õÓ˚ ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈ–
~•z §)e xö%ÎyÎ˚# 1.4 §ü#ܲÓ˚î ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˜fli!ï˛Ü˛ ¢!_´Ó˚ àí˛¸ üyö
( )2 2 2
0 0
1 1 1 1 cos2 2
T T
U kx dt ka t dtT T
= = +∫ ∫ ω θ
= =2 21 1 1. .2 2 4
Tka ka
T................ 1.10
~ÓÇ à!ï˛¢!_´Ó˚ àí˛¸ üyö ( ) ( )2 2 2 2
0 0
1 1 1 1 sin2 2
T T
K k a x dt ka t dtT T
= − = +∫ ∫ ω θ
= =2 21 1 1. .2 2 4
Tka ka
T
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ í˛zû˛Î˚ àí˛¸ üyö•z §üyö ~ÓÇ ˆüyê˛ ¢!_´Ó˚ xˆÏô≈ܲ–
!öˆÏã˛Ó˚ xö%¢#°ö!ê˛ xy˛õöyˆÏܲ àí˛¸ üyˆÏöÓ˚ !Ó£ÏÎ˚!ê˛ Ó%é˛ˆÏï˛ §y•y΃ ܲÓ˚ˆÏÓ–
!fli!ï˛¢!_´ ~ÓÇ à!ï˛ ¢!_´Ó˚ Ó˚y!¢ ò%•z!ê˛ Ü˛yˆÏç °y!àˆÏÎ˚ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î §•ˆÏç•z ≤Ã!ï˛¤˛y
ܲÓ˚y §Ω˛Ó–
ˆüyê˛ ¢!_´ ( )2 21 12 2
= +E m x kx
ˆÎˆÏ•ï%˛ ˆüyê˛ ¢!_´ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ– xï˛~Ó =/ 0dE dt – í˛z˛õˆÏÓ˚y_´ §ü#ܲÓ˚î!ê˛ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚–
= = +2
21 1/ 0 .2 .2 .2 2
d x dx dxdE dt m k x
dt dtdt
xÌÓyñ + =2
2 0d x kx
mdt
xö%¢#°ö xö%¢#°ö xö%¢#°ö xö%¢#°ö xö%¢#°ö : 2 !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö !öî≈Î˚ ܲÓ˚&ö–
S•z!Dï˛ : x Ó˚y!¢Ó˚ §yˆÏ˛õˆÏ«˛ z xˆÏ˛õ«˛ˆÏܲÓ˚ àí˛¸ üyö
1.4 §Ó˚° ˆòy°à!ï˛ §¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î§Ó˚° ˆòy°à!ï˛ §¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î§Ó˚° ˆòy°à!ï˛ §¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î§Ó˚° ˆòy°à!ï˛ §¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î§Ó˚° ˆòy°à!ï˛ §¡õߨ Ó›ï˛ˆÏsfÓ˚ í˛zòy•Ó˚î
xy˛õöyÓ˚y ày!î!ï˛Ü˛ !òܲ ˆÌˆÏܲ §Ó˚° ˆòy°à!ï˛ §¡∫ˆÏ¶˛ xˆÏöܲê˛y•z ˆçˆÏöˆÏåÈö– ~ÓyÓ˚ xyüÓ˚y §Ó˚°
ˆòy°à!ï˛ xö%§Ó˚î ܲˆÏÓ˚ ~üö ܲˆÏÎ˚ܲ!ê˛ ï˛sf §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
1.4.1 !fl±Çû˛Ó˚!fl±Çû˛Ó˚!fl±Çû˛Ó˚!fl±Çû˛Ó˚!fl±Çû˛Ó˚
xy˛õ!ö §y•zˆÏܲˆÏ°Ó˚ !§ˆÏê˛Ó˚ ï˛°yÎ˚ñ ˆ§yú˛yÓ˚ xy§ˆÏöÓ˚ !öˆÏã˛ñ í˛ê‰˛ ˆ˛õˆÏöÓ˚ üˆÏôƒ öyöy xyܲyˆÏÓ˚Ó˚ ˛õƒyÑã˛yˆÏöy
!fl±Ç ˆòˆÏáˆÏåÈö– ~•z !fl±Ç=!°Ó˚ ~ܲ!ê˛ !ö!ò≈‹T ˜òâ≈ƒ ÌyˆÏܲ– Óy•zˆÏÓ˚ ˆÌˆÏܲ Ó° ≤ÈÏÎ˚yà ܲˆÏÓ˚ ≤çy!Ó˚ï˛ Óy
17
§Çö!üï˛ Ü˛Ó˚ˆÏ° ~=!°Ó˚ ˆÎ ˜òâ≈ƒ ≤çyÓ˚î Óy §ÇˆÏܲyã˛ö
âˆÏê˛ñ ï˛y ≤ÃÎ%_´ ӈϰÓ˚ §üyö%˛õyï˛#– ~ܲܲ ≤çyÓ˚î Óy
§ÇˆÏܲyã˛ˆÏöÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ Ó°ˆÏܲ !fl±Ç ô Óܲ (springconstant) Ó Ï°– 1.4 (a) (b) Ä (c) !ã˛ˆÏe ~ܲ!ê˛ !fl±Ç
ÈÙÈû˛Ó˚ ÓƒÓfliy ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö !fl±Ç!ê˛ xö%û)˛!üܲ Ä
ï˛yÓ˚ ~ܲ!òܲ xyÓk˛– xöƒ!òˆÏܲ m û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ Ó› §ÇÎ%_´
xyˆÏåÈ Îy §¡õ)î≈ ü§,î ï˛ˆÏ° xö%û)˛!üܲû˛yˆÏÓ !fl±Ç ÈÙÈ~Ó˚
˜òˆÏâ≈ƒÓ˚ !òܲ Óy x !òˆÏܲ ã˛°yã˛° ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– Ó›!ê˛Ó˚ §yüƒ
!Ó®%Ó˚ fliyöyB˛ xyüÓ˚y x = 0 ӈϰ ôÓ˚Ó– xÓ¢ƒ ~ˆÏ«˛ˆÏe
xyüÓ˚y Ó›!ê˛ˆÏܲ ï˛yÓ˚ û˛Ó˚ˆÏܲˆÏw ˆÜ˛w#û)˛ï˛ ӛܲîy !•§yˆÏÓ
ôˆÏÓ˚ !öˆÏÎ˚!åÈ– !fl±Ç!ê˛ˆÏÜ˛Ä xyüÓ˚y û˛Ó˚•#ö ӈϰ üˆÏö ܲÓ˚Ó–
!ãe 1.4 (a) ˆï˛ Ó›!ê˛ §yüƒ!Ó®%ˆÏï˛ xÓfliyö ܲÓ˚ˆÏåÈ–
~áö !fl±Ç!ê˛ˆÏܲ x ˛õ!Ó˚üyˆÏî ≤çy!Ó˚ï˛ Ü˛ˆÏÓ˚ Ó›!ê˛ˆÏܲ Î!ò
(b) !ã˛ˆÏe ≤Ãò!¢≈ï˛ xÓfliyˆÏö xyöy ÎyÎ˚ñ ï˛ˆÏÓ ï˛yÓ˚ IJõÓ˚
!fl±Ç ÈÙÈ~Ó˚ ≤çyÓ˚ˆÏîÓ˚ ú˛ˆÏ° ˆÎ ≤Ãï˛ƒyöÎ˚ܲ Ó° ܲyç ܲÓ˚ˆÏÓñ
ï˛yÓ˚ üyö Kx(K= !fl±Ç ô ÓܲV– xö%Ó˚&˛õû˛yˆÏÓñ !fl±Ç!ê˛ x˛õ!Ó˚üyˆÏî §Çö!üï˛ •ˆÏ°Ä Kx ≤Ãï˛ƒyöÎ˚ܲ Ó° Ó›!ê˛ˆÏܲ
§yüƒyÓfliyÎ˚ ˆú˛Ó˚yˆÏï˛ Ü˛yç ܲÓ˚ˆÏÓ– ~ˆÏ«˛ˆÏe !fl±Ç ÈÙÈô &Óܲ
K 1.1 §ü#ܲÓ˚ˆÏîÓ˚ k §üyö%˛õyï˛ ô ÓˆÏܲÓ˚ fliyö @ˇÃ•î ܲˆÏÓ˚– à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î!ê˛ ~ˆÏ«˛ˆÏe = − Kx x
m
~ÓÇ Ó›!ê˛
π1
2Km
ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°yã˛° ܲÓ˚ˆÏï˛ ÌyܲˆÏÓ–
1.4.2 !fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚!fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚!fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚!fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚!fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚
˛õ)ˆÏÓ≈Ó˚ (ܲV í˛zòy•Ó˚ˆÏîÓ˚ xyüÓ˚y ˆÎ !fl±Ç!ê˛Ó˚ §¡∫ˆÏ¶˛
xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈñ ôÓ˚y Îyܲ ˆ§!ê˛ í˛zÕ‘¡∫û˛yˆÏÓ xy°!¡∫ï˛ xyˆÏåÈ
~ÓÇ û˛Ó˚!ê˛ ï˛yÓ˚ !ö¡¨≤ÃyˆÏhsˇ xyÓk˛ xyˆÏåÈ S!ã˛e 1.5)– û˛Ó˚!ê˛Ó˚
ˆÜ˛yöÄ Äçö öy ÌyܲˆÏ° ˆ§!ê˛Ó˚ xÓfliyö ˆÎ !Ó®%ˆÏï˛ •ï˛ ï˛yÓ˚
!öˆÏò≈¢yÇܲ z=0 ôÓ˚y Îyܲ S!ã˛e aV– ~áyˆÏö z ò)Ó˚c í˛zÕ‘¡∫û˛yˆÏÓ
!öˆÏã˛Ó˚ !òˆÏܲ üy˛õy •ˆÏÓ– û˛ˆÏÓ˚Ó˚ Mg ÄçˆÏöÓ˚ ú˛ˆÏ° ˆ§!ê˛Ó˚
§yüƒÓfliyÓ˚ !öˆÏò≈¢yÇܲ •ˆÏÓ z = z0 S!ã˛e b)– ˆÎˆÏ•ï%˛ Äçö
Mg !fl±Ç ~Ó˚ z0 ≤çyÓ˚î âê˛yÎ˚ñ xï˛~Ó Mg =Kz0– ~áö
û˛Ó˚!ê˛Ó˚ xÓfliyö Î!ò z !öˆÏò≈¢yLjÏܲ S!ã˛e c) •Î˚ ï˛ˆÏÓ ˆ§!ê˛Ó˚
à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î •ˆÏÓ
§yüƒ xÓfliy
≤çyÓî
§Ç Ïܲyã˛ö
(a)
(b)
(c)
18
= −Mz Mg Kz
xÌÓyñ ˆÎˆÏ•ï%˛
( )0 0,= = − −Mg Kz M z k z z
Î!ò §yüƒ xÓfliy ˆÌˆÏܲ y = z –z0 ˛õ!Ó˚üyˆÏö !fl±Ç!ê˛ ˆê˛ˆÏö ˆåȈÏí˛¸ ˆòÄÎ˚y •Î˚ ï˛ˆÏÓ §ü#ܲÓ˚î!ê˛ •ˆÏÓ
= −M y Ky
xÌ≈yÍñ ~áyˆÏö y Ó˚y!¢ê˛y•z §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î ˛õy°ö ܲˆÏÓ˚– ~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö ˛õ)ˆÏÓ≈Ó˚ üï˛ :
( )= ω + θcosy a t
, ˆÎáyˆÏö ω =2 Km
~ÓÇ
θ,a
x!ö!ò≈‹T ô &Óܲ– ˆÎˆÏ•ï%˛ y =z –z0, z !öˆÏò≈¢yÇܲ!ê˛ ~áö
~•zû˛yˆÏÓ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚:
( )= + ω + θ0 cosz z a t
.......1.11
xÌ≈yÍñ z !öˆÏò≈¢yÇܲ!ê˛ z0 ˆÜ˛ üˆÏôƒ ˆÓ˚ˆÏá π2 KM
ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏï˛ ÌyܲˆÏÓ–
1.4.3 §Ó˚° ˆòy°Ü˛ §Ó˚° ˆòy°Ü˛ §Ó˚° ˆòy°Ü˛ §Ó˚° ˆòy°Ü˛ §Ó˚° ˆòy°Ü˛ (simple pendulum)
xyò¢≈ §Ó˚° ˆòy°Ü˛ Ó°ˆÏï˛ û˛Ó˚ •#ö x≤çyÎ≈ §)ï˛yÓ˚ myÓ˚y xy°!¡∫ï˛ ~ܲ!ê˛ !Ó®%û˛Ó˚ !˛õ[˛(bob) ˆÓyé˛yÎ˚–
ˆòy°ˆÏöÓ˚ §üÎ˚ §)ï˛y!ê˛ ~ܲ!ê˛ í˛zÕ‘¡∫ï˛ˆÏ° xyÓk˛ ÌyˆÏܲ– !˛õ[˛!ê˛ ~ܲ!ê˛ Ó,_ã˛y˛õ ÓÓ˚yÓÓ˚ ã˛°yã˛° ܲÓ˚ˆÏ°Äñ
ˆòy°ˆÏöÓ˚ ˆÜ˛Ô!îܲ !ÓhflÏyÓ˚ θm Î!ò x“ •Î˚ñ ï˛ˆÏÓ !˛õ[˛!ê˛ ˆüyê˛yü%!ê˛û˛yˆÏÓ
xö%û)˛!üܲ §Ó˚°ˆÏÓ˚áyÎ˚ x x«˛ ÓÓ˚yÓÓ˚ ã˛°ö¢#° ӈϰ ôÓ˚y ÎyÎ˚– !˛õˆÏ[˛Ó˚
û˛Ó˚ m ~ÓÇ ˆòy°ˆÏܲÓ˚ §)ï˛yÓ˚ ˆÜ˛Ô!îܲ §Ó˚î Î!ò θ S!ã˛e 1.6) •Î˚ñ ï˛ˆÏÓ
Äçö mg ~Ó˚ ã˛°ö ˛õˆÏÌÓ˚ flõ¢≈ܲ ÓÓ˚yÓÓ˚ í˛z˛õyÇ¢mg sinθ ≤Ãï˛ƒyöÎ˚ܲ
Ó° !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚– §)ï˛yÓ˚ ˜òâ≈ƒ Î!ò l •Î˚ñ ï˛ˆÏÓ !˛õˆÏ[˛Ó˚ xö%û)˛!üܲ
x x«˛ ÓÓ˚yÓÓ˚ §Ó˚î
= θax t
– l ˜òâ≈ƒˆÏܲ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ Ó°y •Î˚–
~áö xyüÓ˚y !˛õ[˛!ê˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î !°áˆÏï˛ ˛õy!Ó˚:
= − θ ≈ − θsinmx mg mg
SÎ!ò θ ΈÏÌ‹T x“üyˆÏöÓ˚ •Î˚V
xÌ≈yÍ
= − θ = − gx g x
l
........1.12
1.1 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓñ ~•z §ü#ܲÓ˚î!ê˛
ˆ§!ê˛Ó˚ xö%Ó˚*˛õ– ~Ó˚ §üyôyöÄ 1.4 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ •ˆÏÓ :
( )= ω + θcosx a t
ˆÎáyˆÏö ω = 8l
ˆòy°Ü˛!ê˛Ó˚ ˆòy°öܲy°
π= = πω02 2 l
Tg
......1.13
xy˛õ!ö •Î˚ï˛ °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ !fl±ÇÈÙÈû˛ˆÏÓ˚Ó˚ ˆ«˛ˆÏe ˆòy°ö ܲy° Óy ܲ¡õyB˛ û˛ˆÏÓ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°
•ˆÏ°Ä §Ó˚° ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe ˆ§=!° !˛õˆÏ[˛Ó˚ û˛ˆÏÓ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– ~Ó˚ ܲyÓ˚îñ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe
19
≤Ãï˛ƒyöÎ˚ܲ Ó°!ê˛ Äçö ˆÌˆÏܲ•z í˛zq(ï˛ •Î˚ñ Îy û˛ˆÏÓ˚Ó˚ §üyö%˛õy!ï˛Ü˛ xöƒ!òˆÏܲ– !fl±Ç ~Ó˚ ˆ«˛ˆÏe ≤Ãï˛ƒyöÎ˚ܲ
Ó°!ê˛ !fl±Ç ~Ó˚ xyܲyÓ˚ñ xyÜ,˛!ï˛ Ä í˛z˛õyòyˆÏöÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°ñ û˛ˆÏÓ˚Ó˚ IJõÓ˚ öÎ˚–
§Ó˚° ˆòy°ˆÏܲÓ˚ à!ï˛ˆÏܲ ˆÜ˛Ó° ï˛áö•z §Ó˚° ˆòy°à!ï˛ Ó°y ÎyÎ˚ñ Îáö
sin θ = θ
xD#ܲyÓ˚!ê˛ §ï˛ƒ •Î˚–
sin θ = θ
~Ó˚ ˆê˛°Ó˚ ˆ◊î# ≤çyÓ˚î (Taylor series expansion) ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ :
θ θθ = θ − +3 5
sin ........6 120
θ ~Ó˚ üyö Îáö 0.245 ˆÓ˚!í˛Î˚yö Óy 14º, ï˛áö θ Ä sin θ ~Ó˚ ˛õyÌ≈ܲƒ ≤ÃyÎ˚ 1% •Î˚– §%ï˛Ó˚yÇñ ˆòy°ˆÏܲÓ˚
ˆÜ˛Ô!îܲ !ÓhflÏyˆÏÓ˚Ó˚ ~•z üyö ˛õÎ≈hsˇ
sin θ = θ
xD#ܲyÓ˚!ê˛ ˆüˆÏö ˆöÄÎ˚y ÎyÎ˚– !Ó¢ò àîöyÎ˚ ˆòáy ÎyÎ˚ ˆÎñ
§Ó˚° ˆòy°ˆÏܲÓ˚ ˆòy°ö ܲy° T ˆÜ˛Ô!îܲ !ÓhflÏyÓ˚ θm SˆÓ˚!í˛Î˚yöV ~Ó˚ IJõÓ˚ ~•zû˛yˆÏÓ !öû≈˛Ó˚ ܲˆÏÓ˚ :
2 4 2
0
912 1 ...... 116 1024 16m m mT T
g⎛ ⎞ ⎛ ⎞θ θ θ= π + + + ≈ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Îáö
.126mθ =
ˆÓ˚!í˛Î˚yö Óy 7.25º, ï˛áö Ó¶˛ö#û%˛_´ Ó˚y!¢!ê˛Ó˚ üyö 1.001 xÌ≈yÍñ ˆòy°ˆÏܲÓ˚ ˆÜ˛Ô!îܲ
!ÓhflÏyÓ˚ Î!ò 7.25º ~Ó˚ üˆÏôƒ ÌyˆÏܲñ ï˛ˆÏÓ ˆòy°öܲy° T0 ˆÌˆÏܲ 0.1 ¢ï˛yLjϢÓ˚ ˆã˛ˆÏÎ˚ !û˛ß¨ •Î˚ öy–
1.4.4 ç!ê˛° ˆòy°Ü˛ç!ê˛° ˆòy°Ü˛ç!ê˛° ˆòy°Ü˛ç!ê˛° ˆòy°Ü˛ç!ê˛° ˆòy°Ü˛ (compound pendulum)
ˆÎ ˆÜ˛yöÄ ò,ì˛¸ Ó› ˆ§!ê˛Ó˚ §ˆÏD xyÓk˛ ~ܲ!ê˛ xö%û)˛!üܲ xˆÏ«˛Ó˚ í˛z˛õÓ˚ ˆòy°ˆÏö §«˛ü •ˆÏ°ñ ï˛yˆÏܲ ç!ê˛°
ˆòy°Ü˛ Ó°y ÎyÎ˚– §Ó˚° ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe ˆÎüö !˛õu˛!ê˛ˆÏܲ !Ó®%
û˛Ó˚ !•§yˆÏÓ ôˆÏÓ˚ ˆöÄÎ˚y ≤ÈÏÎ˚yçö •Î˚ ˆÎÔ!àܲ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe
ˆï˛üö ˆÜ˛yöÄ ¢ï≈˛ xyˆÏÓ˚y˛õ ܲÓ˚ˆÏï˛ •Î˚ öy– 1.7 !ã˛ˆÏe ~ܲ!ê˛
ç!ê˛° ˆòy°Ü˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö 0 xy°¡∫ö !Ó®%ñ
ˆòy°ˆÏöÓ˚ xö%û)˛!üܲ x«˛ ÎyÓ˚ üôƒ !òˆÏÎ˚ !àˆÏÎ˚ˆÏåÈñ G Ä G’
ÎÌye´ˆÏü §yüƒÓfliy Ä !Óã%˛ƒï˛ xÓfliyÎ˚ ˆòy°ˆÏܲÓ˚ û˛yÓ˚ˆÏܲw– θ
ˆÜ˛yî!ê˛ ˆòy°ˆÏܲÓ˚ ˆÜ˛Ô!îܲ !Óã%˛ƒ!ï˛– ôÓ˚y Îyܲñ ˆòy°ˆÏܲÓ˚ û˛Ó˚ m
~ÓÇ ˜òâ≈ƒ OG = OG' =1– !Óã%˛ƒï˛ xÓfliyÎ˚ ˆòy°ˆÏܲÓ˚
û˛yÓ˚ˆÏܲˆÏwÓ˚ üôƒ !òˆÏÎ˚ !e´Î˚y¢#° Äçö mg ˆòy°Ü˛!ê˛Ó˚ í˛z˛õÓ˚
~ܲ!ê˛ ≤Ãï˛ƒyöÎ˚ܲ Ó°Î%ˆÏ@¬Ó˚ (torque) §,!‹T ܲˆÏÓ˚ñ ÎyÓ˚ ºyüܲ
sinmgl θ
SÓ° mg Ä 0 !Ó®% ˆÌˆÏܲ !e´Î˚yˆÏÓ˚áyÓ˚ ò)Ó˚c OP
~Ó˚ =îú˛°V– ~áö O !Ó®%Ó˚ §yˆÏ˛õˆÏ«˛ ˆòy°ˆÏܲÓ˚ çí˛¸ï˛yºyüܲ
Î!ò l •Î˚ñ ï˛ˆÏÓ ~!ê˛Ó˚ ˆÜ˛Ô!îܲ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î •ˆÏÓ
sinI mglθ = − θ
........1.14
20
çí˛¸ï˛y ºyüˆÏܲÓ˚ §üyhsˇÓ˚y° x«˛ í˛z˛õ˛õyòƒ xö%ÎyÎ˚#ñ ˆòy°ˆÏܲÓ˚ û˛Ó˚ˆÏܲˆÏwÓ˚ üˆÏôƒ !òˆÏÎ˚ ÎyÄÎ˚y xy°¡∫ö
xˆÏ«˛Ó˚ í˛z˛õÓ˚ Î!ò ˆòy°ˆÏܲÓ˚ çí˛¸ï˛yºyüܲ l0 ~ÓÇ â)î≈ö ÈÙÈÓƒy§yô≈ (radius of gyration) k •Î˚ ï˛ˆÏÓñ
I = I0 + ml2 = m(k2+l2)
( )20 =∵ I mk
ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ ΈÏÌ‹T x“ ~ÓÇ
sinθ = θ
ôˆÏÓ˚ !öˆÏÎ˚ñ 1.14 §ü#ܲÓ˚î ˆÌˆÏܲ
2 2gl g
Lk lθ = − θ = − θ
+
⎛ ⎞= +⎜ ⎟⎝ ⎠
2kL l
l ......1.15
1.12 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ~•z §ü#ܲÓ˚ˆÏîÓ˚ ï%˛°öy ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆÎñ ò%•z!ê˛ §ü#ܲÓ˚î xö%Ó˚*˛õ– ~áyˆÏö
L ˜òâ≈ƒˆÏܲ §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ (equivalent simple pendulum) ˜òâ≈ƒ Ó°y •Î˚– OG §Ó˚°ˆÏÓ˚áyˆÏܲ
Î!ò C ˛õÎ≈hsˇ Ó!ô≈ï˛ Ü˛Ó˚y •Î˚ñ ÎyˆÏï˛ OC =L •Î˚ñ ï˛ˆÏÓ C !Ó®%ˆÏܲ ˆòy°öÈÙÈˆÜ˛w (centre of oscillation)
Ó°y •Î˚– û˛yÓ˚ˆÏܲw ˆÌˆÏܲ ˆòy°ö ˆÜ˛ˆÏwÓ˚ ò)Ó˚c
= − =2k
CG L ll
–
1.15 §ü#ܲÓ˚î ˆÌˆÏܲ xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ~ˆÏ«˛ˆÏe θ ˆÜ˛yî!ê˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛
xyˆÏ®y!°ï˛ •ˆÏÓ ~ÓÇ ï˛yÓ˚ ˛õÎ≈yÎ˚ܲy° •ˆÏÓ
π= 2 LT
g
– !ܲv §Ó˚° ˆòy°ˆÏܲÓ˚ ˆÌˆÏܲ ç!ê˛° ˆòy°ˆÏܲÓ˚ ~ܲ!ê˛
ˆüÔ!°Ü˛ ≤ÈÏû˛ò xyˆÏåÈ– ç!ê˛° ˆòy°Ü˛ˆÏܲ xy˛õ!ö !Ó!û˛ß¨ !Ó®%ˆÏï˛ xy°!¡∫ï˛ Ü˛Ó˚ˆÏï˛ ˛õyˆÏÓ˚ö ~ÓÇ l ~Ó˚ üyö
ï˛yÓ˚ ú˛ˆÏ° ˛õ!Ó˚Ó!ï≈˛ï˛ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– ~áö →lim 0,l →∝lim l , ~•z ò%•z §#üyˆÏï˛•z → ∞ → ∞,L T –
l ~Ó˚ ˆÎ üyˆÏöÓ˚ çöƒ §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ Ä ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ üyö §Ó≈!ö¡¨ñ ï˛y !öî≈Î˚ ܲÓ˚y Îyܲ–
L Îáö §Ó≈!ö¡¨ñ ï˛áö = 0dLdl
xÌ≈yÍ
⎛ ⎞+ =⎜ ⎟⎝ ⎠
2
0d kl
dl l Óy − =2
2 0kl
l, xÌÓyñ = ±l k
xÌ≈yÍñ xy°¡∫ö !Ó®% Îáö û˛yÓ˚ˆÏܲˆÏwÓ˚ ò%•z ˛õyˆÏ¢ k ò)Ó˚ˆÏc ÌyˆÏܲñ ï˛áö•z §üï)˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ
~ÓÇ ˆÎÔ!àܲ ˆòy°ˆÏܲÓ˚ ˆòy°öܲy° §ü≈!ö¡¨ •Î˚– ~•z xÓfliyÎ˚ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ L =2k ......1.16 (a)
~ÓÇ ˆòy°öܲy°
22m
kT
g= π
.......1.16(b)
Tm ~Ó˚ ˆã˛ˆÏÎ˚ ò#â≈ï˛Ó˚ ˆÜ˛yöÄ ˆòy°öܲyˆÏ°Ó˚ çöƒ ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ l Ü˛ï˛ •ÄÎ˚y ≤ÈÏÎ˚yçö ï˛y ˆòáy Îyܲ–
T ~Ó˚ Ó˚y!¢üy°y ˆÌˆÏܲ xyüÓ˚y ˛õy•z
2 2
2 ,4
k gTL l
l= + =
π xÌ≈yÍ
l gT2 2
24π
.l+k2 = 0 ~!ê˛ ~ܲ!ê˛ !mâyï˛
§ü#ܲÓ˚î– ~Ó˚ ü)° ò%!ê˛ xy˛õ!ö §•ˆÏç•z !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~=!°ˆÏܲ Î!ò l1 Ä l2 Ó°y ÎyÎ˚ñ ï˛ˆÏÓ !mâyï˛
§ü#ܲÓ˚ˆÏîÓ˚ ôü≈ ˆÌˆÏܲ
21
2
21 2 .....1.17( )4gT
l l a+ =π
~ÓÇ
( )= 21 2 .......1.17l l k b
xÌ≈yÍ l ~Ó˚ ~ܲ!ê˛ üyö Î!ò
1l
•Î˚ñ ï˛ˆÏÓ xöƒ!ê˛ •ˆÏÓ
=2
2k
ll
– xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈñ û˛yÓ˚ˆÏܲw ˆÌˆÏܲ
ˆòy°ö ˆÜ˛ˆÏwÓ˚ ò)Ó˚c
2kl
– §%ï˛Ó˚yÇ xy°¡∫ö !Ó®% O ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ ˆòy°ö ˆÜ˛w C -ˆÜ˛ xy°¡∫ö !Ó®% !•§yˆÏÓ
ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° ˆòy°öܲy° ~ܲ•z ÌyܲˆÏÓ– ~ÓÇ ˆ§ˆÏ«˛ˆÏeñ ˆÎˆÏ•ï%˛
=2
121
,/
kl O
k l
!Ó®%!ê˛•z •ˆÏÓ ˆòy°ö ˆÜ˛w–
( )+1 2l l ˜Ïòâ≈ƒ Óy L §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òˆÏâ≈ƒÓ˚
§üyö–
1.8 !ã˛ˆÏe ~ܲ!ê˛ ç!ê˛° ˆòy°ˆÏܲÓ˚ xy°¡∫ö !Ó®%
ˆÌˆÏܲ û˛yÓ˚ˆÏܲˆÏwÓ˚ ò)Ó˚c l-~Ó˚ §ˆÏD ˆòy°öܲy° T ~Ó˚
ˆ°á!ã˛e ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö xö%û)˛!üܲ xˆÏ«˛Ó˚
¢)öƒ!ê˛ û˛yÓ˚ˆÏܲˆÏwÓ˚ xÓfliyö §)!ãï˛ Ü˛ˆÏÓ˚– ˆòy°öܲy°
Îáö
( )> mT T
, ï˛áö ˆòy°ˆÏܲÓ˚ ò%•z!ê˛ ˜òâ≈ƒ l1 Ä l2
§Ω˛Ó– û˛yÓ˚ˆÏܲˆÏwÓ˚ !Ó˛õÓ˚#ï˛ !òˆÏÜ˛Ä xö%Ó˚*˛õ ò%•z!ê˛
xy°¡∫ö !Ó®% ˛õyÄÎ˚y ÎyˆÏÓ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ˆ°á!ã˛e!ê˛Ó˚
~ܲ!ê˛ ¢yáy l ~Ó˚ ˆöˆÏà!ê˛û˛ üyˆÏöÄ ˛õyÄÎ˚y ÎyˆÏÓ– ¢yáy
ò%•z!ê˛ ˛õÓ˚flõˆÏÓ˚Ó˚ ≤Ã!ï˛!Ó¡∫ fl∫Ó˚*˛õ •ˆÏÓ– ˆ°á!ã˛e!ê˛ˆÏï˛
PA Ä PA' ~Ó˚ ˜Ïòâ≈ƒ l1, ~ÓÇ PB Ä PB' ~Ó˚ ˜òâ≈ƒ
l2– A'B xÌÓy AB' ˜òâ≈ƒ
( )+1 2l l
~Ó˚ §üyö ~ÓÇ ˆÎˆÏ•ï%˛ 1.17 (a) §ü#ܲÓ˚î xö%ÎyÎ˚#
1 22l l
Tg+= π
,
~•z ˜òâ≈ƒ ò%!ê˛•z L Óy §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ §)!ã˛ï˛ ܲˆÏÓ˚– ˆòy°öܲy° Îáö §Ó≈!ö¡¨ñ xÌ≈yÍ Tm •Î˚ñ
ï˛áö l1 Ä l
2 í˛zû˛ˆÏÎ˚•z ‘k’ ~Ó˚ §üyö •Î˚ ~ÓÇ ˜òâ≈ƒ CC'(=2k)•z •Î˚ §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ–
1.4.5 ÓƒyÓï≈˛ ˆòy°Ü˛ÓƒyÓï≈˛ ˆòy°Ü˛ÓƒyÓï≈˛ ˆòy°Ü˛ÓƒyÓï≈˛ ˆòy°Ü˛ÓƒyÓï≈˛ ˆòy°Ü˛ (torsional pendulum)
Î!ò ~ܲ!ê˛ §Ó˚& °¡∫y ï˛yˆÏÓ˚Ó˚ ~ܲ≤Ãyhsˇ ò,ì˛¸û˛yˆÏÓ xyÓk˛ ܲˆÏÓ˚ í˛zÕ‘¡∫û˛yˆÏÓ ˆ§!ê˛ˆÏܲ ˆé˛y°yˆÏöy ÎyÎ˚ ~ÓÇ ï˛yÓ˚
!ö¡¨≤ÃyˆÏhsˇ ~ܲ!ê˛ û˛y!Ó˚ ã˛yܲ!ï˛ñ ˆày°Ü˛ Óy ˆÓ°ö ~üöû˛yˆÏÓ xyÓk˛ ܲÓ˚y ÎyÎ˚ ˆÎö ˙ Ó›!ê˛Ó˚ û˛yÓ˚ˆÏܲw ï˛yÓ˚!ê˛Ó˚
§Ó˚°ˆÏÓ˚áyÎ˚ ÌyˆÏܲñ ï˛ˆÏÓ ˙ ÓƒÓfliy!ê˛ˆÏܲ ~ܲ!ê˛ ÓƒyÓï≈˛ ˆòy°Ü˛ Ó°y ÎyÎ˚– 1.9 !ã˛ˆÏe ~ܲ!ê˛ ï˛yÓ˚ Ä ˆÓ°ö
l-t ˆ°á!ã˛e
–l1
22
myÓ˚y à!ë˛ï˛ ÓƒyÓï≈˛ ˆòy°Ü˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˆÓ°ö!ê˛ í˛zÕ‘¡∫ xˆÏ«˛Ó˚ í˛z˛õÓ˚ â%Ó˚ˆÏï˛
˛õyˆÏÓ˚– ˆ§!ê˛Ó˚ ˆÜ˛Ô!îܲ !Óã%˛ƒ!ï˛Ó˚ §ˆÏD §ˆÏD ï˛yÓ˚!ê˛Ó˚ ÓƒyÓï≈˛ö âˆÏê˛ ~ÓÇ
ˆüyã˛í˛¸yˆÏöy ï˛yÓ˚!ê˛ ˛õ)Ó≈yÓfliyÎ˚ ˆú˛Ó˚yÓ˚ çöƒ ˆÓ°ö!ê˛Ó˚ í˛z˛õÓ˚ ˆÜ˛Ô!îܲ !Óã%˛ƒ!ï˛Ó˚
!Ó˛õÓ˚#ï˛ !òˆÏܲ !Óã%˛ƒ!ï˛ ˆÜ˛yˆÏîÓ˚ §üyö%˛õyï˛# ~ܲ!ê˛ ê˛Ü≈˛ Óy Ó°Î%@¬ ≤ÈÏÎ˚yà ܲˆÏÓ˚–
~•z ê˛Ü≈˛!ê˛•z ˆÓ°ˆÏöÓ˚ í˛z˛õÓ˚ ≤Ãï˛ƒyöÎ˚ܲ ê˛Ü≈˛ !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚–
ôÓ˚y Îyܲñ §yüƒyÓfliy ˆÌˆÏܲ ˆÓ°ö!ê˛Ó˚ θ ܲÔ!îܲ !Óã%˛ƒ!ï˛ âˆÏê˛ˆÏåÈ– ï˛yÓ˚!ê˛
ˆÎ ≤Ãï˛ƒyöÎ˚ܲ ê˛Ü≈˛ §,!‹T ܲÓ˚ˆÏÓ ï˛yÓ˚ ˛õ!Ó˚üyîñ ôÓ˚y Îyܲ Tθ– T ~ܲ!ê˛ §üyö%˛õyï˛
ô Óܲ Îy ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒñ Óƒy§yô≈ Ä í˛z˛õyòyˆÏöÓ˚ !fli!ï˛fliy˛õܲ ôˆÏü≈Ó˚ IJõÓ˚ !öû≈˛Ó˚
ܲˆÏÓ˚– ~•z ô ÓܲˆÏܲ ÓƒyÓï≈˛ö ô Óܲ (torsinal constant) Ó°y •Î˚– ï˛áö
ˆÓ°ˆÏöÓ˚ ˆÜ˛Ô!îܲ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î •ˆÏÓ
I Tθ = − θˆÎáyˆÏö I â)î≈öyˆÏ«˛Ó˚ í˛z˛õÓ˚ ˆÓ°ˆÏöÓ˚ çí˛¸ï˛y ÈÙȺyüܲ–
~áö
2
Iτθ = − θ = −ω θ
SˆÎáyˆÏö 2
Iτω = ) ....1.18
~•z §ü#ܲÓ˚î!ê˛ 1.1 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ– ~Ó˚ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ
θ
ˆÜ˛yö!ê˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛
xyˆÏ®y!°ï˛ •ˆÏÓ ~ÓÇ ~Ó˚ ˆòy°ö ܲy° •ˆÏÓ 2 2 IT
π= = πω τ
.....1.19
§Ó˚° ˆòy°ˆÏܲÓ˚ §ˆÏD ÓƒyÓï≈˛ ˆòy°ˆÏܲÓ˚ ~ܲ!ê˛ ˆüÔ!°Ü˛ ˛õyÌ≈ܲƒ !ܲ xy˛õ!ö °«˛ƒ ܲˆÏÓ˚ˆÏåÈö⁄ §Ó˚° ˆòy°ˆÏܲÓ˚
Óy ˆÎÔ!àܲ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe θ ≈ sinθ ~•z §¡õÜ≈˛!ê˛ ôˆÏÓ˚ !öˆÏï˛ •ˆÏÎ˚!åÈ°– !ܲv ÓƒyÓï≈˛ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe
~Ó˚*˛õ ˆÜ˛yöÄ xD#ܲyˆÏÓ˚Ó˚ ≤ÈÏÎ˚yçö •Î˚!ö– xÌ≈yÍñ ÓƒyÓï≈˛ ˆòy°ˆÏܲÓ˚ à!ï˛ θ ˆÜ˛yˆÏîÓ˚ ˆÎ ˆÜ˛yöÄ üyˆÏöÓ˚ çöƒ•z
§Ó˚° ˆòy°à!ï˛– xÓ¢ƒ ~ˆÏ«˛ˆÏe θ ܲyî!ê˛ˆÏܲ ~üö •ˆÏï˛ •ˆÏÓ ÎyˆÏï˛ ï˛yˆÏÓ˚Ó˚ ˆüyã˛í˛¸ !fli!ï˛fliy˛õܲ §#üyÓ˚ üˆÏôƒ
ÌyˆÏܲ–
~ÓyÓ˚ ò%•z!ê˛ xö%¢#°ö#Ó˚ §y•yˆÏ΃ ܲï˛ê˛y !¢áˆÏ°ö ï˛y ˛õÓ˚á ܲˆÏÓ˚ !öö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 3 : ~ܲ!ê˛ û˛Ó˚ M ܲ Ä á !ã˛ˆÏe ˆÎüö ˆòáyˆÏöy xyˆÏåÈ ˆ§•zû˛yˆÏÓ
ò%•z!ê˛ xö%Ó˚*˛õ !fl±Ç myÓ˚y ˆé˛y°yˆÏöy xyˆÏåÈ– ò%•z ˆ«˛ˆÏe í˛z˛õÓ˚ÈÙÈ!öˆÏã˛ ˆòy°ˆÏöÓ˚
ܲ¡õyB˛ Ü˛ï˛ •ˆÏÓ⁄ S!fl±Ç ô Óܲ =k)
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 4 : ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚ ÓƒyÓï≈˛ö âê˛yˆÏï˛ 0.1 Nm/rad ê˛Ü≈˛ ≤ÈÏÎ˚yçö
•Î˚– ï˛yˆÏÓ˚Ó˚ í˛z˛õÓ˚ ≤Ãyhsˇ xyÓk˛ ˆÓ˚ˆÏá !öˆÏã˛Ó˚ ≤Ãyhsˇ ~ܲ!ê˛ 0.5 kg Äç ÏöÓ˚ 50cm
°¡∫y xö%û)˛!üܲ òˆÏ[˛Ó˚ üôƒ!Ó®%ˆÏï˛ °yàyˆÏöy •°– ï˛yÓ˚ Ä òˆÏ[˛Ó˚ ÓƒyÓï≈˛ ˆòy°ˆÏöÓ˚ ˛õÎ≈yÎ˚ܲy° ܲï˛⁄ (M û˛Ó˚
Ä 2l ˜òˆÏâ≈ƒÓ˚ òˆÏ[˛Ó˚ û˛Ó˚ˆÏܲˆÏwÓ˚ üôƒ !òˆÏÎ˚ xyí˛¸yxy!í˛¸ xˆÏ«˛Ó˚ í˛z˛õÓ˚ çí˛¸ï˛y ÈÙȺyüܲ
21 )3
Mt
SܲV
SáV
23
1.4.6 ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö#˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö#˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö#˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö#˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö# (electric inductance-capacitance
circuit)
~ ˛õÎ≈hsˇ xyüÓ˚y ܲˆÏÎ˚ܲ!ê˛ Îy!sfܲ Ó›ï˛ˆÏsfÓ˚ §Ó˚° ˆòy°à!ï˛Ó˚ §¡∫ˆÏ¶˛
xyˆÏ°yã˛öy ܲÓ˚°yü– ~ÓyÓ˚ xyüÓ˚y ~üö ~ܲ!ê˛ í˛zòy•Ó˚î !ÓˆÏÓã˛öy ܲÓ˚Ó
ˆÎáyˆÏö ˆÜ˛yöÄ Îy!sfܲ à!ï˛ öy ÌyܲˆÏ°Ä ï˛!í˛¸Í xyôyö Ä ï˛!í˛¸Í ˛˛≤ÃÓy•
§Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚–
1.10 !ã˛ˆÏe ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ L, ôyÓ˚ܲ C Ä Óƒyê˛y!Ó˚ B ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– S §%•zˆÏã˛Ó˚ ˆüÓ˚&!ê˛ X !Ó®%ˆÏï˛ Î%_´ ܲÓ˚ˆÏ° ôyÓ˚ܲ!ê˛ xy!•ï˛ •Î˚– ~•z
xÓfliyÎ˚ §%•zˆÏã˛Ó˚ ˆüÓ˚& Y !Ó®%ˆÏï˛ Î%_´ ܲÓ˚ˆÏ° L-C Óï≈˛ö#!ê˛ §¡õ)î≈ •Î˚ ~ÓÇ
ôyÓ˚ˆÏܲÓ˚ xyôyö xyˆÏӈϢÓ˚ üôƒ !òˆÏÎ˚ ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ §,!‹T ܲˆÏÓ˚– ôÓ˚y Îyܲñ
ôyÓ˚ˆÏܲÓ˚ xyôyö q S≤ÃyÌ!üܲ xÓfliyÎ˚ q0) ~ÓÇ L-C Óï≈˛ö#ˆÏï˛ ï˛!í˛¸Í ≤ÃÓy•
I, ~•z ï˛!í˛¸Í ≤ÃÓy• §üˆÏÎ˚Ó˚ §ˆÏD ˛õ!Ó˚Óï≈˛ö¢#° ~ÓÇ ~Ó˚ ú˛ˆÏ° xyˆÏӈϢÓ˚ í˛z˛õÓ˚ ˆÎ !Óû˛Ó ˛õï˛ö âˆÏê˛ ï˛yÓ˚
˛õ!Ó˚üyî
= −L
dlV L
dtxyÓyÓ˚ ôyÓ˚ˆÏܲÓ˚ í˛z˛õÓ˚ !Óû˛Ó ˛õï˛ö =1
qV
C
ˆÎˆÏ•ï%˛ Óï≈˛ö#ˆÏï˛ ˆÜ˛yöÄ !Óû˛Ó ÈÙÈí˛zͧ ˆö•zñ Óï≈˛ö#ˆÏï˛ Ü˛yÎ≈ܲÓ˚# ï˛!í˛¸Íã˛y°Ü˛ Ó° (net c.m.f.) •ˆÏÓ–
+ = 0dlq Lc dt
ˆÎˆÏ•ï%˛ ôyÓ˚ܲ ˆÌˆÏܲ xyôyˆÏöÓ˚ ≤ÃÓy••z ï˛!í˛¸Í ≤ÃÓy• §,!‹T ܲˆÏÓ˚ñ = dqI
dt– ~•z §¡õÜ≈˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ í˛z˛õˆÏÓ˚Ó˚
§ü#ܲÓ˚î!ê˛ˆÏܲ §y!çˆÏÎ˚ ˆ°áy ÎyÎ˚ :
22
2 0d qq
dt+ ω =
SˆÎáyˆÏö
2 1LC
ω =
) .......1.19
~!ê˛ 1.1 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ ~ÓÇ ~Ó˚ §üyôyöÄ 1.4 ~Ó˚ xö%Ó˚*˛õ xÌ≈yÍ
q = q0 cos(ωt + θ) xÌÓy q = q0 cos(ωt) Î!ò xy!ò ò¢y θ = 0 •Î˚– xÓ¢ƒ Î!ò = 0t §üˆÏÎ˚
S §%•zˆÏã˛Ó˚ ˆüÓ˚& Y - ˆï˛ ˆÎyà ܲÓ˚y •Î˚ ï˛ˆÏÓ ˙ ü%•)ˆÏï≈˛•z q ~Ó˚ üyö §ˆÏÓùy≈Fã˛ •ˆÏÓ ~ÓÇ §üyôyö!ê˛ ˆ°áy
ÎyˆÏÓ ~•z û˛yˆÏÓ:
= 0q q ........1.20(a)
24
flõ‹Tï˛•z q xyôyö §Ó˚° ˆòy° à!ï˛ xö%§Ó˚î ܲˆÏÓ˚– ~•z ˆòy°à!ï˛Ó˚ ܲ¡õyB˛
1 12 2 LCω =π π
Ä
˛õÎ≈yÎ˚ܲy° 2 LCπ xyÓyÓ˚ ï˛!í˛¸Í ≤ÃÓy• 0 0sin cos2
dqI q t q t
dtπ⎛ ⎞= = − ω ω = ω ω +⎝ ⎠ ........1.20 (b)
xÌ≈yÍ ï˛!í˛¸Í ≤ÃÓy•Ä
0q ω
!ÓhflÏyˆÏÓ˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚ ~ÓÇ ~!ê˛Ó˚ ò¢y ôyÓ˚ˆÏܲ ï˛!í˛¸Í
xyôyˆÏöÓ˚ñ ï%˛°öyÎ˚
2π
ˆÜ˛yˆÏî ~!àˆÏÎ˚ ÌyˆÏܲ–
1.3 xLjϢ xy˛õ!ö Îy!sfܲ §Ó˚° ˆòy°à!ï˛Ó˚ ˆ«˛ˆÏe !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ xyˆÏ®y°ˆÏöÓ˚ !ӣψÏÎ˚
ˆçˆÏöˆÏåÈö– xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ñ ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲ Óï≈˛ö#Ó˚ ˆ«˛ˆÏe ¢!_´Ó˚ xö%Ó˚*˛õ xyˆÏ®y°ö
âˆÏê˛ !ܲ öy– xy˛õ!ö •Î˚ï˛ ˆçˆÏö ÌyܲˆÏÓö ˆÎñ xy!•ï˛ xÓfliyÎ˚ ôyÓ˚ˆÏܲÓ˚ ¢!_´
=2
0 2q
EC
~ÓÇ ï˛!í˛¸Í ≤ÃÓy•
ã˛°yܲy°#ö ˜Óò%ƒ!ï˛Ü˛ xyˆÏӈϢ §!MÈ˛ï˛ ¢!_´
= 212LE LI
– 1.20 §ü#ܲÓ˚î xö%ÎyÎ˚#
220 cos
2c
qE t
C= ω
~ÓÇ
1.21 §ü#ܲÓ˚î xö%ÎyÎ˚#
22 2 2 200
1 sin sin2 2L
qE Lq t t
C= ω ω = ω
xÌ≈yÍ ˆüyê˛ ¢!_´
+ =20
2c L
qE E
C
~ˆÏ«˛ˆÏe ôyÓ˚ˆÏܲÓ˚ ¢!_´ !fli!ï˛¢!_´Ó˚ üï˛ S§ü#ܲÓ˚î 1.7) ~ÓÇ xyˆÏӈϢÓ˚ ¢!_´ à!ï˛¢!_´Ó˚ üï˛ S§ü#ܲÓ˚î
1.8) xyã˛Ó˚î ܲˆÏÓ˚– ôyÓ˚ˆÏܲÓ˚ xyôyö q Îáö §Ó≈y!ôܲñ ï˛áö
=20
2c
qE
C ~ÓÇ
= 0LE
– x˛õÓ˚ ˛õˆÏ«˛ñ Îáö
= 0,q ï˛áö =20
2L
qE
C ~ÓÇ
= 0cE
– ~•z Óï≈˛ö#ˆÏï˛ ˆÜ˛yöÄ ˆÓ˚yô öy ÌyܲyÎ˚ ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ çöƒ ¢!_´
«˛Î˚ •Î˚ öy ~ÓÇ ˆüyê˛ ¢!_´ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ–
~ܲ!ê˛ §yôyÓ˚î §ÇÓ˚«˛î §)e~ܲ!ê˛ §yôyÓ˚î §ÇÓ˚«˛î §)e~ܲ!ê˛ §yôyÓ˚î §ÇÓ˚«˛î §)e~ܲ!ê˛ §yôyÓ˚î §ÇÓ˚«˛î §)e~ܲ!ê˛ §yôyÓ˚î §ÇÓ˚«˛î §)e :
1.3 ~ÓÇ 1.4.6 xLjϢ ÎÌye´ˆÏü Îy!sfܲ ¢!_´ Ä ˜Óò%ƒ!ï˛Ü˛ ¢!_´Ó˚ §ÇÓ˚«˛î ˛õ,Ìܲû˛yˆÏÓ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
§•ˆÏç•z ˆòáyˆÏöy ÎyÎ˚ ˆÎñ §yôyÓ˚îû˛yˆÏÓ ˆÎ ˆÜ˛yöÄ §Ó˚° ˆòy°à!ï˛ˆÏï˛•z ~ܲ!ê˛ xö%Ó˚*˛õ §ÇÓ˚«˛î §)e ÌyܲˆÏÓ–
ˆòy°à!ï˛§¡õߨ Ó˚y!¢!ê˛ˆÏܲ y !òˆÏÎ˚ !öˆÏò≈¢ ܲˆÏÓ˚ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚
2y y= −ω
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î!ê˛ˆÏܲ
y
!òˆÏÎ˚ =î ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚
2yy yy= −ω
xÌÓyñ 2 2 21 1
2 2d d
y ydt dt
⎛ ⎞ ⎛ ⎞= − ω⎝ ⎠ ⎝ ⎠ xÌÓyñ 2 2 21 1 0
2 2d
y ydt
⎛ ⎞+ ω =⎝ ⎠
25
~!ê˛ˆÏܲ §üyܲ°ö ܲÓ˚ˆÏ° xy˛õ!ö ˛õyˆÏÓö
2 2 21 12 2y y+ ω = ô Óܲ– ........ 1.21
~•z §ü#ܲÓ˚î!ê˛ˆÏܲ xy˛õ!ö !öÿ˛Î˚•z ~ܲ!ê˛ §ÇÓ˚«˛î §)e !•§yˆÏÓ !ã˛öˆÏï˛ ˛õyÓ˚ˆÏÓö– xÓ¢ƒ y Ó˚y!¢!ê˛ ~üö
•ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ §ü#ܲÓ˚ˆÏîÓ˚ Ó˚y!¢=!°ˆÏܲ ¢!_´ !•§yˆÏÓ !ã˛!•´ï˛ ܲÓ˚y ÎyˆÏÓ öy– í˛zˆÏÕ‘á ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ
xˆÏöܲ §üÎ˚ 1.21 §ü#ܲÓ˚î!ê˛ˆÏܲ §Ó˚° ˆòy°à!ï˛Ó˚ !Ó!¢‹T §ü#ܲÓ˚î ӈϰ üˆÏö ܲÓ˚y •Î˚–
!öˆÏã˛Ó˚ xö%¢#°ö!ê˛ xy˛õöyˆÏܲ xyˆÏ°y!ã˛ï˛ !Ó£ÏÎ˚!ê˛ Ó%é˛ˆÏï˛ §y•y΃ ܲÓ˚ˆÏÓ–
xö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈ5 : ~ܲ!ê˛ 32 Fμ ôyÓ˚ܲˆÏܲ 6V Óƒyê˛y!Ó˚Ó˚ §y•yˆÏ΃ xy!•ï˛ ܲˆÏÓ˚ 500mH xyˆÏӈϢÓ˚ üôƒ
!òˆÏÎ˚ «˛!Ó˚ï˛ (discharged) ܲÓ˚y •°– xyôyˆÏöÓ˚ ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy° Ä ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ §ˆÏÓù≈yFã˛ üyö ܲï˛
•ˆÏÓ⁄
1.4.7 !fl±Ç myÓ˚y Î%_´ û˛Ó˚Î%@¬!fl±Ç myÓ˚y Î%_´ û˛Ó˚Î%@¬!fl±Ç myÓ˚y Î%_´ û˛Ó˚Î%@¬!fl±Ç myÓ˚y Î%_´ û˛Ó˚Î%@¬!fl±Ç myÓ˚y Î%_´ û˛Ó˚Î%@¬
1.4.1 xLjϢ xy˛õ!ö ˆÎ í˛zòy•Ó˚î!ê˛ ˆòˆÏáˆÏåÈö ˆ§!ê˛ˆÏï˛ !fl±Ç ÈÙÈ~Ó˚ ~ܲ !òܲ xyÓk˛ Ä xöƒ !òܲ ~ܲ!ê˛
û˛ˆÏÓ˚Ó˚ §ˆÏD Î%_´ !åÈ°– ~áö xyüÓ˚y ˆòáÓñ Î!ò !fl±Ç!ê˛Ó˚ ò%•z ≤Ãyhsˇ•z ~ܲ ~ܲ!ê˛ û˛ˆÏÓ˚Ó˚ §ˆÏD Î%_´ ÌyˆÏܲñ ï˛ˆÏÓ
ˆ§!ê˛Ó˚ §ÇˆÏܲyã˛ö Ä ≤çyÓ˚î ܲ#û˛yˆÏÓ •ˆÏÓ–
ôÓ˚y Îyܲñ k !fl±Ç ô ÓˆÏܲÓ˚ !fl±Ç!ê˛Ó˚ ò%•z ≤ÃyˆÏhsˇ m1 Ä m
2 û˛Ó˚ Î%_´ xyˆÏåÈ– xyüÓ˚y m
1 Ä m
2 û˛ˆÏÓ˚Ó˚
û˛Ó˚ˆÏܲw ò%!ê˛Ó˚ üôƒ!òˆÏÎ˚ x x«˛ ܲ“öy ܲÓ˚Ó– û˛Ó˚ˆÏܲw ò%!ê˛Ó˚ !öˆÏò≈¢yÇܲ ÎÌye´ˆÏü x1 Ä x2 ôÓ˚y Îy– !fl±Ç
ÈÙÈ~Ó˚ fl∫yû˛y!Óܲ ˜òâ≈ƒ Î!ò l0 •Î˚ñ ï˛ˆÏÓ ˆÎ ˆÜ˛yöÄ ü%•)ˆÏï≈˛ !fl±Ç ÈÙÈ~Ó˚ ˜òâ≈ƒ ˆÎˆÏ•ï%˛ x
2-x
1, !fl±Ç ÈÙÈ~Ó˚ ≤çyÓ˚î
y= (x2–x
1) –l
0 ≤çy!Ó˚ï˛ !fl±Ç!ê˛ m
1 ~Ó˚ í˛z˛õÓ˚ x !òˆÏܲ ~ÓÇ m
2 ~Ó˚ í˛z˛õÓ˚ ï˛yÓ˚ !Ó˛õÓ˚#ï˛ xÌ≈yÍ -x !ò Ïܲ
ky ˛õ!Ó˚üyî Ó° ≤ÈÏÎ˚yà ܲÓ˚ˆÏÓ– ~áö xyüÓ˚y m1 Ä m2 û˛ˆÏÓ˚Ó˚ à!ï˛ §ü#ܲÓ˚îà%!° !°áˆÏï˛ ˛õy!Ó˚:
1 1 =m x ky
Óyñ
11
= kx y
m
......1.22(a)
~ÓÇ
= −2 2m x ky
Óyñ
= −22
kx y
m
.....1.22(b)
§ü#ܲÓ˚î 1.22 (a) ˆÌˆÏܲ 1.22 (b) !ÓˆÏÎ˚yà ܲˆÏÓ˚
( ) ⎛ ⎞− = +⎜ ⎟⎝ ⎠
2
1 221 2
1 1dx x k y
m mdt
......1.23
ôÓ˚y Îyܲñ 1 2
1 1 1m m
+ = μ – μ ˛Ó˚y!¢!ê˛ˆÏܲ m1 Ä m2 û˛Ó˚Î%ˆÏ@¬Ó˚ §üyö#ï˛ û˛Ó˚ (reduced mass)Ó°y •Î˚–
26
~åÈyí˛¸y 1.23 §ü#ܲÓ˚ˆÏîÓ˚ Óyü!òˆÏܲÓ˚ Ó˚y!¢!ê˛
−2
2d y
dt
~Ó˚ §üyö– §%ï˛Ó˚yÇñ ~•z §ü#ܲÓ˚î!ê˛Ó˚ §Ó˚°#Ü,˛ï˛ Ó˚*˛õ :
2
2d y k
ydt
= − μ
..............1.24
ˆÎ!ê˛ y Ó˚y!¢Ó˚ §Ó˚° ˆòy°à!ï˛ §)!ã˛ï˛ ܲˆÏÓ˚– xÌ≈yÍ (x2 –x
1) Ó˚y!¢, ˆÎ!ê˛ m
1 Ä n
2 ~Ó˚ ˛õyÓ˚flõ!Ó˚ܲ ò)Ó˚cñ
ˆ§!ê˛
0l
üyˆÏöÓ˚ í˛z˛õÓ˚ ö#ˆÏã˛
12
kπ μ
ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ •Î˚–
xy˛õ!ö •Î˚ï˛ 1.4.1 xLjϢ xyˆÏ°y!ã˛ï˛ !fl±Ç ÈÙÈû˛Ó˚ ÓƒÓfliyÓ˚ §ˆÏD ~•z !fl±Ç Î%_´ û˛Ó˚Î%ˆÏ@¬Ó˚ ï%˛°öy ܲÓ˚ˆÏï˛
ã˛y•zˆÏÓö– °«˛ƒ ܲÓ˚&öñ Î!ò
>>1 2m m
•Î˚ñ ï˛ˆÏÓ ˆ°áy ÎyÎ˚
2mμ =
– ~•z ÓƒÓfliy !fl±Ç ~Ó˚ m1 ≤Ãyhsˇ ò,ì˛¸û˛yˆÏÓ
xyÓk˛ ÌyܲyÓ˚ §üï%˛°ƒ ~ÓÇ ~ˆÏ«˛ˆÏe ܲ¡õyB˛ •ˆÏÓ
2
12
kmπ
, Îy !fl±Ç û˛Ó˚ ÓƒÓfliyÓ˚•z xö%Ó˚*˛õ–
≤ÃÜ,˛!ï˛ˆÏï˛ !Ó!û˛ß¨ !mÈÙÈ˛õÓ˚üyî%ˆÏܲ xî% !fl±Ç Î%_´ û˛Ó˚Î%ˆÏ@¬Ó˚ üï˛ xyã˛Ó˚î ܲˆÏÓ˚– í˛zòy•Ó˚î !•§yˆÏÓ CO SܲyÓ≈ö
üˆÏöy:y•zí˛V xî%!ê˛ ˆöÄÎ˚y Îyܲ– 12C Ä
16O
˛õÓ˚üyî%Ó˚ û˛Ó˚ ÎÌye´ˆÏü
− −× × = ×27 2612 1.66 10 2.00 10
kg
ÄÈ
−× × = ×27 2616 166 10 2.66 10 kg
– CO xî%Ó˚ §üyö#ï˛ û˛Ó˚
27 2612 16 1.66 10 1.14 1012 16 kg kg− −×μ = × × = ×+ – C ~ÓÇ O ˛õÓ˚üyî%Ó˚ Ó¶˛ö#Ó˚ !fl±Ç ÈÙÈô &Óܲ 187N/
m ~•z ô &Óܲ!ê˛ˆÏܲ ‘Ó° ô &Óܲ’ (force constant) Ó°y •Î˚– ~•z ï˛Ìƒ ˆÌˆÏܲ CO xî%Ó˚ ܲ¡õyB˛ ˛õyÄÎ˚y
ÎyÎ˚
261 1 1872 2 1.14 10
kv −
⎛ ⎞= =π μ π ⎝ ⎠×
Óy × 13204 10 zH È
˛õÓ˚Óï≈˛# xö%¢#°ö#ˆÏï˛ ~ ôÓ˚ˆÏöÓ˚ xyÓ˚ ~ܲ!ê˛ í˛zòy•Ó˚î ˆòáˆÏï˛ ˛õyˆÏÓö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 6 : •y•zˆÏí»˛yˆÏÜœ˛y!Ó˚ܲ xƒy!§í˛ (HCl) xî%Ó˚ ܲ¡õyB˛ 8.57 ×1013Hz– xî%!ê˛Ó˚ •y•zˆÏí»˛yˆÏçö
Ä ˆÜœ˛y!Ó˚ö ˛õÓ˚üyî%Ó˚ üôƒfli Ó¶˛ö#Ó˚ !fl±Ç ô ÓˆÏܲÓ˚ üyö ܲï˛⁄
1.5 §Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y§Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y§Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y§Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y§Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&c Ä Óƒy˛õܲï˛y
≤ÃÜ,˛!ï˛ˆÏï˛ xˆÏöܲ à!ï˛•z §Ó˚° ˆòy°à!ï˛Ó˚ ˜Ó!¢‹Tƒ ˆüˆÏö ã˛ˆÏ°– ~!ê˛ ˆÜ˛ö •Î˚ ï˛y xy˛õ!ö §•ˆÏç•z Ó%é˛ˆÏï˛
˛õyÓ˚ˆÏÓö– üˆÏö ܲÓ˚&öñ ~ܲ!ê˛ Ü˛îy x0
\fliyöyˆÏB˛ §yüƒyÓfliyÎ˚ xyˆÏåÈ– xÌ≈yÍ ˙ xÓfliyˆÏö ܲîy!ê˛Ó˚ IJõÓ˚ ӈϰÓ˚
üyö ¢)öƒ– ~áö Ó° F-ˆÜ˛ ~ܲ!ê˛ ˆê˛°Ó˚ ˆ◊î# ≤çyÓ˚î ܲˆÏÓ˚ ˆ°áy ÎyÎ˚ :
27
( ) ( ) ( ) ( )= =
⎛ ⎞⎛ ⎞= + − + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠0 0
22
0 0 021 ..........2
x x x x
dF d FF x F x x x x x
dx dx
xyüyˆÏòÓ˚ ¢ï≈˛ xö%ÎyÎ˚#ñ ( ) =0 0F x – ~åÈyí˛¸y Î!ò (x – x0) ~üö «%˛o •Î˚ ÎyˆÏï˛ ï˛yÓ˚ í˛zFã˛ï˛Ó˚ âyï˛=!°
í˛zˆÏ˛õ«˛y ܲÓ˚y ã˛ˆÏ°ñ ï˛ˆÏÓ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ ≠
( ) ( )0
0=
⎛ ⎞= −⎜ ⎟⎝ ⎠x x
dFF x x x
dx
xÌ≈yÍñ Ó° F §Ó˚ˆÏîÓ˚ §üyö%yï˛#– ~áö dFdx x xe j
= 0 Î!ò îydܲ •Î˚ñ ï˛y•ˆÏ° à!ï˛!ê˛ §Ó˚° ˆòy°à!ï˛ •ˆÏÓ–
~ê˛y•z §Ó˚° ˆòy°à!ï˛Ó˚ Óƒy˛õܲï˛yÓ˚ ܲyÓ˚î–
xÓ¢ƒ =
⎛ ⎞⎜ ⎟⎝ ⎠
0x x
dFdx Î!ò ôöydܲ •Î˚ ï˛ˆÏÓ Ó°!ê˛ ≤Ãï˛ƒyöÎ˚ܲ •ˆÏÓ öyñ ÓÓ˚Ç ï˛yÓ˚ ≤Ãû˛yˆÏÓ ≤çyÓ˚î ˆÓà ˆÓˆÏí˛¸
ÎyˆÏÓñ ˆ¢ˆÏ£Ï ( )− 0x x -~Ó˚ í˛zFã˛ï˛Ó˚ âyï˛=!° ˆüyˆÏê˛•z í˛zˆÏ˛õ«˛î#Î˚ •ˆÏÓ öy– xyÓyÓ˚
=
⎛ ⎞⎜ ⎟⎝ ⎠
0x x
dFdx
Î!ò ¢)öƒ •Î˚ñ
ˆ§ˆÏ«˛ˆÏeÄ ≤Ãï˛ƒyöÎ˚ܲ Ó° ÌyܲˆÏÓ öy– ~•z ò%•z ˆ«˛ˆÏe•z à!ï˛ §Ó˚° ˆòy°à!ï˛ •ˆÏÓ öy–
§Ó˚° ˆòy°à!ï˛Ó˚ =Ó˚&ˆÏcÓ˚ xyÓ˚ ~ܲ!ê˛ Ü˛yÓ˚î ~•z ˆÎñ ˆÜ˛yöÄ ˛õÎ≈yÓ,_ à!ï˛ §#!üï˛ Óy x§#ü §ÇáƒÜ˛
§Ó˚° ˆòy°à!ï˛Ó˚ !ü◊î !•§yˆÏÓ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚– ôÓ˚&öñ §Ó˚î x §üÎ˚ t ~Ó˚ ~ܲ!ê˛ ˛õÎ≈yÓ,_ xˆÏ˛õ«˛Ü˛ ÎyÓ˚
˛õÎ≈yÎ˚ܲy° T Sܲ¡õyB˛ = 1v
T), ï˛y•ˆÏ°
( ) ( )= = ±x f t f t T
=
0
2sinn nn
nta
T
∞
=
π⎛ ⎞+ θ⎝ ⎠∑
Sn = ˛õ)î≈ §ÇáƒyV
= ( )0
sin 2 cos2n nn
A nvt B nvt∞
=π + π∑ [ υ = ܲ¡õyB˛]
x -~Ó˚ Ó˚y!¢üy°y ˆÌˆÏܲ §•ˆÏç•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ ~!ê˛ xˆÏöܲ=!° §Ó˚° ˆòy°à!ï˛Ó˚ !ü!°ï˛ ú˛°–
ˆÜ˛yö ˛õÎ≈yÓ,_ à!ï˛ Óy xˆÏ˛õ«˛Ü˛ ~û˛yˆÏÓ !°áˆÏ°ñ ï˛yÓ˚ !ӈϟ’£Ïî §•ç •Î˚–
28
1.6 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~•z ~ܲܲ!ê˛ˆÏï˛ xyüÓ˚y ≤Ã̈Ïü•z §Ó˚° ˆòy°à!ï˛Ó˚ ¢ï≈˛=!°Ó˚ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ~•z à!ï˛ §Ó˚°˜ÏÓ˚!áܲ
~ÓÇ ˙ §Ó˚°ˆÏÓ˚áyÓ˚ í˛z˛õÓ˚ !ö!ò≈‹T !Ó®% x!û˛ü%ˆÏá ˙ !Ó®% ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚ §üyö%˛õyï˛# ~ܲ!ê˛ ≤Ãï˛ƒyöÎ˚ܲ Ó°
ˆÌˆÏܲ í˛zq$ï˛– §Ó˚° ˆòy°à!ï˛Ó˚ §yôyÓ˚î à!ï˛ §ü#ܲÓ˚î 2x x= −ω ~ÓÇ ~Ó˚ §yôyÓ˚î §üyôyö
( )cosx a t= ω + θ
– ~•z à!ï˛Ó˚ ܲ¡õyB˛
2ωπ
, ˆòy°öܲy°
2πω
, !ÓhflÏyÓ˚ a ~ÓÇ ò¢y
( )tω + θ
–
ï˛ˆÏÓ §yôyÓ˚îû˛yˆÏÓ ˆÎ ˆÜ˛yöÄ Ó˚y!¢ x Î!ò xöƒ ˆÜ˛yöÄ Ó˚y!¢ y-~Ó˚ §yˆÏ˛õˆÏ«˛ §y•zö xˆÏ˛õ«˛Ü˛ xö%ÎyÎ˚#
˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ñ ï˛y•ˆÏ°•z Ó°y •Î˚ ˆÎ y -~Ó˚ §yˆÏ˛õˆÏ«˛ x §Ó˚° ˆòy°à!ï˛ xö%§Ó˚î ܲˆÏÓ˚– xÓ¢ƒ ≤ÃyÌ!üܲ
ò¢y ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ §y•zöˆÏܲ ˆÜ˛y§y•zö !òˆÏÎ˚Ä ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚– x ~ÓÇ y ˆÎ ˆÜ˛yöÄ ˆû˛Ôï˛ Óy ày!î!ï˛Ü˛
Ó˚y!¢ •ˆÏï˛ ˛õyˆÏÓ˚– ˆüyˆÏê˛Ó˚ í˛z˛õÓ˚ñ Î!ò
( )α= +sinx A ky
•Î˚ñ ˆÎáyˆÏö a,k ~ÓÇ α ô Óܲñ xÌ≈yÍ Î!ò
= −2
22
d xk x
dy •Î˚ñ ï˛y•ˆÏ°•z Ó°y ÎyÎ˚ y-~Ó˚ §yˆÏ˛õˆÏ«˛ x §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚–
§Ó˚° ˆòy°à!ï˛ˆÏï˛ Ó›ï˛ˆÏsfÓ˚ Îy!sfܲ ¢!_´ ˛õÎ≈yÎ˚e´ˆÏü à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ üˆÏôƒ xyˆÏ®y!°ï˛ •Î˚–
ï˛ˆÏÓ §üˆÏÎ˚Ó˚ §yˆÏ˛õˆÏ«˛ àí˛¸ à!ï˛¢!_´ Ä àí˛¸ !fli!ï˛¢!_´ í˛zû˛Î˚•z ˆüyê˛ ¢!_´Ó˚ xˆÏô≈ˆÏܲÓ˚ §üyö–
~•z ~ܲˆÏܲ §Ó˚° ˆòy°à!ï˛Ó˚ í˛zòy•Ó˚î !•§yˆÏÓ !Ó!û˛ß¨ Ó›ï˛ˆÏsfÓ˚ à!ï˛Ó˚ Óî≈öy ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö
Ó›ï˛sf=!° Ä ï˛yˆÏòÓ˚ à!ï˛Ó˚ ˆòy°ö ܲyˆÏ°Ó˚ ï˛y!°Ü˛y ˆòÄÎ˚y •°–
Ó›ï˛sf Ó›ï˛sf Ó›ï˛sf Ó›ï˛sf Ó›ï˛sf ˆòy°ö ܲy°ˆòy°ö ܲy°ˆòy°ö ܲy°ˆòy°ö ܲy°ˆòy°ö ܲy°
1. !fl±Ç ÈÙÈû˛Ó˚ S!fl±ÇÈÙÈô Óܲ = k, û˛Ó˚ = m) 2 mT
k= π
2. !fl±Ç myÓ˚y xy°!¡∫ï˛ û˛Ó˚ S!fl±Ç ÈÙÈô Óܲ = k, û˛Ó˚ = M) 2 MT
K= π
3. §Ó˚° ˆòy°Ü˛ Sˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ = l, x!û˛Ü˛£Ï≈ç cÓ˚î = g) 2 lT
g= π
4. ˆÎÔ!àܲ ˆòy°Ü˛ S§üï%˛° §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ = L, x!û˛Ü˛£Ï≈ç cÓ˚î = g) 2 LT
g= π
5. ÓƒyÓï≈˛ö ˆòy°Ü˛ (I = çí˛¸ï˛y ºyüܲñ C = ÓƒyÓï≈˛ö ô ÓܲV 2 lT = π τ
6. ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ö Óï≈˛ö# (L = xyˆÏÓ¢ñ C = ôyÓ˚ܲcV
2T LC= π
29
§Ó˚° ˆòy°à!ï˛Ó˚ à%Ó˚&c Ä ~ ôÓ˚ˆÏöÓ˚ à!ï˛Ó˚ Óƒy˛õܲï˛yÓ˚ §Ç!«˛Æ Óƒyáƒy !òˆÏÎ˚ ~ܲܲ!ê˛Ó˚ xyˆÏ°yã˛öy ˆ¢£Ï
•ˆÏÎ˚ˆÏåÈ–
1.6 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ~ܲ!ê˛ §Ó˚° ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ û˛Ó˚ 5 kg ~ÓÇ ˆ§!ê˛ 10m ò#â≈ §Ó˚& ï˛yÓ˚ !òˆÏÎ˚ ˆé˛y°yˆÏöy xyˆÏåÈ– !˛õ[˛!ê˛
§yüƒyÓfliy ˆÌˆÏܲ xm(x<<10) !Óã%˛ƒï˛ •ˆÏ° ˆ§!ê˛Ó˚ í˛z˛õÓ˚ !ܲ ˛õ!Ó˚üyî
≤Ãï˛ƒyöÎ˚ܲ Ó° ܲyç ܲˆÏÓ˚⁄ ôˆÏÓ˚ !öö g=10m/s2–
2. m û˛Ó˚ Ä r Óƒy§yˆÏô≈Ó˚ ~ܲ!ê˛ ã˛yܲ!ï˛ ˆ§!ê˛Ó˚ ˆÜ˛w ˆÌˆÏܲ 2r
ò)Ó˚ˆÏc
ˆ§!ê˛ˆÏܲ °¡∫û˛yˆÏÓ ˆåÈò ܲˆÏÓ˚ ~üö ~ܲ!ê˛ xˆÏ«˛Ó˚ í˛z˛õÓ˚ ò%°ˆÏåÈ– ã˛yܲ!ï˛!ê˛Ó˚
ˆòy°öܲy° Ä ˆòy°öˆÏܲˆÏwÓ˚ xÓfliyö !öî≈Î˚ ܲÓ˚&ö–
3. ˛õ,!ÌÓ#Ó˚ ~ܲ!ê˛ Óƒy§ ÓÓ˚yÓÓ˚ ~ܲ!ê˛ §%í˛¸D áöö ܲˆÏÓ˚ ï˛yÓ˚ üˆÏôƒ
~ܲ!ê˛ ˆày°Ü˛ ˆú˛°y •°– ˆòáyö ˆÎñ ˆày°Ü˛!ê˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛
Äë˛yöyüy ܲÓ˚ˆÏÓ ~ÓÇ ï˛yÓ˚ ˛õÎ≈yÎ˚ܲy° •ˆÏÓ
2 Rg
π, ˆÎáyˆÏö R ˛õ,!ÌÓ#Ó˚
Óƒy§yô≈ Ä g û)˛ÈÙÈ˛õ,ˆÏ¤˛ x!û˛Ü˛£Ï≈ç cÓ˚î– ôˆÏÓ˚ !öö ˛õ,!ÌÓ# ~ܲ!ê˛ §ü§_¥ ˆày°Ü˛–
4– ~ܲ!ê˛ ˆÓ°öyÜ,˛!ï˛ ˆê˛fiê˛ !ê˛í˛zˆÏÓÓ˚ üˆÏôƒ ~ܲ!ê˛ ôyï˛Ó ˆày°Ü˛ ˆÓ˚ˆÏá ˆ§!ê˛ˆÏܲ
çˆÏ° í˛zÕ‘¡∫û˛yˆÏÓ û˛y§yˆÏöy •°– ˆòáy ˆà° ˆ§!ê˛Ó˚ 15 cm ˜òâ≈ƒ çˆÏ° !öü!Iï˛
Ó˚ˆÏÎ˚ˆÏåÈ– ~áö Î!ò ˆê˛fiê˛ !ê˛í˛zÓ!ê˛ˆÏܲ í˛z˛õÓ˚ !òˆÏܲ Óy !öˆÏã˛Ó˚ !òˆÏܲ §yüyöƒ !Óã%˛ƒï˛
ܲÓ˚y ÎyÎ˚ ï˛ˆÏÓ ˆ§!ê˛Ó˚ í˛zÕ‘¡∫ ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy° Ü˛ï˛ •ˆÏÓ⁄ ôˆÏÓ˚ !öö çˆÏ°Ó˚
ˆÓ˚yô í˛zˆÏ˛õ«˛î#Î˚–
5. §Ó˚° ˆòy°à!ï˛§¡õߨ ˆÜ˛yö Ó› Îáö §yüƒ!Ó®% ˆÌˆÏܲ x1 Ä x2 ò)Ó˚ˆÏc ÌyˆÏܲ ï˛áö ï˛yÓ˚ ˆÓà •Î˚
ÎÌye´ˆÏü v1 Ä v
2– ˆòy°à!ï˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ Ä !ÓhflÏyÓ˚ !öî≈Î˚ ܲÓ˚&ö–
30
6. ò%•z!ê˛ xî%Ó˚ ˛õyÓ˚flõ!Ó˚ܲ ˜fli!ï˛Ü˛ ¢!_´ ( )6 12
0 00 2
r rV r V
r r
⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
~áyˆÏö r = xî% ò%•z!ê˛Ó˚ üˆÏôƒ ÓƒÓôyöñ V0Ä r0 ò%•z!ê˛ ô Óܲ– xî%mÎ˚ x“ !ÓhflÏyˆÏÓ˚ ܲ!¡õï˛ •ˆÏ° ܲ¡õyB˛
Ü˛ï˛ •ˆÏÓ⁄
7. ~ܲ!ê˛ !mÈÙÈ˛õÓ˚üyî%ܲ xî%Ó˚ ˛õÓ˚üyî%mˆÏÎ˚Ó˚ ÓƒÓôyö Îáö r, ï˛áö !fli!ï˛¢!_´
( ) 9 ,K CU r
r r= − +
~áy Ïö
K Ä C ò%•z!ê˛ ô &Óܲ– ≤ÃÌü Ó˚y!¢!ê˛ xÌ≈yÍ Kr
− ˛õÓ˚üyî%mˆÏÎ˚Ó˚ üˆÏôƒ xyܲ£Ï≈î Ä !mï˛#Î˚ Ó˚y!¢ xÌ≈yÍ
9Cr
,
!Óܲ£Ï≈î §)!ã˛ï˛ ܲˆÏÓ˚– §yüƒyÓfliyÎ˚ ˛õÓ˚üyî% ò%!ê˛Ó˚ üˆÏôƒ ÓƒÓôyö r0 •ˆÏ° ï˛yˆÏòÓ˚ üˆÏôƒ Ó° ô &ÓˆÏܲÓ˚ Ó˚y!¢!ê˛ !öî≈Î˚
ܲÓ˚&ö– SÓ° ô Óܲ =x!ï˛«%˛o !Óã%˛ƒ!ï˛Ó˚ ú˛ˆÏ° í˛zq(ï˛ ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ §ˆÏD !Óã%˛ƒ!ï˛Ó˚ xö%˛õyï˛–V
8. A Ä B !Ó®%ˆÏï˛ ò%•z!ê˛ §üyö xyôyö + e !fliÓ˚û˛yˆÏÓ xÓ!fliï˛ ~ÓÇ üôƒ!Ó®% O ˆï˛ xyÓ˚ ~ܲ!ê˛ xyôyö
ÈÙÈe ü%_´ xÓfliyÎ˚ xyˆÏåÈ– !öˆÏã˛Ó˚ ò%•z xÓfliyÎ˚ ü%_´ xyôyö!ê˛Ó˚ !fli!ï˛¢!_´ !öî≈Î˚ ܲÓ˚&ö :
(i) -e xyôyö!ê˛ AB ~Ó˚ °¡∫ ÈÙÈÓÓ˚yÓÓ˚ §yüyöƒ §ˆÏÓ˚ xyˆÏåÈ–
(ii) -e ~Ó˚ §Ó˚î AB ˆÓ˚áyÓ˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚–
í˛zû˛Î˚ ˆ«˛ˆÏe•z §Ó˚î x!ï˛ «%˛o– ~Ó˚ üˆÏôƒ ˆÜ˛yöÄ ˆ«˛ˆÏe -e xyôyˆÏöÓ˚ à!ï˛ §Ó˚° ˆòy°à!ï˛ •ˆÏÓ !ܲ⁄
Î%!_´ !òˆÏÎ˚ Ó%!é˛ˆÏÎ˚ Ó°%ö–
1.7 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#
1.SܲV
sin cosx A t B t= ω + ω
cos cosx A t B t∴ = ω ω − ω ω
Îáöñ t = 0, x=x0 = B ~ÓÇ
x x A= =0 ω
§%ï˛Ó˚yÇñ
Ax
B x= =,00ω
31
2. ˜fli!ï˛Ü˛ ¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö˜fli!ï˛Ü˛ ¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö˜fli!ï˛Ü˛ ¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö˜fli!ï˛Ü˛ ¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö˜fli!ï˛Ü˛ ¢!_´Ó˚ ò)Ó˚c §yˆÏ˛õ«˛ àí˛¸ üyö
2
0 0
1 1 12
a a
xU Udx kx dx
a a= =∫ ∫
S~áyˆÏö U ~Ó˚ ≤Ã!ï˛§yüƒ ˆ•ï%˛ x = 0 ˆÌˆÏܲ x = a §#üyÓ˚ çöƒ àí˛¸ !öî≈Î˚ ܲÓ˚y •°V
31 1.2 3
ak
a=
= 216
ka
à!ï˛¢!_´Ó˚ ˆ«˛ˆÏe xö%Ó˚*˛õû˛yˆÏÓ
( )3
2 2 3 2
0
1 1 1 1.2 3 3
a
x
aK k a x dx k a ka
a a⎛ ⎞= − = − =⎜ ⎟⎝ ⎠∫
3. (ܲV !ã˛ˆÏeÓ˚ ˆ«˛ˆÏe M û˛Ó˚!ê˛Ó˚ !öˆÏã˛Ó˚ !òˆÏܲ x !Óã%˛ƒ!ï˛ âê˛ˆÏ° !fl±Ç ò%!ê˛Ó˚ ≤çyÓ˚î ôÓ˚y Îyܲ 1 2,x x –
ˆÎˆÏ•ï%˛ ò%!ê˛ !fl±Ç ~Ó˚ ê˛yö F §üyö •ˆÏÓñ
F = kx1= kx2– xï˛~Óñ x1 = x2
!ܲvñ
1 2 12x x x x= + =
– xï˛~Óñ
12
F kx=
û˛Ó˚ M ~Ó˚ à!ï˛ §ü#ܲÓ˚î Mx = –F=
12
kx−
∴ 2= k
x xM
§Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚
2
2km
ω =
– xÌ≈yÍ 22 M
Tk
π= –
SáV !ã˛ˆÏeÓ˚ ˆ«˛ˆÏe M û˛Ó˚!ê˛Ó˚ !öˆÏã˛Ó˚ !òˆÏܲ x !Óã%˛ƒ!ï˛ âê˛ˆÏ° í˛zû˛Î˚ !fl±Ç ÈÙÈ~Ó˚ ~ܲ•z ≤çyÓ˚î •ˆÏÓ ~ÓÇ
ˆüyê˛ ≤Ãï˛ƒyöÎ˚ܲ Ó° •ˆÏÓ 2kx– ~áö xy˛õ!ö ˆòáyˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ 22M
Tk
π=
32
4. çí˛¸ï˛y ÈÙȺyüܲ ( )22 21 1 10.5 0.253 3 96
I Ml kgm= = × × =
∴ ˛õÎ≈yÎ˚ܲy° = 12 2 2.02896 0.1
= =IS
T xπ π
5. ~ˆÏ«˛ˆÏe ( ) 13 6 21 500 10 32 10 250 /
−− −= = × × =x SLC
ω
∴ ˛õÎ≈yÎ˚ܲy° 2 25.= = lmsπ
ω
ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ §ˆÏÓ≈yFã˛ üyö
0q ω
, ˆÎáyˆÏö q0 = ï˛!í˛¸Í xyôyˆÏöÓ˚ §ˆÏÓ≈Fã˛ üyö–
6 60 32 10 6 192 10q F V C− −= × × = ×
∴ 60 192 10 250 0.048q Aω −= × × = Óy 48mA
6. HCI xî%Ó˚ §üyö#ï˛ û˛Ó˚ 2735 1 1.66 10
35 1kgμ −×= × ×+
( )22 2 13 27354 4 9.87 8.57 10 1.66 1036
−= = × × × × × ×k vπ μ
= 468Nm-2
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. !˛õˆÏ[˛Ó˚ ˆÜ˛Ô!îܲ !Óã%˛ƒ!ï˛ θ = 10x
rad
≤Ãï˛ƒyöÎ˚ܲ Ó° =
sinmg mgθ = θ
ˆÜ˛ööy θ x!ï˛ «%˛o ˆÜ˛yî
5 1010x= × ×
= 5N, Î!ò x = 1m
33
2. xyüÓ˚y çy!ö ã˛yܲ!ï˛Ó˚ ï˛ˆÏ°Ó˚ °¡∫ xˆÏ«˛Ó˚ §yˆÏ˛õˆÏ«˛ ˆ§!ê˛Ó˚ çí˛¸ï˛y ºyüܲ
212
mr
– §üyhsˇÓ˚y° x«˛
í˛z˛õ˛õyòƒ xö%ÎyÎ˚# ˆòy°ˆÏöÓ˚ xˆÏ«˛Ó˚ §yˆÏ˛õˆÏ«˛ çí˛¸ï˛y ÈÙȺyüܲ
22 21 3
2 2 4r
I mr m mr⎛ ⎞= + =⎜ ⎟⎝ ⎠
ã˛yܲ!ï˛!ê˛Ó˚ ˆòy°ˆÏöÓ˚ à!ï˛ §ü#ܲÓ˚î
. sin2r
I mgθ = − θ
xÌÓyñ θ x!ï˛ «%˛o ˆÜ˛yî ôˆÏÓ˚ !öˆÏ°ñ 23
4 2r
mr mgθ = − θ
Óyñ 23gr
θ = − θ
1.12 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˆòáy ÎyÎ˚ ˆòy°öܲy° = 223gr
π
~ÓÇ §üï%˛°ƒ §Ó˚° ˆòy°ˆÏܲÓ˚ ˜òâ≈ƒ = 32
r
§%ï˛Ó˚yÇñ ˆòy°öˆÏܲw xy°¡∫ö !Ó®% ˆÌˆÏܲ 32
r!öˆÏã˛ñ Óy ã˛yܲ!ï˛Ó˚ ˛õ!Ó˚!ôÓ˚ í˛z˛õÓ˚ xÓ!fliï˛–
3. ôˆÏÓ˚ !ööñ ˛õ,!ÌÓ#Ó˚ âöc = p1– ˆày°Ü˛!ê˛ Îáö ˛õ,!ÌÓ#Ó˚ ˆÜ˛w ˆÌˆÏܲ r ò)Ó˚ˆÏc ï˛áö ˆày°Ü˛!ê˛Ó˚ í˛z˛õÓ˚
x!û˛Ü˛£Ï≈ç xyܲ£Ï≈î
3 24 4. . .3 3
= =F m r pGlr m pG rπ π
ˆÜ˛ööy ˆÜ˛Ó°üye ˛õ,!ÌÓ#Ó˚ xû˛ƒhsˇˆÏÓ˚Ó˚ r Óƒy§yˆÏô≈Ó˚ ˆày°Ü˛yÜ,˛!ï˛ xÇ¢•z ˆày°Ü˛!ê˛ˆÏܲ xyܲ£Ï≈î ܲˆÏÓ˚–
û)˛ÈÙÈ˛õ,¤˛ ~•z xyܲ£Ï≈î
mg = m . 343
R pGπ – 2 4 .
3R m pG Rπ=
∴ .= rF mg
R
~áö ˆày°Ü˛!ê˛Ó˚ à!ï˛ §ü#ܲÓ˚î
= − = − rmr F mg
R
34
Óy
= − gr r
R
~•z §ü#ܲÓ˚î!ê˛ §Ó˚° ˆòy°à!ï˛ §)!ã˛ï˛ ܲˆÏÓ˚ ~ÓÇ ~Ó˚ ˛õÎ≈yÎ˚ܲy°
2 Rg
π
.
4. ôÓ˚y Îyܲ ˆê˛fiê˛ !ê˛í˛zˆÏÓÓ˚ ≤ÃfliˆÏFåȈÏòÓ˚ ˆ«˛eú˛° = a, çˆÏ°Ó˚ âöc ρ– ˆê˛fiê˛ !ê˛í˛zˆÏÓÓ˚ ˆüyê˛ û˛Ó˚ È =
0.15×aρ– ~!ê˛ˆÏܲ Î!ò z ˜òˆÏâ≈ƒ öy!üˆÏÎ˚ ˆòÄÎ˚y ÎyÎ˚ ï˛ˆÏÓ ˆÎ Óyí˛¸!ï˛ ˛õ!Ó˚üyî ç° fliyöã% ƒï˛ •ˆÏÓ ï˛yÓ˚ çöƒ
|ôù≈ü%á# ˛≤Ãï˛ƒyöÎ˚ܲ Ó° •ˆÏÓ zaρg– §%ï˛Ó˚yÇ ˆê˛fiê˛ !ê˛í˛zˆÏÓÓ˚ à!ï˛ §ü#ܲÓ˚î
0.15ap z za g= − ρ S~áyˆÏö z ˆöˆÏà!ê˛û˛ Ó˚y!¢V
Óyñ
.15g
z z= −
~!ê˛ ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛ §)!ã˛ï˛ ܲˆÏÓ˚ ÎyÓ˚ ˛õÎ≈yÎ˚ܲy° .152Tg
= π
~áö g = 9.8ms–2 üyö ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ°, T = 0.78s
5. ôÓ˚y Îyܲ ñ Ó›!ê˛Ó˚ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î
( )cosx a t= ω + θ
Ó›!ê˛Ó˚ ˆÓà
( ) 2 2sinx a t a x= − ω ω + θ = −ω −∴ 2 2
1 1v a x= −ω −
Ä
2 22 2v a x= −ω −
∴
21
2 21a x
ϑ−
=
22
2 22a x
ϑ−
– ~Ó˚ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ a =
12 2 2 2 21 2 2 1
2 21 2
x x
v v
⎛ ⎞υ − υ⎜ ⎟−⎝ ⎠
xyÓyÓ˚
2 21 2
2 2
v v−ω ω
= x2
2 – x1
2– xÌ≈yÍ
12 2 2
1 22 2
2 1
v v
x x
⎛ ⎞−ω = ⎜ ⎟−⎝ ⎠
6. xî%mˆÏÎ˚Ó˚ üˆÏôƒ ˛õyÓ˚flõ!Ó˚ܲ Ó° F = –
dVdr
=
12 60 0
0 13 712r r
Vr r
⎛ ⎞−⎜ ⎟⎝ ⎠
§yüƒyÓfliyÎ˚ F = 0 xÌ≈yÍ r = r0– §yüƒ ˆÌˆÏܲ §yüyöƒ !Óã%˛ƒï˛ xÓfliyÎ˚ Î!ò r = r
0 + ∆r •Î˚ ï˛ˆÏÓ ≤Ãï˛ƒyöÎ˚ܲ
Ó°
35
0
.=
⎛ ⎞ Δ⎜ ⎟⎝ ⎠r r
dFr
dr
= ( )0
6 8 12 140 0 012 7 13
r rV r r r r r− −
=− Δ
=
02
0
72.
Vr
r− Δ
xÌ≈yÍ Ó° ô &Óܲ =
02
0
72V
r
xö%mˆÏÎ˚Ó˚ §üyö#ï˛ û˛Ó˚ñ ôÓ˚y Îyܲñ μ– ~ܲܲ !Óã%˛ƒ!ï˛Ó˚ ú˛ˆÏ° cÓ˚î =
72 0
02
V
rμ
1.4.7 xLjϢÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚
ܲ¡õyB˛ v = 02
0
7212
V
rπ μ =
0
0
23 Vrπ μ
7. ˆÎˆÏ•ï%˛ U(r) =
9K Cr r
− +
˛ õyÓ˚flõ!Ó˚ܲ Ó° F =
dUdr
−
= 2 109K C
r r−
Î!ò §yüƒyÓfliyÎ˚ r ~Ó˚ üyö r0 •Î˚ ï˛ˆÏÓ F(r = r
0) = 0
xÌ≈yÍñ 2 100 0
9K C
r r− = 0 Óy
80
9Cr
K=
~áö r = r0 + ∆r •ˆÏ° ≤Ãï˛ƒyöÎ˚ܲ Ó°
F(r = r0 + ∆r) = F(r = r
0) +
0
.r r
dFr
dr =
⎛ ⎞ Δ⎜ ⎟⎝ ⎠
= O + 0
3 112 90 .
=
⎛ ⎞− + Δ⎜ ⎟⎝ ⎠r r
K Cr
r r
= 3 80 0
1 90 2 .⎛ ⎞
− Δ⎜ ⎟⎝ ⎠
CK r
r r
= 20
8 .ΔKr
r
§%ï˛Ó˚yÇñ Ó° ô &Óܲ = FrΔ =
30
8K
r
36
8. (i) ôÓ˚&öñ AO = BO = l
≤Ã!ï˛ +e xyôyö –e xyôyöˆÏܲ
e
l x
2
2 24π ∈ +( )
ӈϰ xyܲ£Ï≈î ܲˆÏÓ˚– ~=!°Ó˚ °!∏˛ O x!û˛ü%á# Ä üyö
( )2 2
3 32 2 2
.4 4e x e x
ll x
=π ∈ π ∈+
ˆÜ˛ööy x << l–
°!∏˛Ó°!ê˛ ≤Ãï˛ƒyöÎ˚ܲ ~ÓÇ §Ó˚ˆÏîÓ˚ §üyö%˛õyï˛# •ÄÎ˚yÎ˚ –e xyôyö!ê˛ AB ˆÓ˚áyÓ˚ °¡∫ ÓÓ˚yÓÓ˚ §Ó˚°
ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ–
(ii) ~ˆÏ«˛ˆÏe –e xyôyö!ê˛Ó˚ í˛z˛õÓ˚ ˆüyê˛ Ó° B x!û˛ü%ˆÏá ( ) ( )2
2 21 1
4 (1e
l x x
⎡ ⎤−⎢ ⎥π ∈ − +⎢ ⎥⎣ ⎦
( )2 2
2 32 2
4 4.4 4e lx e x
ll x= =π ∈ π ∈−
ˆÜ˛ööy x << l
~•z Ó° –e xyôyöˆÏܲ B x!û˛ü%ˆÏá ã˛y!°ï˛ ܲÓ˚ˆÏÓñ O x!û˛ü%ˆÏá ≤Ãï˛ƒyöÎ˚ö ܲÓ˚ˆÏÓ öy– ܲyˆÏç•z ~ˆÏ«˛ˆÏe
–e xyôyö!ê˛Ó˚ §Ó˚° ˆòy°à!ï˛ âê˛ˆÏÓ öy–
37
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 2 §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
àë˛öàë˛öàë˛öàë˛öàë˛ö
2.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
2.2 §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛
2.3 ~ܲˆÏÓ˚á#Î˚ Ä §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
2.4 ~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
2.5 ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚
2.6 ˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
2.6.1 §üܲ¡õyB˛
2.6.2 ~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛
2.6.3 !°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö
2.7 §yÓ˚yÇ¢
2.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
2.9 í˛z_Ó˚üy°y
2.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
xyˆÏàÓ˚ ~ܲˆÏܲ xyüÓ˚y §Ó˚° ˆòy°à!ï˛ Ä ï˛yÓ˚ ày!î!ï˛Ü˛ xÓܲ° §ü#ܲÓ˚î §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ–
§Ó˚° ˆòy°à!ï˛Ó˚ ˆÓ¢ ܲˆÏÎ˚ܲ!ê˛ í˛zòy•Ó˚ˆÏîÓ˚Ä xy˛õ!ö ˛õ!Ó˚ã˛Î˚ ˆ˛õˆÏÎ˚ˆÏåÈö– ≤Ã!ï˛!ê˛ ˆ«˛ˆÏe•z xyüÓ˚y ˆòˆÏá!åÈñ
ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ Ó˚y!¢ xñ ~ܲ!ê˛ !ö!ò≈‹T §ü§_¥ !mï˛#Î˚ e´ˆÏüÓ˚ xÓܲ° §ü#ܲÓ˚îˆÏܲ !§k˛ ܲˆÏÓ˚ñ ˆÎ!ê˛Ó˚ §yôyÓ˚î
Ó˚*˛õ + ω2
22
d x xdt
= 0– ~•z §ü#ܲÓ˚î!ê˛Ó˚ §üyôyö sin ωt Ä cos ωt Ó˚y!¢mˆÏÎ˚Ó˚ ~ܲ!ê˛ ˜Ó˚!áܲ ˆÎyàú˛°
~ÓÇ ~!ê˛•z x Ó˚y!¢Ó˚ §Ó˚° ˆòy°à!ï˛ §)!ã˛ï˛ ܲˆÏÓ˚– ≤ÃÜ,˛!ï˛ˆÏï˛ xˆÏöܲ §üÎ˚ ~ܲ•z §ˆÏD ~ܲy!ôܲ §Ó˚°
ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ âê˛ˆÏï˛ ˆòáy ÎyÎ˚– ˆ§ï˛yˆÏÓ˚Ó˚ ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ ~ܲ!ê˛ !Ó®% ~ܲ•z §ˆÏD ~ܲy!ôܲ ܲ¡õyˆÏB˛Ó˚
§Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •Î˚– !ü!◊ï˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ Îáö ܲyˆÏöÓ˚ ˛õò≈yÎ˚ xyâyï˛ Ü˛ˆÏÓ˚ñ ï˛áö ˛õò≈y!ê˛ ~ܲ•z
§ˆÏD ~ܲy!ôܲ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •Î˚– xyÓyÓ˚ ˆé˛y°yˆÏöy é˛yí˛¸°Z˛ö ~ܲ•z §ˆÏD
í˛z_Ó˚ÈÙÈò!«˛î Ä ˛õ)Ó≈ÈÙÈ˛õ!ÿ˛ˆÏü §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚– ~•z ~ܲˆÏܲ xyüÓ˚y ~ܲ•z !òˆÏܲ Óy ˛õÓ˚flõÓ˚
°¡∫ x!û˛ü%ˆÏá ò%•z ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
í˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒ
~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚ˆÏ° xy˛õ!öÈÙÙÙÈ
38
§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛Ó˚ !ÓÓ,!ï˛ Ä Óƒyáƒy !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
~ܲ•z §Ó˚°ˆÏÓ˚áyÎ˚ §üyö Ä x§üyö ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛° !ӈϟ’£Ïî ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
fl∫Ó˚ܲˆÏ¡õÓ˚ í˛zͲõ!_ Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
˛õÓ˚flõÓ˚ §üˆÏܲyˆÏî ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° í˛zͲõߨ °!∏˛ à!ï˛Ó˚ Óî≈öy !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
2.2 §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛
xy˛õ!ö xyˆÏàÓ˚ ~ܲˆÏܲ !Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ Ó›§ü!‹TÓ˚ §Ó˚° ˆòy°à!ï˛Ó˚ §¡∫ˆÏ¶˛ ˆçˆÏöˆÏåÈö– ~Ó˚ üˆÏôƒ ˆÎüö
§Ó˚° ˆòy°ˆÏܲÓ˚ à!ï˛ Ó˚ˆÏÎ˚ˆÏåÈñ ˆï˛üö•z ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ ôyÓ˚ܲ Óï≈˛ö#ˆÏï˛ ôyÓ˚ˆÏܲÓ˚ xyôyˆÏöÓ˚ •…y§ÈÙÈÓ,!k˛Ä xyˆÏåÈ–
xˆÏöܲ ˆ«˛ˆÏe ˆÜ˛yö Ó›Ó˚ ~ܲ§ˆÏD ~ܲy!ôܲ §Ó˚° ˆòy°à!ï˛ Óï≈˛üyö ÌyܲˆÏï˛ ˛õyˆÏÓ˚–
Îáö ~ܲ•z Ó›Ó˚ ò%•z Óy ï˛ˆÏï˛y!ôܲ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ âˆÏê˛ñ ï˛áö ˆÎ ˆÜ˛yöÄ §üˆÏÎ˚ Ó›Ó˚ §Ó˚î
˙ §üˆÏÎ˚ ≤Ã!ï˛!ê˛ ˆòy°à!ï˛Ó˚ çöƒ Ó›!ê˛Ó˚ ˆÎ §Ó˚î âê˛ï˛ñ ˆ§=!°Ó˚ ˆû˛QÓ˚ ˆÎyàú˛° •Î˚– ~!ê˛•z §Ó˚°
ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛–
§yôyÓ˚îû˛yˆÏÓ §Ó˚ˆÏîÓ˚ ˆÎyàú˛° Ó°ˆÏï˛ xyüÓ˚y ˆû˛QÓ˚ ˆÎyàú˛°•z Ó%!鲖 ï˛ˆÏÓ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ˆòy°à!ï˛=!°
Î!ò ~ܲ•z §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ xÌ≈yÍ ~ܲˆÏÓ˚á#Î˚ •Î˚ñ ï˛ˆÏÓ ˆû˛QÓ˚ ˆÎyàú˛° !•§yˆÏÓ Ó#çà!îï˛#Î˚ ˆÎyàú˛°ˆÏܲ•z
ˆöÄÎ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
~áö xyüÓ˚y §Ó˚° ˆòy°à!ï˛Ó˚ ~ܲ!ê˛ !ӈϢ£Ï ày!î!ï˛Ü˛ ôˆÏü≈Ó˚ !òˆÏܲ öçÓ˚ ˆòÓ– xy˛õ!ö ˛õˆÏí˛¸ˆÏåÈö ˆÎñ
Î!ò ˆÜ˛yöÄ Ó› x x«˛ ÓÓ˚yÓÓ˚ ω ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚ñ ï˛ˆÏÓ ï˛yÓ˚ à!ï˛Ó˚
xÓܲ° §ü#ܲÓ˚î •Î˚
+ ω2
22
d x xdt
= 0 ....... 2.1
Î!ò Ó›!ê˛ ~ܲ•z ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ y x«˛ ÓÓ˚yÓÓ˚ xyˆÏ®y!°ï˛ •Î˚ñ ï˛yÓ˚ à!ï˛ §ü#ܲÓ˚î •ˆÏÓ
+ ω2
22
d y ydt
= 0 ....... 2.2
xö%Ó˚*˛õû˛yˆÏÓñ z x«˛ ÓÓ˚yÓÓ˚ ~ܲ•z ˆÜ˛y!îܲ ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î •ˆÏÓ
+ ω2
22
d z zdt
= 0 ....... 2.3
~áö Î!ò 2.1, 2.2 Ä 2.3 §ü#ܲÓ˚î=!°ˆÏܲ ÎÌye´ˆÏü i ,
j
Ä
k
(x, y, Ä z x«˛ ÓÓ˚yÓÓ˚ ~ܲܲ ˆû˛QÓ˚V
!òˆÏÎ˚ =î ܲˆÏÓ˚ ˆÎyà ܲˆÏÓ˚öñ ï˛ˆÏÓ xy˛õ!ö ˛õyˆÏÓö
+++ω++2
22
ˆˆ ˆˆ ˆˆ ()() dixjykzixjykzdt
= 0
39
Óy + ω2
22
d r rdt
= 0 ....... 2.4
~áyˆÏö r Ó›!ê˛Ó˚ fliyöyB˛ ˆû˛QÓ˚–
~ÓyÓ˚ ôÓ˚&öñ Ó›!ê˛Ó˚ ˆÜ˛yöÄ ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ çöƒ §Ó˚î
1() rt
~ÓÇ ~ܲ•z ˆÜ˛y!îܲ ܲ¡õyˆÏB˛Ó˚
xöƒ ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ çöƒ §Ó˚î
2() rt
– ~=!°Ó˚ çöƒ xyüÓ˚y ò%!ê˛ ˛õ,Ìܲ à!ï˛ §ü#ܲÓ˚î !°áˆÏï˛ ˛õy!Ó˚ñ
+ω=2
2 11 20
drr
dt
~ÓÇ + ω =2
2222 0
d rr
dt
~ ò%!ê˛Ó˚ ˆÎyàú˛° + + + =2
21 2 1 22 ( ) ( ) 0d r r r r
dtω ....... 2.5
xÌ≈yÍñ 1( )r t Ä
2() rt
Î!ò ˛õ,Ìܲû˛yˆÏÓ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚îˆÏܲ !§k˛ ܲˆÏÓ˚ñ ï˛ˆÏÓ ï˛yˆÏòÓ˚
ˆÎyàú˛°
+12 () rr
Ä ˙ xÓܲ° §ü#ܲÓ˚îˆÏܲ !§k˛ ܲÓ˚ˆÏÓ– Ó°y Óy‡°ƒñ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î!ê˛
˜Ó˚!áܲ §ü§_¥ (linear homogeneous) •ÄÎ˚yÓ˚ ú˛ˆÏ°•z ~!ê˛ §Ω˛Ó– §ü#ܲÓ˚î!ê˛ˆÏï˛
⎛⎞⎜⎟ ⎝⎠
2drdt
Óy r2 Ó˚y!¢
ÌyܲˆÏ°
+12 rr Ó˚y!¢!ê˛ §ü#ܲÓ˚î!ê˛ˆÏܲ !§k˛ ܲÓ˚ï˛ öy–
§Ó˚° ˆòy°à!ï˛Ó˚ à!ï˛ §ü#ܲÓ˚ˆÏîÓ˚ ~•z !ӈϢ£Ï ôˆÏü≈Ó˚ ú˛ˆÏ° ò%•z!ê˛ §üܲ¡õyˆÏB˛Ó˚ ˆòy°à!ï˛ í˛z˛õ!Ó˚˛õy!ï˛ï˛
•ˆÏÎ˚ öï%˛ö ~ܲ!ê˛ ˆòy°à!ï˛ í˛zͲõߨ ܲˆÏÓ˚ñ ÎyÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyܲ í˛z˛õ!Ó˚˛õy!ï˛ï˛ ˆòy° à!ï˛ ò%•z!ê˛Ó˚ ܲ¡õyˆÏB˛Ó˚
xö%Ó˚*˛õ– ~áö xyüÓ˚y ~ܲˆÏÓ˚á#Î˚ Ä §üܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛° !Ó¢òû˛yˆÏÓ ˛õÓ˚#«˛y
ܲÓ˚Ó–
2.3 ~ܲˆÏÓ˚á#Î˚ §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° òy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° òy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° òy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° òy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° òy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
ôÓ˚y Îyܲ x x«˛ ÓÓ˚yÓÓ˚ ω ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ çöƒ ˆÜ˛yöÄ Ó›Ó˚ §yüƒyÓfliy ˆÌˆÏܲ
§Ó˚î ÎÌye´ˆÏü
x1 = a1 cos (ωt + θ1)
~ÓÇ x2 = a
2 cos (ωt + θ
2)
ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° °!∏˛ §Ó˚î í˛zû˛ˆÏÎ˚Ó˚ ˆÎyàú˛°
x = x1 + x
2
= a1 cos (ωt + θ1) + a2 cos (ωt + θ2)
= (a1 cos θ
1 + a
2 cos θ
2) cos ωt – (a
1 sin θ
1 + a
2 sin θ
2) sin ωt
....... 2.6
40
~áö Î!ò (a1 cos θ1 + a2 cos θ2) ~Ó˚ çyÎ˚àyÎ˚ A cos θ ~ÓÇ (a1 sin θ1 + a2 sin θ2) ~Ó˚ çyÎ˚àyÎ˚
A sin θ ˆ°áy ÎyÎ˚ ï˛ˆÏÓñ
x = A cos θ cosωt – A sin θ sinωt
= A cos (ωt + θ) (A Ä θ ò%•z!ê˛ öï%˛ö ô &Óܲ) ....... 2.7
~áö A Ä θ ~Ó˚ üyö a1, a
2, θ
1 Ä θ
2 ~Ó˚ !•§yˆÏÓ çyöy ≤ÈÏÎ˚yçö–
A2 = (A cos θ)2 + (A sin θ)2
= (a1
2 cos2 θ1 + a
22 cos2 θ
2 + 2a
1a
2 cos θ
1 cos θ
2)
+ (a12 sin2 θ
1 + a
22 sin2 θ
2 + 2a
1a
2 sin θ
1 sin θ
2)
Óyñ A2 = a12 + a2
2 + 2a1a2 cos (θ1 – θ2) ....... 2.8
~ÓÇ tan θ =
sincos
AA
θθ
=
1122
1122
sinsincoscos
aaaa
++
θθθθ
....... 2.9
~áyˆÏö ~ܲ•z ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° xyüÓ˚y ˙ ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛
öï%˛ö §Ó˚° ˆòy°à!ï˛ ˆ˛õ°yü ÎyÓ˚ !ÓhflÏyÓ˚ A ~ÓÇ ò¢yܲÓ˚î θ–
ˆû˛QÓ˚ !ã˛eˆû˛QÓ˚ !ã˛eˆû˛QÓ˚ !ã˛eˆû˛QÓ˚ !ã˛eˆû˛QÓ˚ !ã˛e
§Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏܲ ˆû˛QÓ˚ !ã˛e myÓ˚y Ó˚*˛õy!Î˚ï˛ Ü˛Ó˚y
ÎyÎ˚– xy˛õ!ö !öÿ˛Î˚•z çyˆÏöö ˆÎñ ~ܲ!ê˛ â)î≈ƒüyö ˆû˛QÓ˚ myÓ˚y ~ܲ!ê˛
§Ó˚° ˆòy°à!ï˛ˆÏܲ §)!ã˛ï˛ ܲÓ˚y ÎyÎ˚– a1 ˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ ˆû˛QÓ˚ Î!ò
t = 0 §üˆÏÎ˚ x xˆÏ«˛Ó˚ §ˆÏD θ1 ˆÜ˛yî Ó˚ã˛öy ܲˆÏÓ˚ Ä ω ˆÜ˛Ô!îܲ ˆÓˆÏà
ÓyüyÓˆÏï≈˛ â%Ó˚ˆÏï˛ ÌyˆÏܲ ï˛ˆÏÓ t §üˆÏÎ˚ x xˆÏ«˛Ó˚ §ˆÏD ˆ§!ê˛ ωt + θ1
ˆÜ˛yö Ó˚ã˛öy ܲÓ˚ˆÏÓ ~ÓÇ x xˆÏ«˛Ó˚ í˛z˛õÓ˚ ˆ§!ê˛Ó˚ x!û˛ˆÏ«˛˛õ (projection) •ˆÏÓ S!ã˛e 2.1V
x1
= a1 cos (ωt + θ
1)
xÌ≈yÍñ ˆû˛QÓ˚!ê˛Ó˚ x!û˛ˆÏ«˛˛õ a1 !ÓhflÏyˆÏÓ˚ ω ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •ˆÏÓ ~ÓÇ
x xˆÏ«˛Ó˚ §ˆÏD ˆû˛QÓ˚!ê˛Ó˚ ˆÜ˛yî (ωt + θ1) •ˆÏÓ t §üˆÏÎ˚ ˙ à!ï˛Ó˚ ò¢yˆÏܲyî–
~ ôÓ˚ˆÏöÓ˚ ˆû˛QÓ˚ !ã˛eˆÏܲ xyüÓ˚y ò%•z!ê˛ §üܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ˆÏܲ §ÇÎ%_´ ܲÓ˚yÓ˚ ܲyˆÏç °yàyˆÏï˛
˛õy!Ó˚– 2.2 !ã˛ˆÏe t = 0 §üˆÏÎ˚ ò%•z!ê˛ â)î≈ƒüyö ˆû˛QˆÏÓ˚Ó˚ xÓfliyö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ˆÎ=!°Ó˚ ˜òâ≈ƒ ÎÌye´ˆÏü
a1 Ä a2 ~ÓÇ ò¢yˆÏܲyî ÎÌye´ˆÏü θ1 Ä θ2– ˆû˛QÓ˚ ò%•z!ê˛Ó˚ °!∏˛Ó˚ ˜òâ≈ƒ A ~ÓÇ ≤ÃyÌ!üܲ ò¢yˆÏܲyî θ1– a1
Ä a2 ˜òˆÏâ≈ƒÓ˚ ˆû˛QÓ˚ ò%•z!ê˛
!ã˛e 2.1§Ó˚° ˆòy°à!ï˛Ó˚ §)ã˛Ü˛ ˆû˛QÓ˚
x1 = a1 cos (ω t + θ1)
41
x1 = a1 cos (ωt + θ1)
Ä x2 = a
2 cos (ωt + θ
2)
~•z ò%•z §Ó˚° ˆòy°à!ï˛ˆÏܲ §)!ã˛ï˛ ܲˆÏÓ˚– °!∏˛ ˆû˛QÓ˚
A
~Ó˚ x xˆÏ«˛Ó˚ í˛z˛õÓ˚ x!û˛ˆÏ«˛˛õ A cos (ωt +
θ) ˙ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° í˛zͲõߨ §Ó˚° ˆòy°à!ï˛!ê˛ !öˆÏò≈¢ ܲÓ˚ˆÏåÈ– !ã˛e 2.2 ˆÌˆÏܲ
xy˛õ!ö §•ˆÏç•z ˆòáyˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ
A cos θ = a1 cos θ
1 + a
2 cos θ
2....... 2.10a
~ÓÇñ A sin θ = a1 sin θ
1 + a
2 sin θ
2....... 2.10b
~•z ò%•z!ê˛ §ü#ܲÓ˚ˆÏîÓ˚ §y•yˆÏ΃ xy˛õ!ö ~áö 2.8 Ä 2.9 §¡õÜ≈˛=!° ˆ˛õˆÏï˛ ˛õyˆÏÓ˚ö–
~•z ˛õk˛!ï˛!ê˛ xy˛õ!ö §üyö ܲ¡õyˆÏB˛Ó˚ ˆÎ ˆÜ˛yöÄ §ÇáƒÜ˛ §Ó˚° ˆòy°à!ï˛ˆÏܲ Î%_´ ܲÓ˚yÓ˚ ܲyˆÏç ÓƒÓ•yÓ˚
ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– ôÓ˚&öñ n §ÇáƒÜ˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î
x1 = a1 cos (ωt + θ1), x2 = a2 cos (ωt + θ2), .........., xn = an cos (ωt + θn)
~•z §Ó˚° ˆòy°à!ï˛=!° a1, a
2, ...., a
n ˜òˆÏâ≈ƒÓ˚ n §ÇáƒÜ˛ â)î≈ƒüyö ˆû˛QÓ˚ myÓ˚y §)!ã˛ï˛ •ˆÏÓ– í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚
ú˛ˆÏ° ˆòy°à!ï˛!ê˛Ó˚ §ü#ܲÓ˚î Î!ò
x = A cos (ωt + θ)
•Î˚ñ ï˛ˆÏÓ 2.10a, b §ü#ܲÓ˚î=!°Ó˚ xö%Ó˚*˛õ §ü#ܲÓ˚î •ˆÏÓ
A cos θ =
1
cosin
iii
a=
=∑θ
A sin θ =
sin ii a ∑θ
xÌ≈yÍ A2 =
()() 22
1 cossin iii aa + ∑∑ θθ
....... 2.11a
~ÓÇ tan θ =
1
sincosin
iiiii
aa=
=∑∑ θθ
....... 2.11b
~áö xy˛õ!ö ܲˆÏÎ˚ܲ!ê˛ xö%¢#°ö#Ó˚ §y•yˆÏ΃ í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öyÓ˚ ≤ÈÏÎ˚yà ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö–
xö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈ1 : ~ܲ!ê˛ ˆòy°ˆÏܲÓ˚ §yôyÓ˚î à!ï˛ §ü#ܲÓ˚î
2
2sin dgt dt
+ θθ
= 0
42
xö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈxö%¢#°ö#ÈÙÈ2 : ò%•z!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ à!ï˛ §ü#ܲÓ˚î x1 = 5 cos (50t + 30º) Ä x2 = 12 cos (50t
+ 120º) ~=!°Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° ˆÎ §Ó˚° ˆòy°à!ï˛ í˛zͲõߨ •ˆÏÓ ï˛yÓ˚ !ÓhflÏyÓ˚ Ä ò¢yˆÏܲyî !öî≈Î˚
ܲÓ˚&ö–
~ ˛õÎ≈hsˇ xy˛õ!ö ~ܲ•z ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ §¡∫ˆÏ¶˛ ˛õí˛¸ˆÏ°ö– !ܲv ܲ¡õyB˛ ò%•z!ê˛
§üyö öy •ˆÏ° ï˛yÓ˚ ú˛° ܲ# •ˆÏÓ⁄ xy§%ö ~ÓyÓ˚ ˆ§!ê˛ ˛õÓ˚#«˛y ܲˆÏÓ˚ ˆòáy Îyܲ–
2.4 ~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛~ܲˆÏÓ˚á#Î˚ Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
§üܲ¡õyˆÏB˛Ó˚ ˆ«˛ˆÏe xyüÓ˚y í˛z˛õ!Ó˚˛õy!ï˛ï˛ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω ôˆÏÓ˚ !öˆÏÎ˚ 2.6
§¡õÜ≈˛=!° !°ˆÏá!åÈ°yü– ~áö ˆÜ˛Ô!îܲ ܲ¡õyB˛ ò%•z!ê˛ !û˛ß¨ ôˆÏÓ˚ !öˆÏ° §ü#ܲÓ˚î=!° •ˆÏÓ
x1 = a
1 cos (ω
1t + θ
1)
~ÓÇ x2 = a2 cos (ω2t + θ2)....... 2.12
í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° °!∏˛ §Ó˚î
x = x1 + x2 = a1 cos (ω1t + θ1) + a2 cos (ω2t + θ2)
~áyˆÏö í˛z˛õ!Ó˚˛õy!ï˛ï˛ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ ò¢yÓ˚ xhsˇÓ˚ (ω1 – ω2)t + (θ1 – θ2)– (ω1 – ω2)t Ó˚y!¢!ê˛
§üÎ˚ !öû≈˛Ó˚– !ܲv (θ1 – θ
2) Ó˚y!¢!ê˛ x˛õ!Ó˚Óï≈˛ö¢#° ~ÓÇ °!∏˛ ˆòy°à!ï˛ˆÏï˛ ~!ê˛Ó˚ ˆÜ˛yöÄ =Ó˚&c˛õ)î≈ û)˛!üܲy
ˆö•z– ~çöƒ xyüÓ˚y θ1
Ä θ2 ≤ÃyÓ˚!Ω˛Ü˛ ò¢yˆÏܲyî ò%!ê˛ˆÏܲ ¢)öƒ ӈϰ ôˆÏÓ˚ ˆöÓ– ~Ó˚ xÌ≈ ~•zñ ˆÎ Îáö x
1
= a1 ~ÓÇ x2 = a2 !ë˛Ü˛ ï˛áö ˆÌˆÏܲ•z xyüÓ˚y §üˆÏÎ˚Ó˚ àîöy ÷Ó˚& ܲÓ˚!åÈ– ~áö xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
x = a1 cos ω1t + a2 cos ω2t ....... 2.13
≤Ã̈Ïü xyüÓ˚y ~ܲ!ê˛ xˆÏ˛õ«˛yÜ,˛ï˛ §•ç xÓfliy !ÓˆÏÓã˛öy ܲÓ˚Ó–
Îáö a1 = a
2 : ôÓ˚y Îyܲ a
1 = a
2 = a ~ÓÇ ω
1 > ω
2
~•z xÓfliyÎ˚ x = a [cos ω1t + cos ω
2t]
=
1212 2coscos 22 att−+ ωωωω
....... 2.14
[ˆÜ˛ööy xyüÓ˚y çy!ö cos A + cos B =
+− 2coscos22
ABAB t
]
43
~áy Ïö +1 2
2ω ω
Ó˚y!¢!ê˛ ω1 Ä ω
2 ~Ó˚ àí˛¸ üyö– ôÓ˚y Îyܲ
12
2+ ωω
= ω–
12
2− ωω
Ó˚y!¢!ê˛Ó˚ üyö
xˆÏ˛õ«˛yÜ,˛ï˛ x“– ~!ê˛Ó˚ fliyˆÏö xyüÓ˚y ω0!°áÓ– ~áö 2.14 ˆÌˆÏܲ xyüÓ˚y ˛õy•z
x = 2a cos ω0t cos ωt– ........ 2.15
2.15 §ü#ܲÓ˚î!ê˛ °«˛ƒ ܲÓ˚ˆÏ° ~!ê˛ Ü˛ï˛Ü˛ê˛y ω ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î ӈϰ üˆÏö
•ˆÏÓ– !ܲv ~áyˆÏö ˆòy°à!ï˛!ê˛Ó˚ !ÓhflÏyÓ˚ 2a cos ω0t §üˆÏÎ˚Ó˚ §ˆÏD ˛õ!Ó˚Óï≈˛ö¢#°– ~!ê˛ !öˆÏç•z 2a !ÓhflÏyˆÏÓ˚
§Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚– ~•z §Ó˚° ˆòy°à!ï˛!ê˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω0ñ xÌ≈yÍ ˛õÎ≈yÎ˚ܲy°
0
2πω
–
2.3 !ã˛ˆÏe 2.15 §ü#ܲÓ˚î myÓ˚y §)!ã˛ï˛ ˆòy°à!ï˛!ê˛ ˆòáyö •ˆÏÎ˚ˆÏåÈ– !ã˛e!ê˛ °«˛ƒ ܲÓ˚ˆÏ° Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ
~ˆÏ«˛ˆÏe
0
πω
§üÎ˚ xhsˇÓ˚ ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ §ˆÏÓ≈yFã˛ üyˆÏö ˆ˛õÑÔåÈÎ˚ xÌÓy ¢)öƒ •Î˚– xy§ˆÏ°ñ í˛z˛õ!Ó˚˛õy!ï˛ï˛
ˆòy°à!ï˛ ò%•z!ê˛ Îáö §üò¢yÎ˚ ÌyˆÏܲñ ï˛áö !ÓhflÏyÓ˚ §ˆÏÓ≈yFã˛ •Î˚ ~ÓÇ ~=!° Îáö !Ó˛õÓ˚#ï˛ ò¢yÎ˚ ÌyˆÏܲñ
ï˛áö !ÓhflÏyÓ˚ §Ó≈!ö¡¨ñ ~ˆÏ«˛ˆÏe ¢)öƒ •Î˚–
~ÓyÓ˚ xyüÓ˚y xyÓ˚Ä §yôyÓ˚î xÓfliyÎ˚ñ Îáö a1 Ä a
2 x§üyöñ ï˛áö ܲ# âˆÏê˛ ï˛y ˆÓyé˛yÓ˚ ˆã˛‹Ty ܲÓ˚Ó–
Îáö a1 ≠ a
2 : ôÓ˚y Îyܲ a
1 > a
2, a
1 = a + b, a
2 = a – b ˆÎáy Ïö a Ä b ò%•z!ê˛ öï%˛ö ô &Óܲ– xyüÓ˚y
üˆÏö Ó˚yáÓ a =
() +121,2
aa
b =
() −1212
aa
–
2.13 §)ˆÏe a1 Ä a
2 ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ a Ä b ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õy•z
x = (a + b) cos ω1t +(a – b) cos ω2t
~áö !ܲå%Èê˛y ày!î!ï˛Ü˛ §Ó˚°#ܲÓ˚î ≤ÈÏÎ˚yçö–
x = a [cos ω1t + cos ω2t] + b [cos ω1t – cos ω2t]
πω0 2o
t = 0
πωo
πω32o
πω2
o
t = 0
πω2
oπωo
πω32o
πω2
o
–2a
O
–2a2acosωot
t
xa1+a2
a1–a
2
44
à!îˆÏï˛Ó˚ §)e cos A + cos B =
+− 2coscos22
ABAB
~ÓÇ cos A – cos B =
+− 2sinsin22
ABBA
ÓƒÓ•yÓ˚ ܲˆÏÓ˚ñ
x =
12121221 2coscos2sinsin 2222 attbtt+−+− + ωωωωωωωω
˛õ)ˆÏÓ≈Ó˚ üï˛
12
2+ ωω
= ω ~ÓÇ È
12
2− ωω
= ω0 !°áˆÏ° ˛õyÄÎ˚y ÎyˆÏÓ
x = 2a cos ω0t . cos ωt – 2b sin ω0t . sin ωt
xyÓ˚Ä §Ó˚°#ܲÓ˚ˆÏîÓ˚ çöƒ 2a cos ω0t ~Ó˚ fliyˆÏö A cos φ ~ÓÇ 2b cos ω0t ~Ó˚ fliyˆÏö A sin φ ˆ°áy
ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~Ó˚ ú˛ˆÏ° ˛õyÄÎ˚y ÎyÎ˚
x = A cos φ cos ωt – A sin φ sin ωt ....... 2.16
= A cos (ω t + φ)
~•z §)ˆÏe A2 = (A cos φ)2 + (A sin φ)2
= (2a cos ω0t)2 + (2b sin ω0t)
2
= (a1 + a 2)2 cos2 ω0t + (a1 – a 2)
2 sin2 ω0t
= a12 + a
22 + 2a
1a
2cos2ω0t
Óyñ A =
⎡⎤ ++− ⎣⎦1
222121212 2cos() aaaat ωω
....... 2.17
~åÈyí˛¸y tan φ = sincos
AA
φφ =
0
0
2sin2cos
btat
ωω
=
−−+
1212
12
tan2
aat
aaωω
....... 2.18
2.17 §ü#ܲÓ˚î!ê˛ ˛õÓ˚#«˛y ܲÓ˚ˆÏ° xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ °!∏˛ ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ A §üˆÏÎ˚Ó˚ §ˆÏD
(a1 + a2) Ä (a1 – a2) §#üyÓ˚ üˆÏôƒ Äë˛yöyüy ܲˆÏÓ˚– Sω1 – ω2Vt Ó˚y!¢Ó˚ üyö Îáö 0, 2π, 4π, ..... ï˛áö
A = a1 + a
2 ~ÓÇ Sω1 – ω2Vt ~Ó˚ üyö Îáö π, 3π, 5π..., ï˛áö A = a
1 – a
2– §%ï˛Ó˚yÇñ !ÓhflÏyˆÏÓ˚Ó˚
~•z ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˛õÎ≈yÎ˚ܲy°
−12
2πωω
– 2.4 !ã˛ˆÏe 2.16 §ü#ܲÓ˚î xö%ÎyÎ˚# x ~Ó˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
xÓ¢ƒ ~áyˆÏö ôˆÏÓ˚ ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈ ˆÎ ω1 Ä ω2 ~Ó˚ üˆÏôƒ ˛õyÌ≈ܲƒ á%Ó ˆÓ¢# öÎ˚ ~ÓÇ Sω1 – ω2V <<
45
12
2+ ⎛⎞
⎜⎟ ⎝⎠ωω
xÌ≈yÍ ω0 << ω– ~Ó˚ ú˛ˆÏ° A ~ÓÇ φ §üˆÏÎ˚Ó˚ §ˆÏD ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏ°Ä ~•z ˛õ!Ó˚Óï≈˛ö
xï˛ƒhsˇ ô#Ó˚– °!∏˛ ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy°
2πω
§üˆÏÎ˚Ó˚ üˆÏôƒ ˆ§!ê˛Ó˚ !ÓhflÏyˆÏÓ˚Ó˚ !ӈϢ£Ï ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö âˆÏê˛
öy–
°!∏˛ xyˆÏ®y°ˆÏöÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ~•z ˛õÎ≈yÎ˚e´ü# Äë˛yöyüyˆÏܲ Ó#ê˛ (beat) Ó°y •Î˚– ¢∑ !ÓK˛yˆÏö ˆòáy ÎyÎ˚
ˆÎñ ܲyåÈyܲy!åÈ Ü˛¡õyˆÏB˛Ó˚ ò%•z!ê˛ fl∫Ó˚ ~ܲ§ˆÏD ôù!öï˛ •ˆÏ° ¢ˆÏ∑Ó˚ ˛≤Ãyӈϰƒ ˛õÎ≈yÎ˚e´ˆÏü ï˛yÓ˚ï˛üƒ âˆÏê˛– ~ˆÏܲ
xyüÓ˚y fl∫Ó˚ܲ¡õ Ó!°– ~ §¡∫ˆÏ¶˛ xy˛õ!ö 7ü ~ܲˆÏܲ xyÓ˚Ä ˆÓ!¢ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
~Ó˚˛õÓ˚ xyüÓ˚y ç!ê˛° Ó˚y!¢Ó˚ §y•yˆÏ΃ ò%•z ~ܲˆÏÓ˚á#Î˚ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy
ܲÓ˚Ó– !ܲv ï˛yÓ˚ xyˆÏà xy˛õ!ö •Î˚ï˛ ~ܲ!ê˛ §yÇ!áƒÜ˛ (numerical) ≤Èϟ¿Ó˚ §üyôyö ܲˆÏÓ˚ !öˆÏï˛
ã˛y•zˆÏÓö–
xö%¢#°ö#ÙÈxö%¢#°ö#ÙÈxö%¢#°ö#ÙÈxö%¢#°ö#ÙÈxö%¢#°ö#ÙÈ3 :
ò%•z!ê˛ ~ܲˆÏÓ˚á#Î˚ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î x1 = 20 sin (52πt) ˆ§!ü Ä x
2 = 24 sin (48πt) ˆ§!üñ
ˆ§áyˆÏö tÈÙÈ~Ó˚ ~ܲܲ ˆ§ˆÏܲu˛– °!∏˛ ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î!ê˛ !öî≈Î˚ ܲÓ˚&ö–
2.5 ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚ç!ê˛° Ó˚y!¢Ó˚ ÓƒÓ•yÓ˚
xy˛õöyÓ˚ •Î˚ï˛ üˆÏö xyˆÏåÈ ˆÎñ –1 Ó˚y!¢!ê˛Ó˚ Óà≈ü)°ˆÏܲ xyüÓ˚y j !ã˛•´ !òˆÏÎ˚ !öˆÏò≈¢ ܲ!Ó˚ ~ÓÇ ˆ§!ê˛ xÌÓy
ï˛yÓ˚ ˆÜ˛yöÄ =!îï˛Ü˛ˆÏܲ ܲy“!öܲ Ó˚y!¢ Ó!°– ܲy“!öܲ Ä ÓyhflÏÓ Ó˚y!¢Ó˚ §üß∫ˆÏÎ˚ à!ë˛ï˛ ˆÜ˛yöÄ §ÇáƒyˆÏܲ
SˆÎüö 2 + 3i) xyüÓ˚y ç!ê˛° Ó˚y!¢ ӈϰ Ìy!ܲ– ej.θ ~Ó˚*˛õ ~ܲ!ê˛ ç!ê˛° Ó˚y!¢– ~!ê˛Ó˚ üyö cos θ + j sin
θ xÌ≈yÍñ ~!ê˛Ó˚ ÓyhflÏÓ xÇ¢ cos θ Ä Ü˛y“!öܲ xÇ¢ j sin θ– xyüÓ˚y ˛õÓ˚Óï≈˛# xyˆÏ°yã˛öyÎ˚ Óy
() j eθ
Ó°ˆÏï˛ ejθ Ó˚y!¢Ó˚ ÓyhflÏÓ xÇ¢ ~ÓÇ Ü˛y
() j eθ
Ó°ˆÏï˛ Ó˚y!¢!ê˛Ó˚ ܲy“!öܲ xÇ¢ ˆÓyé˛yÓ–
ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ ç!ê˛° Ó˚y!¢ Z (= X + jY) ˆÜ˛ Aejθ Ó˚*ˆÏ˛õ ˆ°áy ÎyÎ˚– §•ˆÏç•z ˆòáyˆÏöy ÎyÎ˚ ˆÎñ
~ˆÏ«˛ˆÏe A2 = X2 + Y2 = ZZ*, ˆÎáyˆÏö Z* ç!ê˛° Ó˚y!¢ Z ~Ó˚ Î%@¬ ç!ê˛° Ó˚y!¢ (complex conjugate)
~ÓÇ tan θ =
YX
– à!îˆÏï˛Ó˚ ~•z ï˛Ìƒ=!° xyüÓ˚y ~áyˆÏöÏ Ü˛yˆÏç °yàyÓ–
xyüÓ˚y ~ÓyÓ˚ 2.6 §ü#ܲÓ˚î=!°ˆÏï˛ !ú˛ˆÏÓ˚ Îy•z– ~=!°ˆÏܲ xyüÓ˚y xöƒû˛yˆÏÓ !°áˆÏï˛ ˛õy!Ó˚– Î!ò Z1 =
+1 ()1
jt aeωθ
~ÓÇ Z2 =
2 ()2
jt ae+ ωθ
ôÓ˚y ÎyÎ˚ñ ï˛ˆÏÓ x1 = Óy (Z
1), x
2 = Óy (Z
2) ~ÓÇ x
1 + x
2 = Óy (Z
1
+ Z2)– ~áö Î!ò Z
1 + Z
2 =
() jt Ae+ ωθ
ˆ°áy •Î˚ñ ï˛ˆÏÓ ˛õ)ˆÏÓ≈Ó˚ üï˛ S§ü#ܲÓ˚î 2.7)
46
x = x1+ x
2 = A cos (ωt + θ)
A Ó˚y!¢Ó˚ üyö !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
(Z1 + Z2)e–jωt = Aejθ §%ï˛Ó˚yÇñ
A2 = (Z1 + Z
2)e–jωt.(Z
1 + Z
2)*ejωt
= (Z1 + Z2)(Z1 + Z2)*
=
()() ++−+−+ ++ 1212 ()()()()1212
jtjtjtjt aeaeaeae ωθωθωθωθ
(a1, a
2 ÓyhflÏÓ §ÇáƒyV
= ( ) ( )( )− − −+ + +1 2 1 22 21 2 1 2
j ja a a a e eθ θ θ θ
Óyñ A2 = + + −2 21 2 1 2 1 22 cos( )a a a a θ θ
Îy 2.8 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ
xyÓyÓ˚ ˆÎˆÏ•ï%˛ñ (Z1 + Z
2)e–jωt
=
+ 1122jj aeae θθ
= (a1 cos θ
1+a
2 cos θ
2) + j(a
1 sin θ
1+a
2 sin θ
2)
~•z Ó˚y!¢ˆÏܲ (X + jY) Ó˚y!¢Ó˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ xy˛õ!ö !°áˆÏï˛ ˛õyˆÏÓ˚ö
tan θ =
++
1122
1122
sinsincoscos
aaaa
θθθθ
– ~!ê˛ 2.9 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ–
xy˛õ!ö !öÿ˛Î˚•z xö%û˛Ó ܲÓ˚ˆÏåÈö ˆÎñ ç!ê˛° §ÇáƒyÓ˚ §y•yˆÏ΃ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ày!î!ï˛Ü˛
!ӈϟ’£Ïî !ܲå%Èê˛y §•ˆÏç ܲÓ˚y ÎyÎ˚– ~ÓyÓ˚ xyüÓ˚y §üˆÏÓ˚á Ä x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚
ˆ«˛ˆÏe ~•z ˛õk˛!ï˛Ó˚ ≤ÈÏÎ˚yà ܲÓ˚Ó–
ç!ê˛° §ÇáƒyÓ˚ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ 2.12 §ü#ܲÓ˚î ò%!ê˛ˆÏï˛ ˆ°áy ÎyÎ˚ ≠
x1 = Óy
() + 11 ()1
jt aeωθ
Ä x2 = Óy ( )+2 2( )
2j ta e ω θ
˛õ)ˆÏÓ≈Ó˚ üï θ1 Ä θ
2 ò¢yˆÏܲyî=!°ˆÏܲ ¢)öƒ ôˆÏÓ˚ !öˆÏÎ˚ ˆ°áy ˆÎˆÏï˛ ˛õyˆÏÓ˚
x = Óy ( )+1 21 2
j t j ta e a eω ω Óy (Z) ˆÎáyˆÏö Z = +1 2
1 2j t j ta e a eω ω
47
Î!ò a1 = a2 = a ôÓ˚y •Î˚ ï˛ˆÏÓ
Z =
() + 12 jtjt aee ωω
°!∏˛ ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ Î!ò A •Î˚ ï˛ˆÏÓ
A2 = ZZ* = ( )( )− −+ +1 2 1 22 j t j t j t j ta e e e eω ω ω ω [a ~ܲ!ê˛ ÓyhflÏÓ §Çáƒy]
= ( )− − −+ +1 2 1 2( ) ( )2 2 j t j ta e eω ω ω ω
= 2a2[1 + 2 cos (ω1 – ω
2)t]
= −⎛ ⎞
⎜ ⎟⎝ ⎠2 2 1 24 cos
2a t
ω ω
Óyñ A = −⎛ ⎞
⎜ ⎟⎝ ⎠1 22 cos .
2a t
ω ωS§ü#ܲÓ˚î 2.15 ï%˛°ö#Î˚V
x˛õÓ˚˛õˆÏ«˛ñ a1 > a
2 ôˆÏÓ˚ !öˆÏ°ñ ˛õyÄÎ˚y ÎyˆÏÓ
A2 = ZZ* = ( )( )− −+ +1 2 1 21 2 1 2
j t j t j t j ta e a e a e a eω ω ω ω
= ( ) ( )( )− − −+ + +1 2 1 22 21 2 1 2
j t j ta a a a e eω ω ω ω
= a a a a t12
22
1 2 1 22+ + −cos( )ω ω
xÌ≈yÍ A = ( )⎡ ⎤+ + −⎣ ⎦1
2 2 21 2 1 2 1 22 cosa a a a tω ω S§ü#ܲÓ˚î 2.17 ï%˛°ö#Î˚V
~áyˆÏö xy˛õ!ö ˆòáˆÏï˛ ˆ˛õˆÏ°ö ˆÎñ ç!ê˛° §Çáƒy ÓƒÓ•yÓ˚ ܲˆÏÓ˚ x§ü ܲ¡õyˆÏB˛Ó˚ ˆòy°à!ï˛Ó˚
í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ !ӈϟ’£ÏîÄ §•ˆÏç ܲÓ˚y ÎyÎ˚– ~ÓyÓ˚ ~ ôÓ˚ˆÏöÓ˚ ~ܲ!ê˛ xö%¢#°ö# xy˛õ!ö !öˆÏç•z ܲˆÏÓ˚ ˆòá%ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -4 : ~ܲ•z !ÓhflÏyˆÏÓ˚Ó˚ !ï˛ö!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ ܲ¡õyB˛ §üyö !ܲv ò¢yˆÏܲyî=!°Ó˚ ÓƒÓôyö
φ– °!∏˛ ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ !öî≈Î˚ ܲÓ˚&ö–
~ ˛õÎ≈hsˇ xyüÓ˚y ~ܲˆÏÓ˚á#Î˚ ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ~áö xyüÓ˚y ˛õÓ˚flõÓ˚
§üˆÏܲyˆÏî ~ܲ•z ܲ¡õyˆÏB˛Ó˚ xÌÓy §Ó˚° xö%˛õyˆÏï˛ ò%•z !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ö §¡∫ˆÏ¶˛
xyˆÏ°yã˛öy ܲÓ˚Ó–
2.6 ˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛
xy˛õ!ö !öÿ˛Î˚•z ~ܲ!ê˛ §Ó˚° ˆòy°ˆÏܲÓ˚ à!ï˛ °«˛ƒ ܲˆÏÓ˚ ÌyܲˆÏÓö– ~ܲ!ê˛ §)ï˛yÓ˚ ~ܲ≤ÃyˆÏhsˇ ~ܲ!ê˛ ˆåÈyê˛
!ì˛° ˆÓшÏô ˆ§!ê˛ˆÏܲ é%˛!°ˆÏÎ˚ xy˛õ!ö ~ܲ!ê˛ §Ó˚° ˆòy°Ü˛ ˜ï˛!Ó˚ ܲˆÏÓ˚ !öˆÏï˛ ˛õyˆÏÓ˚ö– xyò¢≈ §Ó˚° ˆòy°ˆÏܲÓ˚
48
à!ï˛ ~ܲ §Ó˚°ˆÏÓ˚áyÎ˚ xyÓk˛ ÌyܲˆÏ°Ä ˆòy°ˆÏܲÓ˚ !˛õ[˛!ê˛ ˆüyê˛yü%!ê˛û˛yˆÏÓ xö%û)˛!üܲ ï˛ˆÏ° !Óã˛Ó˚î ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚–
~•z ï˛°!ê˛ˆÏܲ x-y ï˛° !•§yˆÏÓ !ÓˆÏÓã˛öy ܲÓ˚ˆÏ° ˆòáy ÎyÎ˚ ˆÎñ x Ä yñ í˛zû˛Î˚ !òܲ ÓÓ˚yÓÓ˚ !˛õu˛!ê˛ ~ܲ•z ܲ¡õyˆÏB˛
§Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°yˆÏú˛Ó˚y ܲˆÏÓ˚– ~ çyï˛#Î˚ ˆòy°Ü˛ˆÏܲ xyüÓ˚y ˆày°#Î˚ ˆòy°Ü˛ (spherical pendulum)
Ó!° ~ÓÇ ~!ê˛ ˛õÓ˚flõÓ˚ §üˆÏܲyˆÏî ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ~ܲ!ê˛ §•ç í˛zòy•Ó˚î– Îáö ˆÜ˛yöÄ
Ó› ~ܲ•z §ˆÏD ˛õÓ˚flõÓ˚ §üˆÏܲyˆÏî ò%•z §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •Î˚ñ ï˛áö Ó›!ê˛Ó˚ à!ï˛˛õÌ ˆÜ˛üö
•Î˚ñ ˆ§!ê˛•z ~•z xLjϢÓ˚ xyˆÏ°yã˛ƒ !Ó£ÏÎ˚– ≤Ã̈Ïü xyüÓ˚y ò%•z ˆòy°à!ï˛Ó˚ ܲ¡õyB˛ §üyö ӈϰ ôˆÏÓ˚ ˆöÓ–
2.6.1 §üܲ¡õyB˛§üܲ¡õyB˛§üܲ¡õyB˛§üܲ¡õyB˛§üܲ¡õyB˛
ôÓ˚y Îyܲñ ˆÜ˛yöÄ Ó› x Ä y !òܲ ÓÓ˚yÓÓ˚ ~ܲ•z ܲ¡õyˆÏB˛Ó˚ ˆÎ §Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°yã˛° ܲˆÏÓ˚ñ ˆ§=!°Ó˚
§ü#ܲÓ˚î
x = a cos ωt
Ä y = b cos (ωt + φ)....... 2.19
~áyˆÏö ò%•z ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ a Ä b– ò¢yÓ˚ xhsˇÓ˚ φ– ~áö xyüÓ˚y a, b Ä φ ~Ó˚ í˛z˛õÓ˚ !Ó!û˛ß¨ ¢ï≈˛
xyˆÏÓ˚y˛õ ܲˆÏÓ˚ ≤Ã!ï˛ ˆ«˛ˆÏe Ó›!ê˛Ó˚ à!ï˛ !öî≈Î˚ ܲÓ˚Ó–
(i) a = b, φ = 0 ≠ ~ˆÏ«˛ˆÏe x = a cos ωt, y = a cos ωt xÌ≈yÍñ y = x– ~!ê˛ x xˆÏ«˛Ó˚ §ˆÏD
45º ˆÜ˛yˆÏî xyöï˛ §Ó˚°ˆÏÓ˚áyÓ˚ §ü#ܲÓ˚î– Ó›!ê˛Ó˚ à!ï˛˛õÌ 2.5(i) !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
(ii) a ≠ b, φ = 0 ≠ ~ˆÏ«˛ˆÏe x = a cos ωt, y = b cos ωt xÌ≈yÍñ y = ba
. x– ~•z §ü#ܲÓ˚î!ê˛
x xˆÏ«˛Ó˚ §ˆÏD
−1 tanba
ˆÜ˛yˆÏî xyöï˛ §Ó˚°ˆÏÓ˚áy !öˆÏò≈¢ ܲˆÏÓ˚– 2.5 (ii) !ã˛ˆÏe ~•z à!ï˛˛õˆÏÌ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
(iii) a = b, φ = π ≠ ~ˆÏ«˛ˆÏe x = a cos ωt, y = a cos (ωt + π) = – a cos ωt– xï˛~Óñ y
= –x, ˆÎ!ê˛ x xˆÏ«˛Ó˚ §ˆÏD –45º ˆÜ˛yˆÏî xyöï˛ §Ó˚°ˆÏÓ˚áyÓ˚ §ü#ܲÓ˚î S!ã˛e 2.5(iii))–
(iv) a ≠ b, Q = π : x = a cos ωt, y = b cos (ωt + π) = – b cos ωt xï˛~Ó y =
bxa
−
ˆÎ!ê˛ x xˆÏ«˛Ó˚ §ˆÏD 1tan ba
−− ˆÜ˛yˆÏî xyöï˛ §Ó˚°ˆÏÓ˚áy !öˆÏò≈¢ ܲˆÏÓ˚ S!ã˛e 2.5(iv))
−1btan
a
−1btan
a
49
(v) a = b, φ =
2π
: ~ˆÏ«˛ˆÏe x = a cos ωt, y =
cos2
atπ ω⎛⎞ + ⎜⎟ ⎝⎠
= – a sin ωt x˛õöÎ˚ö ܲˆÏÓ˚
˛õyÄÎ˚y ÎyÎ˚ ≠
x2 + y2 = a2
Îy ~ܲ!ê˛ Ó,ˆÏ_Ó˚ §ü#ܲÓ˚î– ~•z ˆ«˛ˆÏe Ó›!ê˛ ~ܲ!ê˛ Ó,_yܲyÓ˚ ˛õˆÏÌ ã˛°ˆÏï˛ ÌyˆÏܲ S!ã˛e 2.5(v))– °«˛ƒ
ܲÓ˚yÓ˚ !Ó£ÏÎ˚ ~•z ˆÎñ x xˆÏ«˛Ó˚ §ˆÏD Ó›!ê˛Ó˚ xÓfliyö ˆû˛QÓ˚ ˆÎ ˆÜ˛yî Ó˚ã˛öy ܲˆÏÓ˚ñ ï˛y •°
1 tanyx
−
= –
ωt, xÌ≈yÍñ Ó›!ê˛ â!í˛¸Ó˚ ÜÑ˛yê˛yÓ˚ !òܲ ÓÓ˚yÓÓ˚ xÌ≈yÍ ò!«˛îyÓˆÏï≈˛ ã˛°ˆÏï˛ ÌyˆÏܲ– Î!ò y = cos2
a t πω⎛ ⎞−⎜ ⎟⎝ ⎠
•ï˛ñ ï˛ˆÏÓ Ó›!ê˛Ó˚ à!ï˛Ó˚ ˆÜ˛yöÄ ˛õyÌ≈ܲƒ âê˛ï˛ñ ï˛y xy˛õ!ö ˆû˛ˆÏÓ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö–
(vi) a ≠ b, φ =
2π
: ~áyˆÏö x = a cos ωt, y =
cos2
btπ ω⎛⎞ + ⎜⎟ ⎝⎠
= – b sin ωt
ˆÎˆÏ•ï%˛ñ
xa
= cos ωt,
yb
= – sin ωt,
2 2
22y x
ab+
= 1,
ˆÎ!ê˛ ~ܲ!ê˛ í˛z˛õÓ,ˆÏ_Ó˚ (ellipse) §ü#ܲÓ˚î– ˛õ)ˆÏÓ≈Ó˚ üï˛ ˆòáy ÎyˆÏÓ ˆÎ Ó›!ê˛ ~ܲ!ê˛ í˛z˛õÓ,_yܲyÓ˚ ˛õˆÏÌ
â!í˛¸Ó˚ ÜÑ˛yê˛yÓ˚ !òܲ ÓÓ˚yÓÓ˚ ã˛°ˆÏï˛ ÌyˆÏܲ S!ã˛e 2.5(vi))–
(vii) ¢ï≈˛•#ö xÓfliyÎ˚ñ xÌ≈yÍ a ≠ b, φ x!ö!ò≈‹T ≠ x = a cos ωt, y = b cos (ωt +φ)– ~•z ò%!ê˛
§ü#ܲÓ˚î ˆÌˆÏܲ ωt x˛õöÎ˚ö ܲÓ˚y Îyܲ–
y = b (cos ωt cos φ – sin ωt sin φ)
=
2
2 ..cos1sin xx bbaa
φφ −−
50
xÌÓyñ cosy xb a
− φ =
2
2 1.sin xa
−−φ
xÌÓy í˛zû˛Î˚ !òˆÏܲÓ˚ Óà≈ !öˆÏÎ˚ñ
222
222 coscos yxxyab ba
+− φφ
=
22
2 1sin xa
⎛⎞ − ⎜⎟ ⎝⎠φ
xÌÓyñ
22
222cos xyxyab ab
+−φ
= sin2 φ
~!ê˛ ~ܲ!ê˛ ˆ•°yˆÏöy í˛z˛õÓ,ˆÏ_Ó˚ (inclined ellipse) §ü#ܲÓ˚î– xy˛õ!ö §•ˆÏç•z ˛õÓ˚#«˛y ܲˆÏÓ˚ ˆòˆÏá !öˆÏï˛
˛õyˆÏÓ˚ö ˆÎñ φ ~Ó˚ üyö ¢)öƒ •ˆÏ° ~!ê˛ (ii) öÇ xÓfliyÎ˚ ~ÓÇ φ ~Ó˚ üyö
2π
•ˆÏ° ~!ê˛ (vi) öÇ xÓfliyÓ˚ §ü#ܲÓ˚ˆÏî
˛õ!Ó˚îï˛ •Î˚– Îáö π > φ > 0, ï˛áö Ó›Ó˚ à!ï˛ â!í˛¸Ó˚ ÜÑ˛yê˛yÓ˚ !òܲ ÓÓ˚yÓÓ˚ xÌ≈yÍñ ò!«˛îyÓˆÏï≈˛ •Î˚– x˛õÓ˚˛õˆÏ«˛
– π < φ < 0 •ˆÏ°ñ à!ï˛ â!í˛¸Ó˚ ÜÑ˛yê˛yÓ˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ Óy ÓyüyÓˆÏï≈˛ •Î˚–
~ÓyÓ˚ !öˆÏã˛Ó˚ xö%¢#°ö#!ê˛Ó˚ í˛z_Ó˚ !òˆÏï˛ xy˛õöyÓ˚ ˆÓyô•Î˚ û˛y°•z °yàˆÏÓ–
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#-5 : §ˆÏD ˆòÄÎ˚y !ã˛e=!°ˆÏï˛ ˛õÓ˚flõÓ˚ §üˆÏܲyˆÏî Ìyܲy §üܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚
í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ° í˛zq(ï˛ à!ï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ≤ÈÏï˛ƒÜ˛ ˆ«˛ˆÏe•z í˛zÕ‘¡∫ ˆòy°à!ï˛Ó˚ ò¢yˆÏܲyî xö%û)˛!üܲ
ˆòy°à!ï˛Ó˚ ï%˛°öyÎ˚ ¢)öƒ xÌÓy 4π
~Ó˚ ˆÜ˛yö =!îï˛ˆÏܲ ~!àˆÏÎ˚ xyˆÏåÈ– ≤Ã!ï˛!ê˛ !ã˛ˆÏeÓ˚ çöƒ ò¢yˆÏܲyî!ê˛Ó˚ üyö
!°á%ö–
~•z xLjϢ xyüÓ˚y §üyö ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛ˆÏï˛•z xyˆÏ°yã˛öy §#!üï˛ ˆÓ˚ˆÏá!åÈ– ˛õÓ˚flõÓ˚
§üˆÏܲyˆÏî ˆÎ ˆÜ˛yöÄ ò%‰•z ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ ~ܲ!eï˛ •ˆÏ°ñ Ó›Ó˚ à!ï˛˛õÌ §yôyÓ˚îû˛yˆÏÓ xï˛ƒhsˇ ç!ê˛°
•Î˚– ܲ¡õyB˛ ò%•z!ê˛Ó˚ xö%˛õyï˛ §Ó˚° •ˆÏ° SÎÌy 1:1, 1:2, 1:3 •zï˛ƒy!òV Ó›!ê˛ x-y ï˛ˆÏ° ~ܲ !ö!ò≈‹T xyÓk˛
˛õˆÏÌ ã˛°ˆÏï˛ ÌyˆÏܲ– ~•z xyÓk˛ à!ï˛˛õÌ=!° !°§yç%§‰ Sí˛zFã˛yÓ˚î !°§yç% zV !ã˛e (Lissajous figures) öyˆÏü
˛õ!Ó˚!ã˛ï˛– ~áö xyüÓ˚y ~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ §Ó˚° =!îï˛Ü˛ ôˆÏÓ˚ !öˆÏÎ˚ à!ï˛˛õÌ !öî≈Î˚ ܲÓ˚Ó–
51
2.6.2 ~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛~ܲ!ê˛ Ü˛¡õyB˛ xöƒ!ê˛Ó˚ =!îï˛Ü˛ (Frequencies in the ratio of 1:2)
SܲV ôÓ˚y Îyܲñ y !òܲ ÓÓ˚yÓÓ˚ §Ó˚° ˆòy°à!ï˛Ó˚ ܲ¡õyB˛ x !òܲ ÓÓ˚yÓÓ˚ §Ó˚° ˆòy°à!ï˛Ó˚ ܲ¡õyˆÏB˛Ó˚ !m=î–
ܲ¡õyB˛ !û˛ß¨ •ÄÎ˚yÎ˚ ò%•z ˆòy°à!ï˛Ó˚ ò¢yhsˇÓ˚ §Ó≈òy•z ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏï˛ ÌyܲˆÏÓ– t = 0 §üˆÏÎ˚ ≤ÃyÌ!üܲ ò¢yhsˇÓ˚
φ ôˆÏÓ˚ !öˆÏÎ˚ xyüÓ˚y ˆòy°à!ï˛=!°Ó˚ §ü#ܲÓ˚î !°áˆÏï˛ ˛õy!Ó˚ ≠
x = a cos ωt
y = b cos (2ωt +φ)....... 2.20
~áö xyüÓ˚y φ ~Ó˚ ܲˆÏÎ˚ܲ!ê˛ !ö!ò≈‹T üyˆÏöÓ˚ çöƒ à!ï˛˛õˆÏÌÓ˚ §ü#ܲÓ˚î !öî≈Î˚ ܲÓ˚Ó–
(i) φ = 0 : x = a cos ωt ~ÓÇ y = b cos 2ωt ˆÌˆÏܲ ˛õyÄÎ˚y
ÎyˆÏÓ
y = b(2 cos2ωt – 1)
Óyñ y =
2
2 21 x ba
⎛⎞ − ⎜⎟ ⎝⎠
~!ê˛ ~ܲ!ê˛ x!ôÓ,ˆÏ_Ó˚ (Parabola) §ü#ܲÓ˚î– ~Ó˚ ˆ°á!ã˛e!ê˛
2.5(i) !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
(ii) φ =
2π: ~ˆÏ«˛ˆÏe x = a cos ωt ~ÓÇ y =
cos22
btπ ω ⎛⎞ + ⎜⎟ ⎝⎠
= – b sin 2ωt–
§%ï˛Ó˚yÇñ y = –2b sin ωt cos ωt =
2
2 21 xx baa
−−
xÌÓyñ
2
2yb
=
22
2241 xxaa
⎛⎞ − ⎜⎟ ⎝⎠
~Ó˚ ˆ°á!ã˛e!ê˛ 2.6(ii) !ã˛ˆÏe ˆòáyˆÏöy •°– ~!ê˛Ó˚ xyÜ,˛!ï˛ Ú4Û
§Çáƒy!ê˛Ó˚ üï˛–
(iii) φ = π : ~•z ˆ«˛ˆÏe x = a cos ωt
y = b cos (2ωt +π) = – b cos 2ωt = – b(2 cos2 ωt
–1)
Óyñ y =
2
2 12x ba
⎛⎞ − ⎜⎟ ⎝⎠
~!ê˛Ä ~ܲ!ê˛ x!ôÓ,ˆÏ_Ó˚ §ü#ܲÓ˚î ~ÓÇ 2.6(iii) !ã˛ˆÏe ~Ó˚
ˆ°á!ã˛e!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
!ã˛e 2.6(i)
52
≤Ã!ï˛!ê˛ ˆ«˛ˆÏe•z xy˛õ!ö °«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ ˆòy°à!ï˛ˆÏï˛ §MÈ˛yÓ˚üyö !Ó®%!ê˛ Îáö x !òˆÏܲ ~ܲ!ê˛ §¡õ)î≈
ˆòy°ö ˆ¢£Ï ܲˆÏÓ˚ñ ï˛ï˛«˛ˆÏî y !òˆÏܲ ò%•z!ê˛ ˆòy°ö §üyÆ •Î˚–
φ ˆÜ˛yˆÏîÓ˚ xöƒyöƒ üyˆÏöÓ˚ çöƒÄ !mü%á# ˆòy°à!ï˛Ó˚ à!ï˛˛õˆÏÌÓ˚ §ü#ܲÓ˚î !öî≈Î˚ §Ω˛Ó– ï˛ˆÏÓ ˆ§=!° xï˛ƒhsˇ
ç!ê˛° •ÄÎ˚yÎ˚ xyüÓ˚y φ ~Ó˚ x!ö!ò≈‹T üyˆÏöÓ˚ çöƒ à!ï˛˛õÌ !öî≈Î˚ ܲÓ˚°yü öy–
SáV ~ÓyÓ˚ ôÓ˚&öñ y !òˆÏܲÓ˚ ˆòy°à!ï˛Ó˚ ܲ¡õyB˛ x !òˆÏܲÓ˚ !ï˛ö =î– ~•z ˆ«˛ˆÏe ≤ÃyÌ!üܲ ò¢yhsˇˆÏÓ˚Ó˚ üyö
¢)öƒ ôˆÏÓ˚ !öˆÏÎ˚ xyüÓ˚y !mü%á# ˆòy°à!ï˛Ó˚ ˛õÌ!ê˛ ÓyÓ˚ ܲÓ˚Ó– x Ä y §ü#ܲÓ˚î=!° ˆ°áy Îyܲ ≠
x = a cos ωt
y = b cos3 ωt....... 2.21
à!îˆÏï˛Ó˚ §)e xö%ÎyÎ˚# y = b (4 cos 3ωt –3 cos ωt)
=
2
2 .43 xx baa
⎛⎞ − ⎜⎟ ⎝⎠
~!ê˛ ~ܲ!ê˛ !eâyï˛ §ü#ܲÓ˚î– ~Ó˚ ˆ°á!ã˛e!ê˛ ˛õyˆÏ¢ x!B˛ï˛
•°– °«˛ƒ ܲˆÏÓ˚ ˆòá%öñ §MÈ˛Ó˚üyö !Ó®%!ê˛ t = 0 §üˆÏÎ˚ A !Ó®%
(x = a, y = b) ˆÌˆÏܲ Îyey ÷Ó˚& ܲˆÏÓ˚
2πω
§üˆÏÎ˚Ó˚ üˆÏôƒ
ABCDEDCBA ˛õÌ x!ï˛e´ü ܲˆÏÓ˚– ~•z §üˆÏÎ˚ x !òˆÏܲ ~ܲ!ê˛ Ä y !òˆÏܲ !ï˛ö!ê˛ §¡õ%)î≈ ˆòy°ö (ABD,
DED Ä DBA) §üyÆ •Î˚–
SàV §yôyÓ˚îû˛yˆÏÓ Ó°y ÎyÎ˚ ˆÎñ x Ä y !òˆÏܲÓ˚ ܲ¡õyB˛ ò%!ê˛Ó˚ xö%˛õyï˛ Î!ò ò%•z ˛õ)î≈§ÇáƒyÓ˚ xö%˛õyˆÏï˛Ó˚
§üyö •Î˚ñ ï˛ˆÏÓ §MÈ˛Ó˚üyö !Ó®%!ê˛ ~ܲ!ê˛ Ók˛˛õˆÏÌ ã˛°yã˛° ܲˆÏÓ˚– xÓ¢ƒ ˛õ)î≈§Çáƒy ò%!ê˛ Îï˛ Óí˛¸ •Î˚ñ Ók˛˛õÌ!ê˛Ä
ï˛ï˛•z ç!ê˛° •Î˚–
2.6.3 !°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö!°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö!°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö!°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö!°§yç%§ !ã˛ˆÏeÓ˚ ≤Ãò¢≈ö
~ܲ!ê˛ Ü˛ƒyˆÏÌyí˛ÈÙȈÓ˚ x!§ˆÏ°yˆÏflÒy˛õ (cathode-
ray oscilloscope) Ä ò%•z!ê˛ x!í˛ÄÈÙÈx!§ˆÏ°ê˛ˆÏÓ˚Ó˚
(audio-oscillator) §y•yˆÏ΃ x!ï˛ §%®Ó˚û˛yˆÏÓ
!°§yç%§ !ã˛e ≤Ãò¢≈ö ܲÓ˚y ÎyÎ˚– x!í˛Ä x!§ˆÏ°ê˛Ó˚=!°
ˆÎ ˆÜ˛yöÄ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚ ˜Óò%ƒ!ï˛Ü˛ §ˆÏB˛ï˛
(signal) í˛zͲõyòö ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– ~Ó˚*˛õ ~ܲ!ê˛
x!§ˆÏ°ê˛yˆÏÓ˚Ó˚ §ˆÏB˛ï˛ˆÏܲ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ x xˆÏ«˛Ó˚
53
•zö‰˛õ%ê˛ ~ÓÇ x˛õÓ˚ ~ܲ!ê˛ x!§ˆÏ°ê˛yˆÏÓ˚Ó˚ §ˆÏB˛ï˛ˆÏܲ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ y xˆÏ«˛Ó˚ •zö‰˛õ%ê˛ !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y
•Î˚– x •zö‰˛õ%ê˛ §ˆÏB˛ˆÏï˛Ó˚ ú˛ˆÏ° x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ ˛õò≈yÎ˚ •zˆÏ°Ü˛ê˛Δö !Óü (beam) myÓ˚y §,‹T í˛zIμ° !Ó®%!ê˛
xö%û)˛!üܲ !òˆÏܲ §ˆÏB˛ï˛ÈÙÈ!Óû˛Ó xö%ÎyÎ˚# §Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°yã˛° ܲˆÏÓ˚– xyÓyÓ˚ñ y •zö‰˛õ%ê˛ §ˆÏB˛ï˛!ê˛ ˙ !Ó®%ˆÏܲ
í˛zÕ‘¡∫!òˆÏܲ §ˆÏB˛ï˛ÈÙÈ!Óû˛Ó xö%ÎyÎ˚# Äë˛yöyüy ܲÓ˚yÎ˚– ~áö Î!ò ò%•z §ˆÏB˛ˆÏï˛Ó˚ ܲ¡õyB˛ §Ó˚° xö%˛õyˆÏï˛ ÌyˆÏܲñ
ï˛ˆÏÓ xö%û)˛!üܲ Ä í˛zÕ‘¡∫ ~•z ò%•z à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° í˛zIμ° !Ó®%!ê˛ x!ï˛ o&ï˛ !°§yç%§ !ã˛e xö%§Ó˚î
ܲˆÏÓ˚ ôy!Óï˛ •ˆÏÓ ~ÓÇ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ ˛õò≈yÎ˚ !°§yç%§ !ã˛e!ê˛ ˆòáy ÎyˆÏÓ–
í˛z˛õˆÏÓ˚Ó˚ ˛õÓ˚#«˛y!ê˛ xy˛õ!ö ˆÜ˛yö •zˆÏ°Ü˛ê˛Δ!ö܉˛ˆÏ§Ó˚ ˛õÓ˚#«˛yàyˆÏÓ˚ ܲyç ܲÓ˚yÓ˚ §%ˆÏÎyà ˆ˛õˆÏ° ܲˆÏÓ˚ ˆòáˆÏï˛
˛õyÓ˚ˆÏÓö– !ܲv ~ÓyÓ˚ xy˛õ!ö !öˆÏç•z ~ܲ!ê˛ !ӈϢ£Ï ˆ«˛ˆÏe !°§yç%§ !ã˛e xB˛ˆÏöÓ˚ ˆã˛‹Ty ܲÓ˚&ö–
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#-6 : ôˆÏÓ˚ !ööñ x = a sin 2ωt ~ÓÇ x = a sin ωt ~•z ò%•z §Ó˚° ˆòy°à!ï˛ í˛z˛õ!Ó˚˛õy!ï˛ï˛
•ˆÏÎ˚ˆÏåÈ– °!∏˛ ˆòy°à!ï˛Ó˚ !°§yç%§ !ã˛e!ê˛Ó˚ §ü#ܲÓ˚î !öî≈Î˚ ܲÓ˚&ö ~ÓÇ !ã˛e!ê˛ xÑyÜ%˛ö–
2.7 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~ܲ•z §Ó˚°ˆÏÓ˚áyÎ˚ ò%•z Óy ï˛ˆÏï˛y!ôܲ §ÇáƒÜ˛ §Ó˚° ˆòy°à!ï˛ í˛z˛õ!Ó˚˛õy!ï˛ï˛ •ˆÏÎ˚ öï%˛ö ~ܲ ôÓ˚ˆÏöÓ˚ ˆòy°à!ï˛
í˛zͲõߨ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– xyÓyÓ˚ ˛õÓ˚flõÓ˚ §üˆÏܲyî# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏö ~ܲ!ê˛ !müy!eܲ à!ï˛Ó˚
§,!‹T •ˆÏï˛ ˛õyˆÏÓ˚– ~•z ~ܲˆÏܲ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ö#!ï˛ ~ÓÇ ˆÎ ò%•z ôÓ˚ˆÏöÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚
í˛zˆÏÕ‘á ܲÓ˚y •°ñ ˆ§=!° xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ– §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ !ӈϟ’£ÏˆÏî ç!ê˛° Ó˚y!¢Ó˚
ÓƒÓ•yˆÏÓ˚Ó˚ §ˆÏDÄ xy˛õ!ö ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö–
xyüÓ˚y ˆòˆÏá!åÈ ˆÎñ
~ܲ•z §Ó˚°ˆÏÓ˚áyÎ˚ §üyö ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ §Ç!ü◊ˆÏî ˙ ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ öï%˛ö §Ó˚°
ˆòy°à!ï˛Ó˚ í˛zqÓ •Î˚–
~ܲ•z §Ó˚°ˆÏÓ˚áyÎ˚ x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏö ~üö ~ܲ!ê˛ ˛õÎ≈yÓ,_ à!ï˛Ó˚
§,!‹T •Î˚ñ ÎyˆÏܲ §Ó˚° ˆòy°à!ï˛ Ó°y ÎyÎ˚ öy ~ÓÇ ÎyÓ˚ ˛õÎ≈yÎ˚ܲy° ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲy°=!°Ó˚
°.§y.=– ï˛ˆÏÓ Ü˛¡õyÏB˛mˆÏÎ˚Ó˚ ÓƒÓôyö Îáö ï˛yˆÏòÓ˚ àí˛¸ üyˆÏöÓ˚ ï%˛°öyÎ˚ x!ï˛ «˛%oñ ï˛áö àí˛¸ ܲ¡õyˆÏB˛Ó˚
~ܲ ˆòy°à!ï˛ í˛zͲõߨ •Î˚ñ ÎyÓ˚ !ÓhflÏyÓ˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Äë˛yöyüy ܲˆÏÓ˚–
˛õÓ˚flõÓ˚ §üˆÏܲyî# ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ ܲ¡õyB˛ Î!ò §Ó˚° xö%˛õyˆÏï˛ ÌyˆÏܲñ ï˛ˆÏÓ ˆ§=!°Ó˚
í˛z˛õ!Ó˚˛õyï˛ˆÏö !°§yç%§ !ã˛e öyˆÏü xyÓk˛ Óe´ˆÏÓ˚áy ÓÓ˚yÓÓ˚ à!ï˛Ó˚ §,!‹T •Î˚–
2.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. Î!ò r1(t) = r10 sin ωt ~ÓÇ r2(t) = r20 sin ωt ˆÜ˛yöÄ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚îˆÏܲ !§k˛ ܲˆÏÓ˚ñ
ï˛ˆÏÓ r1(t) + r
2(t) ˙ §ü#ܲÓ˚î!ê˛ˆÏܲ !§k˛ ܲÓ˚ˆÏÓ– ~Ó˚ ܲyÓ˚î ܲ#⁄ §Ó˚° ˆòy°à!ï˛ ò%•z!ê˛Ó˚ ܲ¡õyB˛ §üyö
öy •ˆÏ° ܲ# âˆÏê˛⁄
54
2. ò%•z!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î x1 = A cos (ωt – 30º) Ä x2 = A cos (ωt + 30º)– ˆû˛QÓ˚
˛õk˛!ï˛ˆÏï˛ ~ÓÇ ç!ê˛° Ó˚y!¢Ó˚ ˛õk˛!ï˛ˆÏï˛ ~•z ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ °!∏˛ !öî≈Î˚ ܲÓ˚&ö–
3. ˛õyˆÏ¢ ˆÎ !ã˛e!ê˛ ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ ï˛yˆÏï˛ ~ܲ!ê˛ !˛õ[˛ B
P1 ˛õ%!° ˆÌˆÏܲ ˆé˛y°yˆÏöy xyˆÏåÈ– X Ä Y ≤Ãyhsˇ!Ó!¢‹T ~ܲ!ê˛
§%ï˛y P2 Ä P
3 ˛õ%!°Ó˚ í˛z˛õÓ˚ !òˆÏÎ˚ !àˆÏÎ˚ˆÏåÈ ~ÓÇ ˆ§!ê˛ P
1
˛õ%!°Ó˚ ï˛°y !òˆÏÎ˚ !àˆÏÎ˚ ˆ§!ê˛ˆÏܲ xy°!¡∫ï˛ ˆÓ˚ˆÏáˆÏåÈ– P2 Ä
P3 ˛õ%!° ˆÌˆÏܲ §%ï˛yÓ˚ X Ä Y ≤ÃyhsˇmˆÏÎ˚Ó˚ ò)Ó˚hsˇ ÎÌye´ˆÏü x
Ä y– ~áö Î!ò X Ä Y ≤Ãyhsˇ ò%•z!òˆÏܲ ~üöû˛yˆÏÓ xyˆÏ®y!°ï˛
ܲÓ˚y •Î˚ ÎyˆÏï˛ x Ä y ÎÌye´ Ïü 5Ä6 ˆ§ˆÏܲu˛ ˛õÎ≈yÎ˚ܲyˆÏ°
§üyö !ÓhflÏyˆÏÓ˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Äë˛yöyüy ܲˆÏÓ˚ñ ï˛ˆÏÓ B
!˛õ[˛!ê˛Ó˚ à!ï˛ ˆÜ˛üö •ˆÏÓ xyˆÏ°yã˛öy ܲÓ˚&ö–
4. ˆÜ˛yöÄ Ó› x Ä y !òܲ ÓÓ˚yÓÓ˚ ~ܲ•z !ÓhflÏyÓ˚ Ä Ü˛¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ ã˛°yã˛° ܲÓ˚ˆÏåÈ– x !òˆÏܲÓ˚
ˆòy°à!ï˛Ó˚ ò¢yˆÏܲyî y !òˆÏܲÓ˚ ï%˛°öyÎ˚ 60º ~!àˆÏÎ˚ xyˆÏåÈ– Ó›!ê˛Ó˚ à!ï˛˛õÌ xB˛ö ܲÓ˚&ö–
5. §)ˆÏÎ≈Ó˚ ü•yܲˆÏ£Ï≈Ó˚ ≤Ãû˛yˆÏÓ ˛õ,!ÌÓ#Ó˚ ܲ«˛˛õÌ ~ܲ!ê˛ í˛z˛õÓ,_– ˛õ,!ÌÓ#Ó˚ à!ï˛ !ܲ ò%•z!ê˛ §üܲ¡õyˆÏB˛Ó˚ §Ó˚°
ˆòy°ˆÏöÓ˚ !ü!°ï˛ ú˛° ӈϰ üˆÏö ܲÓ˚y ÎyÎ˚⁄
6. ˛õÓ˚flõÓ˚ °¡∫y!û˛ü%á# ò%!ê˛ x§üyö ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ í˛z˛õ!Ó˚˛õy!ï˛ï˛ •ˆÏÎ˚ˆÏåÈñ ÎyˆÏòÓ˚ üôƒ!Ó®%
~ܲ•z– ~ˆÏ«˛ˆÏe ӛܲîyÓ˚ í˛z˛õÓ˚ °!∏˛ Ó° !ܲ §Ó˚ˆÏîÓ˚ !Ó˛õÓ˚#ï˛ x!û˛ü%á# ~ÓÇ §üyö%˛õyï˛# •ˆÏÓ⁄
2.9 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#
1. ôˆÏÓ˚ !öö θ1(t) ~ÓÇ θ
2(t) í˛zû˛ˆÏÎ˚•z ≤Ãò_ §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö– §%ï˛Ó˚yÇñ
2121 sin
dgl dt
+ θθ
= 0 ~ÓÇ
2222 sin
dgl dt
+ θθ
= 0
ò%!ê˛ §ü#ܲÓ˚ˆÏîÓ˚ ˆÎyàú˛°
212
212
()(sinsin)
dgl dt
+++ θθθθ
= 0.
!ܲv ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛ ≤ÈÏÎyçƒ •ˆÏ° ≤ÈÏÎ˚yçö#Î˚ ¢ï≈˛
212
212
()sin()
dgl dt
+++ θθθθ
= 0.
55
ˆÎˆÏ•ï%˛ sin θ1 + sin θ
2 = sin (θ
1 + θ
2) §ü#ܲÓ˚î!ê˛ §yôyÓ˚îû˛yˆÏÓ §ï˛ƒ öÎ˚ñ xï˛~Ó ~ˆÏ«˛ˆÏe
í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ö#!ï˛ ≤ÈÏÎ˚yà ܲÓ˚y ÎyÎ˚ öy–
2. ≤Ã̈Ïü•z ôÓ˚y Îyܲñ 50t + 30º = 50t', ÎyÓ˚ xÌ≈ ~•z ˆÎñ xyüÓ˚y t = – 30º/50 ü%•)ï≈˛ ˆÌˆÏܲ t' §üÎ˚
àîöy ܲÓ˚!åÈ– t ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ t' ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° §Ó˚° ˆòy°à!ï˛ ò%•z!ê˛Ó˚ §ü#ܲÓ˚î òÑyí˛¸yÎ˚
x1 = 5 cos 50t'
Ä x2 = 12 cos (50t' + 90º)
~áö 2.8 Ä 2.9 §)e ÓƒÓ•yÓ˚ ܲˆÏÓ˚ñ ˆÎˆÏ•ï%˛ θ1 = 0, θ2 = 90º
!ÓhflÏyÓ˚ A =
()1
222 512 +
= 13
~ÓÇ ò¢yˆÏܲyî θ = 1 12tan5
− ≈ 67º
3. °!∏˛ ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î
x = x1 + x
2 = 20 sin (52πt) + 24 sin (48πt)
= 22[sin (52πt) + sin (48πt)] – 2 [sin (52πt) –sin (48πt)]
= 52 48 52 48 52 48 52 4844sin cos 4sin cos2 2 2 2
t t t tπ π π π+ − − +−
= 44 sin 50πt cos 2πt – 4 sin 2πt cos 50πt
ôˆÏÓ˚ !ööñ 44 cos 2πt = A cos φ
~ÓÇñ 4 sin 2πt = A sin φ
∴ x = A cos φ sin 50πt – A sin φ cos 50πt
= A sin (50πt – φ)
~áyˆÏöñ A2 = (A cos φ)2 + (A sin φ)2
= (20 + 24)2 cos2 2πt + (24 – 20)2 sin2 2πt
= 202 + 242 + 2. 20. 24 cos 4πt
=976 + 960 cos 4πt
~ÓÇ tan φ = sincos
AA
φφ =
1tan211
t π
.
56
4. ôÓ˚y Îyܲ ˆòy°à!ï˛=!°Ó˚ §ü#ܲÓ˚î
x1 = a cos (ωt – φ )
x2 = a cos ωt
~ÓÇñ x3 = a cos (ωt + φ )
§%ï˛Ó˚yÇñ x = x1+ x2 + x3 = Óy (Z) ˆÎáyˆÏö
Z =
()() jtjtjt aeaeae ωφωωφ −+ ++
Z =
1 jtjqjq aeee ω− ⎡⎤ ++ ⎣⎦
=
[] 12cos jt aeωφ +
§%ï˛Ó˚yÇñ °∏˛ ˆòy°à!ï˛ (Z Ó˚ ÓyhflÏÓ xÇ¢V
x = a cos ωt (1+ 2 cos φ)
~ÓÇ °!∏˛ ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ A
A = a (1 + 2 cos φ)
5. ò¢yˆÏܲyˆÏîÓ˚ üyö SܲV π, SáV 74π , SàV 0, SâV
2π
, SäV
34π
ñ Sã˛V
54π
, SåÈV
32π
, SçV
4π
6. ~ˆÏ«˛ˆÏe sin ωt =
ya
~ÓÇ cos ωt =
2
2 1ya
−
x = a sin 2ωt = 2a sin ωt cos ωt = 2
22 1 yya
−
!°§yç%§ !ã˛e xÑyܲyÓ˚ çöƒ §yÓ˚î# ≠
y = 0 a±
2a ±
2a ±
x = 0 0 32
a± a±
57
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ≤Èϟ¿Ó˚ ≤ÃÌü xÇ¢!ê˛ 2.2 xLjϢ xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ– !mï˛#Î˚ xÇ¢!ê˛Ó˚ !ӣψÏÎ˚ xy˛õ!ö 2.4 xÇ Ï¢
˛õˆÏí˛¸ˆÏåÈö– xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈö ˆÎñ x§ü ܲ¡õyˆÏB˛Ó˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏö ˆÎ
˛õÎ≈yÎ˚à!ï˛ í˛zͲõߨ •Î˚ñ ï˛y §Ó˚° ˆòy°à!ï˛ öÎ˚– 2.16, 2.17, 2.18 §)e í˛zk,˛ï˛ ܲˆÏÓ˚ ~•z xLjϢÓ˚ í˛z_Ó˚ !òˆÏï˛
˛õyÓ˚ˆÏÓö–
2. ˆû˛QÓ˚ ˛õk˛!ï˛ˆÏï˛ ≠ x1 Ä x
2 ˆòy°à!ï˛ ò%!ê˛ˆÏܲ xyüÓ˚y A ˜òˆÏâ≈ƒÓ˚ ò%•z!ê˛ ˆû˛QÓ˚ myÓ˚y §)!ã˛ï˛ ܲÓ˚ˆÏï˛ ˛õy!Ó˚–
x xˆÏ«˛Ó˚ §ˆÏD ~=!° ÎÌye´ˆÏü ωt – 30º Ä ωt + 30º ˆÜ˛yˆÏî xyöï˛– °!∏˛ ˆòy°à!ï˛ R ~Ó˚ ˜òâ≈ƒ
R = ()1
222 2...cos60º AAAA ++ ˆÜ˛ööy x1 Ä x
2 ~Ó˚ xhsˇÓï≈˛# ˆÜ˛yî 60º
= 3A
ç!ê˛° Ó˚y!¢Ó˚ ˛õk˛!ï˛ˆÏï˛ ≠ x1 + x
2 = Óy (Z)
ˆÎáyˆÏöñ Z =
()() 30º30º jtjt AeAe ωω −+ +
= Aejωt (ej.30º + e–j.30º)
= 2aejωt cos 30º
∴ x1 + x
2 = 2A cos 30º cos ωt =
3 2..cos2
At ω
=
3cos At ω
ÎyÓ˚ !ÓhflÏyÓ˚
3A
58
3. ˆüyê˛ §%ï˛yÓ˚ ˜òâ≈ƒ Î!ò 2L •Î˚ ~ÓÇ P1 ˛õ%!° Î!ò P2 xÌÓy P3 ˆÌˆÏܲ z ò)Ó˚ˆÏc !öˆÏã˛ ÌyˆÏܲñ ï˛ˆÏÓ
2L = x + y + 2z–
Î!ò x Ä y ~Ó˚ !ÓhflÏyÓ˚ a •Î˚ ï˛ˆÏÓ í˛zû˛ˆÏÎ˚Ó˚ ˆ«˛ˆÏe ≤ÃyÌ!üܲ ò¢y ¢)öƒ ôˆÏÓ˚ ˆ°áy ÎyÎ˚
x =
2 cos.5
at π
Ä y =
2 cos.6
at π
∴ x + y =
22 cos.cos.56
att ⎛⎞ + ⎝⎠ππ
= 1 2 2 1 2 22 cos .cos2 5 6 2 5 6
a t t⎛ ⎞ ⎛ ⎞+ −⎝ ⎠ ⎝ ⎠π π π π
=
111 2cos.cos3030
att ⎛⎞⎛⎞⎝⎠⎝⎠ ππ
∴ z = 2
xy L+ − = 1 11cos .cos
30 30L a t t⎛ ⎞ ⎛ ⎞− ⎝ ⎠ ⎝ ⎠π π
B !˛õ[˛!ê˛Ó˚ à!ï˛ P1 ˛õ%!°Ó˚ à!ï˛Ó˚ xö%Ó˚)˛õ ~ÓÇ z Ó˚y!¢!ê˛ ~•z à!ï˛ !öˆÏò≈¢ ܲÓ˚ˆÏåÈ– ~áyˆÏö L ~ܲ!ê˛ ô &Óܲ–
1 cos30
at ⎛⎞⎝⎠ π
Ó˚y!¢!ê˛Ó˚ §ˆÏD
2 cost aTπ
Ó˚y!¢Ó˚ ï%˛°öy ܲÓ˚ˆÏ° ˆÓyé˛y ÎyÎ˚ñ ~!ê˛ 60 ˆ§ˆÏܲu˛ ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚
§Ó˚° ˆòy°à!ï˛ˆÏï˛ ˛õ!Ó˚Óï≈˛ö¢#°– x˛õÓ˚˛õˆÏ«˛ñ
11 cos30
t ⎛⎞⎝⎠ π
Ó˚y!¢
6011
ˆ§ˆÏܲu˛ ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ §Ó˚° ˆòy°à!ï˛
ˆÓyé˛yÎ˚– §%ï˛Ó˚yÇñ z Ó˚y!¢!ê˛
6011
ˆ§ˆÏܲu˛ ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ ˆòy°à!ï˛ˆÏï˛ Äë˛yöyüy ܲˆÏÓ˚ ~ÓÇ ~•z ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚
60 ˆ§ˆÏܲu˛ ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚–
4. ~áyˆÏö Ó›!ê˛Ó˚ à!ï˛ §ü#ܲÓ˚î ˆ°áy ÎyÎ˚ ≠
x = a cos (ωt + 60º)
y = a cos (ωt)
ωt Ó˚y!¢Ó˚ !Ó!û˛ß¨ üyˆÏöÓ˚ çöƒ ~áö xy˛õ!ö x Ä y ~Ó˚ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö ≠
59
wt = 0º 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360º
x =
12
a
0
12
a −32
a −
– a 32
a− 12
a− 0 12
a
32
a
a
32
a12
a
y = a
32
a12
a
0
12
a −
32
− – a 32
a− 12
a− 0 12
a
32
a
a
!ã˛ˆÏe ωt Ó˚y!¢Ó˚ !Ó!û˛ß¨ üyˆÏöÓ˚ çöƒ Ó›!ê˛Ó˚
xÓfliyö Ä à!ï˛˛õÌ ˆòáyˆÏöy •°– !ã˛e!ê˛ ˆÌˆÏܲ
xy˛õ!ö §•ˆÏç•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ ~ˆÏ«˛ˆÏe Ó›!ê˛
ÓyüyÓˆÏï≈˛ xÌ≈yÍ â!í˛¸Ó˚ ܲÑyê˛yÓ˚ !Ó˛õÓ˚#ˆÏï˛ ã˛°ˆÏï˛
ÌyˆÏܲ–
5. öyñ ܲyÓ˚î ~áyˆÏö ü•yܲ£Ï≈ Ó° §Ó !Ó®%ˆÏï˛ §)Î≈ x!û˛ü%á# •ˆÏ°Ä ˙ ӈϰÓ˚ üyö ò)Ó˚ˆÏcÓ˚ ÓˆÏà≈Ó˚
ÓƒhflÏyö%˛õyï˛#ñ ò)Ó˚ˆÏcÓ˚ §üyö%˛õyï˛# öÎ˚–
6. öyñ Ó° ò%!ê˛ Î!ò
21x ω−
Ä 22− yω •Î˚ ï˛ˆÏÓ ï˛yˆÏòÓ˚ °!∏˛Ó˚ üyö 4 2 4 2
1 2x yω ω+ ñ Îy 2 2x y+
~Ó˚ §üyö%˛õyï˛# öÎ˚– ï˛yåÈyí˛¸y ~•z °!∏˛ x xˆÏ«˛Ó˚ §ˆÏD
212
21
tanyx
ωω
−
ˆÜ˛yî Ó˚ã˛öy ܲÓ˚ˆÏÓñ ˆÎáyˆÏö °!∏˛ §yôyÓ˚î
üôƒ!Ó®% x!û˛ü%ˆÏá ܲyç ܲÓ˚ˆÏ° ~•z ˆÜ˛yî 1tan y
x−
•ï˛–
60
~ܲܲ ~ܲܲ ~ܲܲ ~ܲܲ ~ܲܲ 3 xÓü!®ï˛ ˆòy°à!ï˛ xÓü!®ï˛ ˆòy°à!ï˛ xÓü!®ï˛ ˆòy°à!ï˛ xÓü!®ï˛ ˆòy°à!ï˛ xÓü!®ï˛ ˆòy°à!ï˛ (Damped Oscillations)
àë˛öàë˛öàë˛öàë˛öàë˛ö
3.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
3.2 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y
3.3 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
3.3.1 x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ
3.3.2 e´yhsˇ#Î˚ xÓü®ö
3.3.3 °â% xÓü®ö
3.4 °â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛¢!_´
3.5 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹Tƒ
3.5.1 °à#Î˚ •…y§
3.5.2 Ÿ’Ìö ܲy°
3.5.3 !ܲí˛z (Q)ÈÙÈ=îyB˛
3.6 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î
3.7 §yÓ˚yÇ¢
3.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
3.9 í˛z_Ó˚üy°y
3.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
~•z ˛õÎ≈yˆÏÎ˚Ó˚ ≤ÃÌü ~ܲˆÏܲ xy˛õöyÓ˚y §Ó˚° ˆòy°à!ï˛Ó˚ ˛õ!Ó˚ã˛Î˚ ˆ˛õˆÏÎ˚ˆÏåÈö– !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ
~ܲ!ê˛ §%§üO§ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´ §òy§Ó≈òy x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ ~ÓÇ §üÎ˚ÈÙȧÓ˚î ˆ°á!ã˛e!ê˛ §y•zˆÏöÓ˚ SÓy
ˆÜ˛y§y•zˆÏöÓ˚V ˆ°á!ã˛ˆÏeÓ˚ xö%Ó˚*˛õ •Î˚– ï˛y•ˆÏ° !ܲ xyüÓ˚y ôˆÏÓ˚ ˆöÓ ˆÎñ §Ó˚° ˆòy°à!ï˛Ó˚ ~•z ˜Ó!¢‹Tƒ
!ã˛Ó˚ܲy° x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏÓ⁄ ï˛yÓ˚ ¢!_´Ó˚ ˆÜ˛yöÄ «˛Î˚ •ˆÏÓ öy ~ÓÇ ï˛y ~ܲÓyÓ˚ ˆòy°à!ï˛ ÷Ó˚& ܲÓ˚ˆÏ°
xyÓ•üyö ܲy° ôˆÏÓ˚ ã˛°ˆÏï˛•z ÌyܲˆÏÓ⁄ xyüyˆÏòÓ˚ x!û˛K˛ï˛y !ܲv xöƒÓ˚ܲü ӈϰñ ï˛y•z öy⁄
xy§ˆÏ° ˆÎ §Ó˚° ˆòy°à!ï˛Ó˚ §ˆÏD xyüyˆÏòÓ˚ ≤ÃÌü ˛õ!Ó˚!ã˛!ï˛ âˆÏê˛!åÈ°ñ ˆ§áyˆÏö §ÓÓ˚ܲˆÏüÓ˚ x˛õã˛Î˚# Ó°
xö%˛õ!fliï˛ ÓˆÏ° ôÓ˚y •ˆÏÎ˚!åÈ°– ~•z Ó˚ܲˆÏüÓ˚ ˆòy°à!ï˛ˆÏܲ Ó°y •Î˚ ü%_´ (free) xÌÓy xöÓü!®ï˛
(undamped)– §!ï˛ƒ•z ˆï˛yñ ˆÜ˛yöÄÓ˚ܲü x˛õã˛Î˚# Óy ÓyôyòyöܲyÓ˚# Ó° Î!ò í˛z˛õ!fliï˛ öy ÌyˆÏܲñ ï˛ˆÏÓ §Ó˚°
ˆò°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ ܲîyÓ˚ ¢!_´ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏÓ ~ÓÇ !ã˛Ó˚ܲy° ï˛yÓ˚ ~ܲ•z à!ï˛ °«˛ƒ ܲÓ˚y ÎyˆÏÓ–
!ܲv ÓyhflÏÓ ˆ«˛ˆÏe xyüÓ˚y ˆò!á ˆÎñ §Ó˚° ˆòy°ˆÏܲÓ˚ !ÓhflÏyÓ˚ §üˆÏÎ˚Ó˚ §ˆÏD e´ü¢ ܲüˆÏï˛ ÌyˆÏܲ ~ÓÇ ï˛y ô#ˆÏÓ˚
ô#ˆÏÓ˚ ˆÌˆÏü ÎyÎ˚– !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy áy!öܲ«˛î ܲ!¡õï˛ •ÄÎ˚yÓ˚ ˛õÓ˚ !fliÓ˚ •ˆÏÎ˚ ÎyÎ˚ñ ÓƒyÓï≈˛ ˆòy°Ü˛Ä (torsionalpendulum) ~ܲ•zÓ˚ܲü xyã˛Ó˚î ܲˆÏÓ˚– §Ó ˆ«˛ˆÏe•z ˆòáy ÎyˆÏÓ ˆÎñ xyüÓ˚y !ӈϢ£Ï ˆÜ˛yöÄ Ó°≤ÈÏÎ˚yà ܲˆÏÓ˚
61
˙ §ühflÏ Ü˛¡õöˆÏܲ ܲüyˆÏöyÓ˚ Óy Ó¶˛ ܲÓ˚yÓ˚ ˆã˛‹Ty ܲ!Ó˚!öñ ï˛Ó% ≤Ã̈Ïü !ÓhflÏyÓ˚ ܲüyˆÏöyÓ˚ §ˆÏD §ˆÏD ï˛yˆÏòÓ˚
¢!_´Ó˚ •…y§ •ˆÏÎ˚ˆÏåÈ (1.9 §ü#ܲÓ˚î!ê˛ ~ܲÓyÓ˚ ˆòˆÏá !ööV ~ÓÇ ˆ¢£Ï ˛õÎ≈hsˇ ˆ§•z ¢!_´ §¡õ)î≈ !öɈϢ!£Ïï˛ •ˆÏÎ˚
ˆàˆÏåÈ– ÓyhflÏÓ çàˆÏï˛ ~üöê˛y•z •ˆÏÎ˚ ÌyˆÏܲ–
xy§ˆÏ° ÓyhflÏÓ çàˆÏï˛ xyüÓ˚y ≤ÃÜ,˛ï˛ ü%_´ ܲ¡õö ˛õy•z öy– ò%!ê˛ Ü˛!ë˛ö ï˛ˆÏ°Ó˚ üˆÏôƒ â£Ï≈îñ ≤ÃÓy•#Ó˚ SˆÎüö
Óyï˛y§V üˆÏôƒ !òˆÏÎ˚ §Çâ!ê˛ï˛ ˆÜ˛yöÄ ˆÜ˛yöÄ ˆòy°à!ï˛Ó˚ ˆ«˛ˆÏe §ywï˛y ç!öï˛ Óyôy Óy ܲ¡õö¢#° Ó›Ó˚
!öçfl∫ í˛z˛õyòyˆÏöÓ˚ ôü≈ §Ó•z ˆòy°à!ï˛ˆÏï˛ xÓü®ö âê˛yÎ˚– ï˛y•z ÓyhflÏÓ çàˆÏï˛Ó˚ ˆòy°à!ï˛ xÓü®ö ç!öï˛
x˛õã˛Î˚# ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ à!ï˛ ÓçyÎ˚ Ó˚yáˆÏï˛ !àˆÏÎ˚ e´üyàï˛ !öçfl∫ ¢!_´ •yÓ˚yÎ˚ ~ÓÇ ˆ¢£Ï ˛õÎ≈hsˇ §¡õ)î≈ hflÏ∏˛
•ˆÏÎ˚ ÎyÎ˚– Î!ò ~•z xÓü®ˆÏöÓ˚ üyey xï˛ƒ!ôܲ •ˆÏÎ˚ ˛õˆÏí˛¸ñ ï˛y•ˆÏ° ˆòy°à!ï˛ ~ˆÏܲÓyˆÏÓ˚ ÷Ó˚&•z ܲÓ˚y ÎyÎ˚ öy
~ÓÇ §yüƒyÓfliy ˆÌˆÏܲ !Óã%˛ƒï˛ ܲ¡õö«˛ü Ó›!ê˛ ˆÜ˛yöÄ ˆòy°à!ï˛ ≤Ãò¢≈ö öy ܲˆÏÓ˚•z §yüƒyÓfliyÎ˚ xyÓyÓ˚ ˆú˛Ó˚ï˛
ã˛ˆÏ° xyˆÏ§– Îáö xÓü®ˆÏöÓ˚ üyey ΈÏÌ‹T ܲüñ ï˛áö•z ˆÜ˛Ó° ˆòy°à!ï˛ ˛õyÄÎ˚y ÎyÎ˚– xyÓ˚ ~•z ˆòy°à!ï˛Ó˚
!ÓhflÏyÓ˚ ï˛Ìy ¢!_´ §üˆÏÎ˚Ó˚ §ˆÏD ܲüˆÏï˛ ÌyˆÏܲ– ~•z ~ܲˆÏܲ S~ܲܲ 3V xyüÓ˚y xÓü!®ï˛ ˆòy°à!ï˛ !öˆÏÎ˚
xyˆÏ°yã˛öy ܲÓ˚Ó– !ӈϢ£Ï ˆçyÓ˚ ˆòÄÎ˚y •ˆÏÓ ˆ§•z ˆ«˛e!ê˛ˆÏï˛ ˆÎáyˆÏö xÓü®ö ï%˛°öyÎ˚ ò%Ó≈° ~ÓÇ ˆòy°à!ï˛
˛õyÄÎ˚y §Ω˛Ó– ~•z xÓü®öˆÏܲ §ã˛Ó˚yã˛Ó˚ °â% xÓü®ö !•ˆÏ§ˆÏÓ !ã˛!•´ï˛ ܲÓ˚y •Î˚–
xyüÓ˚y ~áyˆÏö °â% xÓü®ˆÏöÓ˚ ˜Ó!¢‹Tƒ§ü)• xyˆÏ°yã˛öy ܲÓ˚yÓ˚ §ˆÏD §ˆÏD ÓyhflÏÓ ˆòy°à!ï˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛
•Ó– ï˛ˆÏÓ xÓü®ˆÏöÓ˚ í˛z˛õ!fli!ï˛ §ˆÏ_¥Ä ˆòy°à!ï˛ ÓçyÎ˚ Ó˚yáˆÏï˛ ˆàˆÏ° ܲ¡õö¢#° ÓƒÓfliy!ê˛ˆÏï˛ Óy•zˆÏÓ˚ ˆÌˆÏܲ
¢!_´Ó˚ ˆÎyàyö ˆòÄÎ˚y ≤ÈÏÎ˚yçö– ~çöƒ ˆÜ˛yöÄ ˛õÎ≈yÓ,_ Ó° ≤ÈÏÎ˚yˆÏàÓ˚ òÓ˚ܲyÓ˚ •ˆÏÎ˚ ˛õˆÏí˛¸– ú˛ˆÏ° ˆÎ ôÓ˚ˆÏöÓ˚
ܲ¡õˆÏöÓ˚ §,!‹T •Î˚ñ ï˛yˆÏܲ Ó°y •Î˚ ≤ÈÏîy!òï˛ Ü˛¡õö Óy ˆòy°à!ï˛ (forced vibration)– ˛õÓ˚Óï≈˛# ~ܲˆÏܲ
S~ܲܲ 4V !Ó£ÏÎ˚!ê˛Ó˚ !ÓhflÏy!Ó˚ï˛ xyˆÏ°yã˛öy •ˆÏÓ–
í˛ zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒ
~•z ~ܲܲ ˛õyˆÏë˛Ó˚ üôƒ !òˆÏÎ˚ !ö¡¨!°!áï˛ !Ó£ÏÎ˚=!° xy˛õöyÓ˚ xyÎ˚_ •ˆÏÓÈÙÙÙÈ
xÓü!®ï˛ ˆòy°à!ï˛Ó˚ Óy xÓü!®ï˛ §üO§ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î àë˛ö Ä ï˛yÓ˚ §üyôyö–
ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ñ ¢!_´ ~ÓÇ ˆòy°öܲy° ܲ#û˛yˆÏÓ xÓü®ˆÏöÓ˚ ú˛ˆÏ° ≤Ãû˛y!Óï˛ ~ÓÇ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ñ
ï˛yÓ˚ Óƒyáƒy–
xÓü®ˆÏöÓ˚ üyey xö%ÎyÎ˚# ï˛yˆÏòÓ˚ °â% SÓy ò%Ó≈°Vñ e´yhsˇ#Î˚ (critical) ~ÓÇ x!ï˛ (heavy) ~•z !ï˛ö!ê˛
ˆ◊î#ˆÏï˛ !Óû˛_´ ܲˆÏÓ˚ ï˛yˆÏòÓ˚ ˜Ó!¢‹Tƒ§ü)ˆÏ•Ó˚ !ӈϟ’£Ïî–
°â% xÓü®öÎ%_´ ˆòy°ö¢#° ÓƒÓfliyÓ˚ ˆ«˛ˆÏe ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)• !ӈϟ’£ÏˆÏîÓ˚ çöƒ °à#Î˚ •…y§ñ Ÿ’Ìö ܲy°
(relaxation time) ~ÓÇ !ܲí˛zÈÙÈ=îyB˛ (Q-factor) !öî≈Î˚–
˛õòyÌ≈!ÓòƒyÓ˚ !Ó!û˛ß¨ ¢yáyÎ˚ xÓü!®ï˛ ˆòy°à!ï˛Ó˚ û)˛!üܲyÓ˚ ˛õÎ≈yˆÏ°yã˛öy Ä àîöy–
3.2 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y
xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤çD í˛zay!˛õï˛ •ˆÏ° fl∫û˛yÓï˛•z üˆÏö •Î˚ ˆÎñ §Ó˚° ˆòy°à!ï˛Ó˚
ˆÎ §ü#ܲÓ˚î xyüÓ˚y ~ܲܲ1ÈÙÈ~ ˆ˛õˆÏÎ˚!åÈ S§ü#ܲÓ˚î 1.1Vñ ï˛yÓ˚ üˆÏôƒ !ܲå%È ˛õ!Ó˚Óï≈˛ö â!ê˛ˆÏÎ˚ xÓü!®ï˛
ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î ˛õyÄÎ˚y §Ω˛Ó– xy˛õöyÓ˚ ~•z xö%üyö §!ë˛Ü˛ñ ï˛ˆÏÓ ˆ§•z xyˆÏ°yã˛öyÎ˚ ÎyÄÎ˚yÓ˚ xyˆÏà í˛zˆÏÕ‘á
62
ܲÓ˚y ≤ÈÏÎ˚yçö ˆÎñ ~áö ˆÌˆÏܲ §ÓÓ˚ܲü xÓü®ö ü%_´ ˆÎ ~ܲ xyò¢≈ ˆòy°à!ï˛Ó˚ !Ó£ÏÎ˚ ≤ÃÌü ~ܲˆÏܲ
xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈñ ï˛yˆÏܲ xyüÓ˚y ü%_´ ˆòy°à!ï˛ ÓˆÏ° í˛zˆÏÕ‘á ܲÓ˚Ó–
xyˆÏà Ó°y •ˆÏÎ˚ˆÏåÈ ˆÎñ xÓü®ˆÏöÓ˚ xÌ≈ •ˆÏFåÈ Ü˛¡õö¢#° Óy ܲ¡õö«˛ü ÓƒÓfliyâ!ê˛ï˛ x˛õã˛Î˚# ӈϰÓ˚
í˛z˛õ!fli!ï˛– ~•z x˛õã˛Î˚# Ó° §Ó≈òy•z à!ï˛Ó˚ !ÓÓ˚&ˆÏk˛ ܲyç ܲÓ˚ˆÏÓñ ˆÎ
û)˛!üܲy xyüyˆÏòÓ˚ x!ï˛ ˛õ!Ó˚!ã˛ï˛ â£Ï≈î Ó° ˛õy°ö ܲˆÏÓ˚ ÌyˆÏܲ– ï˛y•z
xÓü®öç!öï˛ Ó°ñ à!ï˛Ó˚ x!û˛ü%ˆÏáÓ˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ ï˛yÓ˚
x!û˛ü%á ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ÌyˆÏܲ– ~áö ≤ß¿ •ˆÏFåÈ ˆÎñ ~•z x˛õã˛Î˚#
Ó°!ê˛ˆÏܲ ܲ#û˛yˆÏÓ ˛õyÄÎy ÎyˆÏÓ⁄ ~Ó˚ í˛z_ˆÏÓ˚Ó˚ çöƒ xyüÓ˚y !ã˛e 3.1
~Ó˚ !òˆÏܲ ò,!‹T ˆòÓ–
3.1 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ ~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy ~üöû˛yˆÏÓ Ü˛!¡õï˛ •ˆÏFåÈ ˆÎñ !fl±ÇÈÙÈ~Ó˚ §ˆÏD Î%_´
û˛Ó˚!ê˛ ~ܲ!ê˛ ˆã˛yˆÏäÓ˚ üˆÏôƒ !òˆÏÎ˚ ã˛°yˆÏú˛Ó˚y ܲÓ˚ˆÏåÈ ~ÓÇ ˆã˛yˆÏäÓ˚ üˆÏôƒ â£Ï≈î xÌ≈yÍ xÓü®ö ç!öï˛ Óyôy ӈϰÓ˚
§¡ø%á#ö •ˆÏFåÈ– xyüÓ˚y ôˆÏÓ˚ !ö•z ˆÎñ û˛Ó˚!ê˛ ˆÎ ÓyôyÈÙÈӈϰÓ˚ §¡ø%á#ö •ˆÏFåÈ ï˛yÓ˚ ˛õ!Ó˚üyî Fd ~ÓÇ Ó°!ê˛Ó˚
üyö ˆÜ˛yöÄ !ӈϢ£Ï ü%•)ˆÏï≈˛ û˛Ó˚!ê˛Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ §üyö%˛õy!ï˛Ü˛– ~áö xy˛õ!ö !°áˆÏï˛ ˛õyˆÏÓ˚ö ≠
Fd = dx
dtγ− Ó Ïú ...... 3.1
ˆÎáyˆÏö γ ~ܲ!ê˛ ôöydܲ ô &Óܲ Ó˚y!¢– γ Ó˚y!¢!ê˛ˆÏܲ Ó°y •Î˚ xÓü®ö =îyB˛ (damping coefficient)–
~!ê˛ ~ܲܲ à!ï˛ˆÏÓˆÏàÓ˚ çöƒ xÓü®ö Ó° ~ÓÇ ~Ó˚ ~ܲܲ ~•zû˛yˆÏÓ ˛õyÄÎ˚y ÎyÎ˚–
γ ~Ó˚ ~ܲܲ = ÓˆúÏ Ó ~Ñ˛Ñ˛
Ü!ì˛ˆÓÏ ˆÜÏ Ó ~Ñ˛Ñ˛ =
1N
ms−
= kg s–1
á%Ó fl∫yû˛y!Óܲû˛yˆÏÓ•z ≤ß¿ í˛zë˛ˆÏÓ ˆÎñ xÓü®öç!öï˛ Ó°ˆÏܲ à!ï˛ˆÏÓˆÏàÓ˚ §üyö%˛õy!ï˛Ü˛ ôˆÏÓ˚ ˆöÄÎ˚y ܲï˛áy!ö
§Dï˛⁄ ܲ¡õö¢#° Ó›Ó˚ à!ï˛ˆÏÓà Î!ò á%Ó ˆÓ!¢ öy •Î˚ñ ï˛y•ˆÏ° ~•z §üyö%˛õy!ï˛c ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚ ~ÓÇ
!Ó!û˛ß¨ ˛õÓ˚#«˛yÓ˚ üyôƒˆÏü ï˛y ˆòáyˆÏöy §Ω˛Ó •ˆÏÎ˚ˆÏåÈ– xyÓ˚ §ywï˛yç!öï˛ Óyôy ˆÎ à!ï˛ˆÏÓˆÏàÓ˚ §üyö%˛õy!ï˛Ü˛
•ˆÏÎ˚ ÌyˆÏܲñ ï˛y ~ܲê%˛ xyˆÏ°yã˛öy ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
§ywï˛yç!öï˛ ÓyôyÓ˚ çöƒ ˆÜ˛yˆÏöy §ywï˛y!Ó!¢‹T ≤ÃÓy•#Ó˚ üˆÏôƒ !òˆÏÎ˚ ˛õï˛ö¢#° Óy ã˛°üyö ˆày°Ü˛yÜ,˛!ï˛
Ó› ~ܲ!ê˛ Óyôy ӈϰÓ˚ §¡ø%á#ö •Î˚– ~•z !ӣψÏÎ˚ !ÓK˛yö# ˆfiê˛yܲ§ ~ܲ!ê˛ §)e ˆòö (Stokes’ law)– ~•z
§)eyö%ÎyÎ˚#
Fd = 6πηrv
~áy Ïö η üyôˆÏüÓ˚ §ywï˛yB˛ñ r ã˛°üyö Ó›!ê˛Ó˚ Óƒy§yô≈ ~ÓÇ v Ó›Ó˚ à!ï˛ˆÏÓà– ~•z §)eyö%ÎyÎ˚# §yw üyôƒˆÏü
ã˛°üyö Ó›Ó˚ à!ï˛Ó˚ !ÓÓ˚&ˆÏk˛ !e´Î˚y¢#° Óyôy Ó° Ó›Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ §üyö%˛õy!ï˛Ü˛–
ï˛y xyüÓ˚y çy!ö– xï˛~Óñ xÓü®öç!öï˛ Ó° ܲ¡õö¢#° Ó›Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ §üyö%˛õy!ï˛Ü˛ñ ~•z!ê˛ ôˆÏÓ˚
!öˆÏÎ˚•z xyüÓ˚y x@ˇÃ§Ó˚ •Ó–
63
!ã˛e 3.1 ÈÙÈ~ ˆòáyˆÏöy !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy!ê˛ ˆÎ §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ã˛°yˆÏú˛Ó˚y ܲÓ˚ˆÏåÈñ ï˛yˆÏܲ x x«˛ ~ÓÇ
ÓƒÓfliy!ê˛Ó˚ §yüƒyÓfliyÎ˚ û˛Ó˚!ê˛Ó˚ û˛Ó˚ˆÏܲwˆÏܲ ü)°!Ó®% Óy x = 0 !Ó®% ôˆÏÓ˚ ˆöÄÎ˚y •°– ~áö û˛Ó˚!ê˛ˆÏܲ í˛yö!òˆÏܲ
xö%û)˛!üܲ ˆÓ˚áy ÓÓ˚yÓÓ˚ !Óã%˛ƒï˛ ܲˆÏÓ˚ ˆåȈÏí˛¸ !òˆÏ° ˆ§!ê˛ Ü˛!¡õï˛ •ˆÏï˛ ÷Ó˚& ܲÓ˚ˆÏÓ– ˆÜ˛yöÄ ü%•)ˆÏï≈˛ ü)°!Ó®%
(x = 0) Óy §yüƒyÓfliy ˆÌˆÏܲ x ò)Ó˚ˆÏc xÓfliyöܲyˆÏ° û˛Ó˚!ê˛Ó˚ IJõÓ˚ ˆÎ Ó°=!° !e´Î˚y¢#° •ˆÏÓñ ï˛y •°ñ
(i) ≤Ãï˛ƒyöÎ˚ܲ Ó° (restoring force)ÈÙÙÙÈkx, ˆÎáyˆÏö k •ˆÏFåÈ !fl±ÇÈÙÈô &Óܲ (spring constant)
(ii) xÓü®öç!öï˛ Ó°ÈÙÙÙÈγv Óy
dxdt
γ−
ñ ˆÎáyˆÏö γ •ˆÏFåÈ xÓü®ö =îyB˛ (damping coefficient)–
~áyˆÏö v = dxdt
xÌ≈yÍñ !ӈϢ£Ï ü%•)ˆÏï≈˛ à!ï˛ˆÏÓà–
~áö !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚ à!ï˛§)ˆÏeÓ˚ §y•yˆÏ΃ ˆ°áy ÎyÎ˚ ˆÎñ
Óyñ
2
2dx mdt
=
dx kxdt
γ −−
...3.2
≤Ã!ï˛!ê˛ Ó˚y!¢ˆÏܲ m myÓ˚y û˛yà ܲˆÏÓ˚ ˛õy•zñ
2
2d x dx k x
m dt mdtγ+ + = 0
Óyñ
22
02 2d x dxb xdtdt
ω+ + = 0 ...3.3
~áy Ïö 2
0ω =
km
~ÓÇ 2b =
0 .mγω
~•z ˆòy°à!ï˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
mγ
ˆÜ˛ b öy
!°ˆÏá 2b ˆ°áy •ˆÏÎ˚ˆÏåÈ Ü˛yÓ˚î ~Ó˚ ú˛ˆÏ° ˛õÓ˚Óï≈˛# ˛õÎ≈yˆÏÎ˚ 3.3 §ü#ܲÓ˚î!ê˛Ó˚ §üyôyˆÏöÓ˚ Ó˚*˛õ!ê˛ §Ó˚°ï˛Ó˚ •ˆÏÓ–
°«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ ˆÎ ~•zñ 3.3 §ü#ܲÓ˚î!ê˛ ü%_´ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ˆÌˆÏܲ !û˛ß¨– xÓü®ö ç!öï˛
ӈϰÓ˚ çöƒ §ü#ܲÓ˚ˆÏî
2dx bdt
˛õò!ê˛Ó˚ í˛z˛õ!fli!ï˛Ó˚ ú˛ˆÏ° ~•z xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ˆÎ §üyôyö ˛õyÄÎy ÎyˆÏÓñ
ï˛y fl∫û˛yÓï˛•z ü%_´ ˆòy°à!ï˛Ó˚ ˆ«˛ˆÏe ≤ÃyÆ §üyôyö ˆÌˆÏܲ !û˛ß¨ •ˆÏÓ– 3.3 §ü#ܲÓ˚î!ê˛ ~ܲ!ê˛ !mï˛#Î˚ üyeyÓ˚
(second order) ˜Ó˚!áܲ (linear) §ü§_¥ (homogeneous) xÓܲ° §ü#ܲÓ˚î ~ÓÇ §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛!ê˛
˛õˆÏòÓ˚ §•à (coefficient) ô &Óܲ– ~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö xyüÓ˚y !öî≈Î˚ ܲÓ˚Ó– ï˛yÓ˚ xyˆÏà ~ܲ!ê˛ !ӣψÏÎ˚Ó˚
!òˆÏܲ xy˛õöyÓ˚ ò,!‹T xyܲ£Ï≈î ܲÓ˚ˆÏï˛ ã˛y•z– ü%_´ ˆòy°à!ï˛Ó˚ ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ ≤Ãû˛yˆÏÓ §Ó˚î 'x' §üˆÏÎ˚Ó˚ §ˆÏD
ˆÎ û˛yˆÏÓ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ˆÏåÈ
() 0 cos xAt ⎡⎤ =+ ⎣⎦ ωφ
ï˛y x˛õ!Ó˚Ó!ï≈˛ï˛ !ÓhflÏyˆÏÓ˚Ó˚ ܲ¡õö §)!ã˛ï˛ ܲˆÏÓ˚ˆÏåÈ–
xÓü®ö !öÿ˛Î˚•z ˆ§•z à!ï˛ˆÏܲ ≤Ãû˛y!Óï˛ Ü˛Ó˚ˆÏÓ– ܲ# âê˛ˆÏï˛ ˛õyˆÏÓ˚ xö%üyö ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö⁄ xÓü®ö ˆÎˆÏ•ï%˛
ÓyhflÏÓ ˆ«˛ˆÏe ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ˆÏܲ e´ü¢ •…y§ ܲˆÏÓ˚ ˆòÎ˚ñ ï˛y•z ˆ§•z âê˛öyÓ˚ ≤Ã!ï˛ú˛°ö 3.3 §ü#ܲÓ˚ˆÏîÓ˚
§üyôyˆÏö Ìyܲy ≤ÈÏÎ˚yçö– ï˛y•ˆÏ° !ܲ xyüÓ˚y xy¢y ܲÓ˚ˆÏï˛ ˛õy!Ó˚ ˆÎñ 3.3 ~Ó˚ §üyôyö ~ܲ!ê˛ e´ü•…y§üyö
64
!ÓhflÏyˆÏÓ˚Ó˚ ˆòy°à!ï˛ §)!ã˛ï˛ ܲÓ˚ˆÏÓ⁄ •Ñƒyñ !öÿ˛Î˚•z ï˛y á%Ó §Dï˛ ≤Ãï˛ƒy¢y– !ܲv xÓü®ˆÏöÓ˚ üyey xï˛ƒ!ôܲ
•ˆÏÎ˚ ˆàˆÏ° ܲ# •ˆÏÓ⁄ ˆòy°à!ï˛ !ܲ ˆ§ˆÏ«˛ˆÏe xyòˆÏ˛õ•z §Ω˛Ó •ˆÏÓ⁄ xyüyˆÏòÓ˚ ~•z ≤ß¿=!°Ó˚ í˛z_Ó˚ ˛õyÄÎ˚yÓ˚
çöƒ ˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏò xyüÓ˚y §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö !öî≈Î˚ ܲˆÏÓ˚ ï˛yÓ˚ !Ó!û˛ß¨ ˜Ó!¢‹Tƒ=!° ˛õÎ≈yˆÏ°yã˛öy ܲÓ˚Ó–
≤çDï˛ Ó°y òÓ˚ܲyÓ˚ñ 3.3 §ü#ܲÓ˚î ~ܲ!ê˛ xÓü!®ï˛ ˆòy°à!ï˛Ó˚ à!ï˛¢#° xÓfliy §)!ã˛ï˛ ܲÓ˚ˆÏåÈ xyÓ˚ ~Ó˚
§üyôyö•z •ˆÏÓ ˆòy°à!ï˛Ó˚ §üÎ˚ÈÙÈò)Ó˚c §ü#ܲÓ˚î–
3.3 xÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöxÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
3.3 xÓܲ° §ü#ܲÓ˚î!ê˛Ó˚ §üyôyˆÏöÓ˚ çöƒ xyüÓ˚y §Ç!Ÿ’‹T à!ï˛Ó˚ ÓyhflÏÓ ã˛!Ó˚e üˆÏö ˆÓ˚ˆÏá ~üö ~ܲ!ê˛
˛õÓ˚#«˛yü)°Ü˛ §üyôyö (trial solution) ôˆÏÓ˚ !öˆÏÎ˚ ~ˆÏàyˆÏÓy ÎyÓ˚ üˆÏôƒ §üˆÏÎ˚Ó˚ §)ã˛Ü˛ xˆÏ˛õ«˛Ü˛ (exponential
function) xyˆÏåÈ– ܲyÓ˚îñ xyüÓ˚y xy¢y ܲÓ˚!åÈ ˆÎñ ~Ó˚ §yôyÓ˚î §üyôyˆÏö ~ܲyôyˆÏÓ˚ §)ã˛Ü˛ Ä ˛õÎ≈yÓ,_ ã˛!Ó˚e
ÌyܲˆÏÓ– §)ã˛Ü˛ Ó˚y!¢!ê˛ §üˆÏÎ˚Ó˚ §ˆÏD !ÓhflÏyˆÏÓ˚Ó˚ •…y§ §)!ã˛ï˛ ܲÓ˚ˆÏÓ ~ÓÇ ˛õÎ≈yÓ,_ Ó˚y!¢ §)!ã˛ï˛ ܲÓ˚ˆÏÓ ÓƒÓfliy!ê˛Ó˚
ܲ¡õö¢#° ã˛!Ó˚e– ~ˆÏ«˛ˆÏe xyüÓ˚y ˛õÓ˚#«˛yü)°Ü˛ §üyôyö !•ˆÏ§ˆÏÓ !ö¡¨!°!áï˛ §üÎ˚ÈÙÈò)Ó˚c §ü#ܲÓ˚î!ê˛ !öˆÏï˛
˛õy!Ó˚ ≠
x(t) = aeαt ...3.4
~áyˆÏö a ~ÓÇ α ò%!ê˛ xK˛yï˛ ô &Óܲ Ó˚y!¢–
°«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ ˆÎñ (3.4) §ü#ܲÓ˚ˆÏîÓ˚ ÓÑy !òˆÏܲÓ˚ Ó˚y!¢!ê˛ ˜òâ≈ƒ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– xï˛~Óñ ï˛yÓ˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚
~ܲܲ– §%ï˛Ó˚yÇñ í˛yö !òܲ!ê˛Ó˚ §yü!@ˇÃܲ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ ~ܲܲ •ˆÏÓ– ~áö e Ó˚y!¢Ó˚ §)ã˛Ü˛ !•ˆÏ§ˆÏÓ ÓƒÓ•*ï˛
§Çáƒy!ê˛ xÓ¢ƒ•z ~ܲܲ ¢)öƒ •ˆÏÓ– ï˛y•z ~ˆÏ«˛ˆÏe αtÈÙÈÓ˚ ˆÜ˛yöÄ ~ܲܲ ÌyܲˆÏÓ öy– ˆÎˆÏ•ï%˛ tÈÙÈ~Ó˚ ~ܲܲ §üˆÏÎ˚Ó˚
~ܲܲ Óy ˆ§ˆÏܲu˛ (s) ï˛y•z ‘α’ ~Ó˚ ~ܲܲ •ˆÏÓ §üˆÏÎ˚Ó˚ !Ó˛õÓ˚#ï˛ (s-1)– xyÓ˚ ï˛y•z 3.4 §ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚
˛õyˆÏ¢Ó˚ ~ܲˆÏܲÓ˚ §üï˛yÓ˚ çöƒ ‘a’ ~Ó˚ ~ܲܲ •ˆÏÓ ˜òˆÏâ≈ƒÓ˚ ~ܲܲ (m)–
3.4 §ü#ܲÓ˚îˆÏܲ ˛õÓ˚˛õÓ˚ ò%ÓyÓ˚ xÓܲ°ö ܲˆÏÓ˚ ˛õy•zñ
dxdt
= aαeαt
∴
2
2dxdt
= aα2eαt
~•z §¡õÜ≈˛ ò%!ê˛ 3.3 öÇ §ü#ܲÓ˚ˆÏî ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õy•zñ
() 220 2t baeα ααω ++
= 0 ... 3.5
~•z §ü#ܲÓ˚î!ê˛ ‘t’ ~Ó˚ ˆÎ ˆÜ˛yö üyˆÏöÓ˚ çöƒ•z §!ë˛Ü˛– ï˛y•z eαt ≠ 0 S§Ü˛° ‘t’ ~Ó˚ çöƒV
~ÓÇ a ≠ 0 ܲyÓ˚î a = 0 •ˆÏ° 3.4 §ü#ܲÓ˚î xö%ÎyÎ˚# x(t)ÈÙÈÓ˚ üyö §Ó≈òy•z ¢)öƒ •ˆÏÓ– ï˛y•z 3.5 §ü#ܲÓ˚ˆÏî
Ó¶˛ö#û%˛_´ xÇ¢!ê˛•z ¢)öƒ xÌ≈yÍ
( )2 202bα α ω+ + = 0 ... 3.6
~!ê˛ α Ó˚ ~ܲ!ê˛ !mâyï˛ §ü#ܲÓ˚î– ~•z §ü#ܲÓ˚ˆÏîÓ˚ ò%!ê˛ Ó#ç (root) Î!ò α1 ~ÓÇ α2 •Î˚ñ ï˛y•ˆÏ° !mâyï˛
§ü#ܲÓ˚ˆÏîÓ˚ ï˛_¥ xö%ÎyÎ˚# ˆ°áy ÎyÎ˚ ˆÎñ
65
α1 = 2 2
0b b ω− + − ...3. 7a
α2 = 2 20b b ω− − − ...3.7b
Ó›ï˛ ~•z ò%!ê˛ Ó#ç α1 ~ÓÇ α2ÈÙÈÓ˚ ≤ÃÜ,˛!ï˛ Ü˛¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ !öô≈yÓ˚î ܲˆÏÓ˚–
α ~Ó˚ ò%•z!ê˛ üyö α1 ~ÓÇ Èα
2 ÙÈÓ˚ çöƒ xyüÓ˚y §ü#ܲÓ˚ˆÏîÓ˚ ò%•z!ê˛ §üyôyö ˛õyÓ– ~•z ò%•z!ê˛ §üyôyöˆÏܲ x
1(t)
~ÓÇ x2(t) !•ˆÏ§ˆÏÓ !ã˛!•´ï˛ ܲˆÏÓ˚ ˆ°áy ÎyÎ˚ ˆÎñ
x1(t) = ( )2 21 0expa b b t⎡ ⎤− − −⎢ ⎥⎣ ⎦
ω ... 3.8a
x2(t) = ( )2 2
2 0expa b b t⎡ ⎤− + −⎢ ⎥⎣ ⎦ω ... 3.8b
ˆÎ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ 3.3 §üyôyö !•ˆÏ§ˆÏÓ x1 ~ÓÇ x
2 ˆÜ˛ ˛õyÄÎ˚y ˆàˆÏåÈ ˆ§•z §ü#ܲÓ˚î!ê˛ ˜Ó˚!áܲ–
xï˛~Ó í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ (superposition) ö#!ï˛ yÓ°¡∫ö ܲˆÏÓ˚ x1 ~ÓÇ x
2 ˆÜ˛ Î%_´ ܲˆÏÓ˚ ~ܲ!ê˛ §yôyÓ˚î §üyôyö
x(t) ˛õyÄÎ˚y ÎyÎ˚ ≠
x(t) = x1 (t) + x
2(t)
= ( ) ( )1 1
2 2 2 22 21 0 2 0exp( ) exp expbt a b t a b t
⎡ ⎤⎧ ⎫ ⎧ ⎫− − + − −⎨ ⎬ ⎨ ⎬⎢ ⎥
⎩ ⎭ ⎩ ⎭⎣ ⎦ω ω ...3.9
~áyˆÏö a1 ~ÓÇ a2 ò%!ê˛ ˆfl∫FåÈ xã˛Ó˚ (arbitrary constant) ~ÓÇ ~ˆÏòÓ˚ üyö !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ ˆòy°ˆÏöÓ˚
≤ÃyÌ!üܲ ¢ï≈˛yÓ°# (initial condition) çyöy ≤ÈÏÎ˚yçö– xÌ≈yÍñ ˆÎ ˆÜ˛yö §üˆÏÎ˚ñ ôÓ˚y Îyܲ Îáö t = 0ñ
ï˛áö §Ó˚î Óy à!ï˛ˆÏÓˆÏàÓ˚ üyö ܲ# !åÈ° ï˛y çyöy ÌyܲˆÏ° a1, a
2 !öî≈Î˚ ܲÓ˚y §Ω˛Ó–
§ü#ܲÓ˚ˆÏî (b2 – ω02) xÇ¢!ê˛ !ӈϢ£Ï =Ó˚&c˛õ)î≈– ~•z xÇ¢!ê˛Ó˚ üyö b ~ÓÇ ω0ÈÙÈÓ˚ ï%˛°öyü)°Ü˛ üyˆÏöÓ˚
IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ ôöydܲñ îydܲñ xÌÓy ¢)öƒ •ˆÏï˛ ˛õyˆÏÓ˚ ~ÓÇ ~Ó˚ üˆÏôƒ !òˆÏÎ˚ !ï˛ö!ê˛ !û˛ß¨ xÓfliyÓ˚ §,!‹T
•ˆÏÓ– ~•z !ï˛ö!ê˛ ˆ«˛ˆÏe ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ xyüÓ˚y !ï˛ö!ê˛ !û˛ß¨ xÓfliy ˆòáˆÏï˛ ˛õy•z– ày!î!ï˛Ü˛
!ӈϟ’£ÏˆÏîÓ˚ xyˆÏà ~=!°Ó˚ §Ç!«˛Æ ˛õ!Ó˚ã˛Î˚ ˆöÄÎ˚y Îyܲ–
SܲV SܲV SܲV SܲV SܲV Îáö b2 > ω0
2 ï˛áö α1 ~ÓÇ α
2ÈÙÈÓ˚ üyö ÓyhflÏÓ ~ÓÇ Ü˛¡õö¢#° ÓƒÓfliy!ê˛ˆÏï˛ xÓü®ö á%Ó•z ˆÓ!¢–
Ó›ï˛ ~•z xÓü®ˆÏöÓ˚ üyey ~ï˛ê˛y•z ˆÎñ ~Ó˚ ú˛ˆÏ° ܲ¡õö §Ω˛Ó •Î˚ öy– ܲ¡õö«˛ü Ó›!ê˛ˆÏܲ ï˛yÓ˚ §yüƒ
xÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ ܲÓ˚ˆÏ° ï˛y ܲ!¡õï˛ öy •ˆÏÎ˚ ô#ˆÏÓ˚ ô#ˆÏÓ˚ §yüƒ xÓfliyˆÏö !ú˛ˆÏÓ˚ xyˆÏ§– ~•z xÓü®öˆÏܲ
xyüÓ˚y Úx!ï˛ xÓü®öÛ (heavy damping) ӈϰ Ìy!ܲ– 3.3.1 xö%ˆÏFåȈÏò xy˛õ!ö ~ !ӣψÏÎ˚ xyÓ˚Ä çyöˆÏï˛
˛õyÓ˚ˆÏÓö–
66
SáV Î!ò b2 = ω02 •Î˚ ï˛y•ˆÏ° α1 ~ÓÇ α2ÈÙÈÓ˚ üyö §üyö xÌ≈yÍ α1 = α2 – ~•z ˆ«˛ˆÏe ˆÎ ôÓ˚ˆÏöÓ˚
xÓü®ö °«˛ƒ ܲÓ˚y ÎyÎ˚ ï˛yˆÏܲ Ó°y •Î˚ Úe´yhsˇ#Î˚ xÓü®öÛ (critical damping)– ˆÜ˛yöÄ Ü˛¡õö¢#° ÓƒÓfliyÎ˚
e´yhsˇ#Î˚ xÓü®ö í˛z˛õ!fliï˛ ÌyܲˆÏ° ˆ§áyˆÏöÄñ ܲ¡õö âê˛ˆÏï˛ ˛õyˆÏÓ˚ öy– ï˛y•ˆÏ° !ܲ x!ï˛ xÓü®ˆÏöÓ˚ §ˆÏD
~Ó˚ ˆÜ˛yö ˛õyÌ≈ܲƒ ˆö•z⁄ öyñ ò%!ê˛ ˆ«˛e !ܲv ~ܲ öÎ˚– ~ˆÏ«˛ˆÏe ܲ¡õö¢#° Ó›ˆÏܲ §yüƒyÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛
ܲÓ˚ˆÏ° ï˛y x!ï˛ xÓü®ˆÏöÓ˚ ï%˛°öyÎ˚ o&ï˛ §yüƒyÓfliyˆÏö !ú˛ˆÏÓ˚ xy§ˆÏÓñ !ܲv ï˛y x!ï˛e´ü ܲˆÏÓ˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ
ˆÜ˛yöÄ §Ó˚î âê˛ˆÏÓ öyñ xÌ≈yÍ §yüƒyÓfliyˆÏö ~ˆÏ§ ï˛y à!ï˛•#ö •ˆÏÎ˚ ˛õí˛¸ˆÏÓ– 3.3.2 xö%ˆÏFåȈÏò xyüÓ˚y e´yhsˇ#Î˚
xÓü®ö §¡õˆÏÜ≈˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
SàV Î!ò b2 < ω02 •Î˚ xÌ≈yÍ Îáö α1 ~ÓÇ α2ÈÙÈÓ˚ üyö ò%!ê˛ ç!ê˛° (complex) Óy ܲy“!öܲ (imaginary)
•Î˚ñ ï˛y•ˆÏ° ˆÎ xÓfliy §)!ã˛ï˛ •Î˚ñ ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ ï˛y §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ =Ó˚&c˛õ)î≈– ~ˆÏ«˛ˆÏe
xÓü®ö ÌyܲˆÏ°Ä ܲ¡õö §Ω˛Ó •Î˚– ~•z xÓü®öˆÏܲ Ó°y •Î˚ °â% xÓü®ö (weak damping)– ÓyhflÏÓ
çàˆÏï˛ ˆÎ §Ó ܲ¡õö xyüÓ˚y ˆòáˆÏï˛ ˛õy•zñ ï˛yÓ˚ §Ó=!°ˆÏï˛•z °â% xÓü®ö Óï≈˛üyö– ï˛ˆÏÓ °â% xÓü®ˆÏöÓ˚
ú˛ˆÏ° §üˆÏÎ˚Ó˚ §ˆÏD ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ ï˛Ìy ¢!_´ e´ü¢ •…y§ ˛õyÎ˚– ~•z xÓü®ˆÏöÓ˚ xyÓ˚Ä Ü˛ï˛=!° ˜Ó!¢‹Tƒ
xyüÓ˚y ≤Ã̈Ïü 3.3.3 ~ÓÇ ˛õˆÏÓ˚ 3.5 xö%ˆÏFåȈÏò !ÓhflÏy!Ó˚ï˛û˛yˆÏÓ xyˆÏ°yã˛öy ܲÓ˚Ó–
3.3.1 x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)•x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)•x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)•x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)•x!ï˛ xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢‹Tƒ§ü)•
x!ï˛ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe b2 > ω02 (b2 – ω0
2 ) Ó˚y!¢!ê˛ ôöydܲ ~ÓÇ 3.7a, b §ü#ܲÓ˚ˆÏîÓ˚ α1 ~ÓÇ
α2 ò%!ê˛ ÓyhflÏÓ §Çáƒy–
ôÓ˚y Îyܲñ β = 2 20b ω−
ï˛y•ˆÏ° 3.9 §ü#ܲÓ˚ˆÏî í˛z!Õ‘!áï˛ xÓü!®ï˛ ܲ¡õˆÏöÓ˚ §yôyÓ˚î §üyôyö ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yÎ˚
x(t) = 1 2exp( )[ exp( ) exp( )]bt a t a t− + −β β ... 3.10
~•z §ü#ܲÓ˚î myÓ˚y §)!ã˛ï˛ à!ï˛ Ü˛¡õöà!ï˛ öÎ˚– ~•z ôÓ˚ˆÏöÓ˚ à!ï˛ˆÏܲ Ó˚&k˛ ˆòy° (dead beat) Ó°y •Î˚–
ï˛ˆÏÓ Ü˛¡õö §¡õߨ öy •ˆÏ°Ä §üˆÏÎ˚Ó˚ §ˆÏD ò)Ó˚c ܲ#û˛yˆÏÓ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÓñ ï˛y !ܲv !öû≈˛Ó˚ ܲÓ˚ˆÏÓ ≤ÃyÌ!üܲ
¢ˆÏï≈˛Ó˚ (initial condition) IJõÓ˚– ~•z ≤ÃyÌ!üܲ ¢ˆÏï≈˛Ó˚ §y•yˆÏ΃ ò%!ê˛ ˆfl∫FåÈ xã˛Ó˚ a1 ~ÓÇ a
2 Ä !öô≈yÓ˚î
ܲÓ˚y ÎyÎ˚–
ˆÎüö ôÓ˚y Îyܲñ ≤ÃyÌ!üܲ xÓfliyÎ˚ Ó›!ê˛ ï˛yÓ˚ §yüƒyÓfliyˆÏö Ó˚ˆÏÎ˚ˆÏåÈ xÌ≈yÍñ Îáö t = 0, x = 0– ~ÓyÓ˚
~Ó˚ IJõÓ˚ ~ܲ!ê˛ âyï˛ Sx!ï˛ «%˛o §üÎ˚ ôˆÏÓ˚ §•§y ≤ÃÎ%_´ Ó,•Í Ó°V ≤ÈÏÎ˚yà ܲÓ˚y •°– ï˛yÓ˚ ú˛ˆÏ° ≤ÃyÌ!üܲ
§üˆÏÎ˚ xÌ≈yÍñ t = 0ÈÙÈˆï˛ Ó›!ê˛ v0 ˆÓà °yû˛ ܲÓ˚°– ~áö ~•z ò%!ê˛ ¢ï≈˛ ≤ÈÏÎ˚yà ܲÓ˚y Îyܲ–
≤Ãò_ ò%!ê˛ ¢ï≈˛ •°ñ Îáö t = 0, x = 0; Îáö t = 0,
dxdt
Óy v = v0
≤ÃÌü ¢ï≈˛ 3.10 §ü#ܲÓ˚ˆÏî Ó!§ˆÏÎ˚ ˛õy•zñ
0 = a1 + a
2... (i)
67
3.10 §ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚ ˛õ«˛ˆÏܲ xÓܲ°ö ܲÓ˚ˆÏ° xy˛õ!ö ˛õyˆÏÓö ≠
dxdt
=
12 exp()[exp()exp()] bbtatat −−+− ββ 12 exp()[exp()exp()] btatat +−−− βββ
~áö !mï˛#Î˚ ¢ï≈˛!ê˛ Ó§yˆÏ° ˛õyÄÎ˚y ÎyˆÏÓ
v0 = –b[a
1 + a
2] + β [a
1 – a
2] ... (ii)
(i) Ä (ii) ÓƒÓ•yÓ˚ ܲˆÏÓ˚ §•ˆÏç•z ˛õyÄÎ˚y ÎyˆÏÓ
a1 = – a
2 =
0
2v
β
~Ó˚ §y•yˆÏ΃ 3.10 §ü#ܲÓ˚î ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yÎ˚
x(t) =
0exp()[exp()exp()] 2v
bttt −−− ββ β= 0 ( ). ( )
vt tφ ψβ ... 3.11
ˆÎáyˆÏö φ(t) = exp (–bt)
~ÓÇ ψ (t) = exp( ) exp( )2
t t− −β β = sin h (βt)
§ü#ܲÓ˚î 3.11 ~ ˆòáy ÎyˆÏFåÈ ˆÎñ x(t), ò%!ê˛ xˆÏ˛õ«˛Ü˛ xÌ≈yÍ φ(t) ~ÓÇ ψ (t) ~Ó˚ =îú˛ˆÏ°Ó˚ §üyö–
~•z ò%!ê˛ xˆÏ˛õ«˛ˆÏܲÓ˚ ≤ÃÌü!ê˛ xÌ≈yÍ φ(t)[= exp (–bt)], §üˆÏÎ˚Ó˚ §ˆÏD •…y§ ˛õyÎ˚ ~ÓÇ !mï˛#Î˚!ê˛ xÌ≈yÍ ψ(t)[= sinh(βt)], §üˆÏÎ˚Ó˚ §ˆÏD Ó,!k˛ ˛õyÎ˚– x!ï˛ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe b ~Ó˚ üyö β xˆÏ˛õ«˛y Óí˛¸ •ÄÎ˚yÎ˚ φ(t)
~Ó˚ •…y§ ˛õyÄÎ˚yÓ˚ •yÓ˚ ˆÓ!¢– ï˛y•z ò%!ê˛ xˆÏ˛õ«˛ˆÏܲÓ˚ =îú˛° ≤Ã̈Ïü §yüyöƒ Ó,!k˛ ˆ˛õˆÏ°Ä ï˛y o&ï˛ •…y§ ˛õyÎ˚–
ú˛ˆÏ° x(t) ≤Ã̈Ïü §üˆÏÎ˚Ó˚ §ˆÏD §yüyöƒ Ó,!k˛ ˆ˛õˆÏ°Ä á%Ó ï˛yí˛¸yï˛y!í˛¸ ï˛y ¢)öƒ •ˆÏÎ˚ ÎyÎ˚– 3.2 !ã˛ˆÏe §üˆÏÎ˚Ó˚
§ˆÏD ~•z ò%!ê˛ xˆÏ˛õ«˛ˆÏܲÓ˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ñ 3.2(b)ÈÙÈˆï˛ ˆòáyˆÏöy ˆ°á!ã˛ˆÏe
sinh(βt) §üˆÏÎ˚Ó˚ §ˆÏD ˆÓˆÏí˛¸ ã˛ˆÏ°ñ sin (βt)ÈÙÈÓ˚ üï˛ ï˛yÓ˚ ˛õÎ≈yÓ,_ üyö ˛õyÄÎ˚y ÎyÎ˚ öy– Ó›ï˛ ~çöƒ ~ˆÏ«˛ˆÏe
x(t)ÈÙÈÓ˚ üyöÄ ˛õÎ≈yÓ,_ •Î˚ öy ~ÓÇ Ü˛¡õö §Ω˛Ó •Î˚ öy– 3.2(a) !ã˛ˆÏe xy˛õ!ö ˆòáˆÏï˛ ˛õyˆÏÓö ˆÎ φ(t) §üˆÏÎ˚Ó˚
§ˆÏD ܲˆÏü ÎyˆÏFåÈ– ï˛ˆÏÓ ~•z ܲüyÓ˚ •yÓ˚ bÈÙÈ~Ó˚ üyˆÏöÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°– 3.3 !ã˛ˆÏe 3.11 §ü#ܲÓ˚î xö%ÎyÎ˚#
68
tÈÙÈ~Ó˚ §ˆÏD x(t) ~Ó˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– Îáö Ó›!ê˛ˆÏܲ ~ܲ!ê˛ âyï˛ ≤ÈÏÎ˚yà ܲˆÏÓ˚ §yüƒyÓfliy ˆÌˆÏܲ
!Óã%˛ƒï˛ ܲÓ˚y •Î˚ñ ï˛áöܲyÓ˚ x!ï˛ xÓü®öç!öï˛ §üÎ˚ ÈÙÈ ò)Ó˚c ˆÓ˚á!ã˛e ~áyˆÏö ˆòáy ÎyˆÏFåÈ– ~•z !ã˛ˆÏe ˆòáyˆÏöy
x(t) ~Ó˚ üyö 3.2 (a) Ä (b) !ã˛ˆÏeÓ˚ ò%!ê˛ xˆÏ˛õ«˛Ü˛Ä
0 vβ
Ó˚y!¢Ó˚ =îˆÏöÓ˚ myÓ˚y ˛õyÄÎ˚y ˆàˆÏåÈ–
3.3.2 e´yhsˇ#Î˚ xÓü®öe´yhsˇ#Î˚ xÓü®öe´yhsˇ#Î˚ xÓü®öe´yhsˇ#Î˚ xÓü®öe´yhsˇ#Î˚ xÓü®ö
e´yhsˇ#Î˚ xÓü®ö ≤ÃÜ,˛ï˛˛õˆÏ«˛ x!ï˛ xÓü®ö Ä °â% xÓü®ˆÏöÓ˚ §#üyˆÏÓ˚áy §)!ã˛ï˛ ܲˆÏÓ˚– ~•z xÓü®ˆÏöÓ˚
í˛z˛õ!fli!ï˛ˆÏï˛ ˆÜ˛yöÄ Ü˛¡õö«˛ü Ó›ˆÏܲ Îáö ï˛yÓ˚ §yüƒyÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ ܲÓ˚y •Î˚ñ ï˛áö ï˛y §Ó≈yˆÏ˛õ«˛y
o&ï˛ §yüƒyÓfliyˆÏö ˆú˛Ó˚ï˛ xyˆÏ§ ~ÓÇ ˆ§áyˆÏö•z !fliÓ˚ •ˆÏÎ˚ ÎyÎ˚ ˆÜ˛yöÄ Ü˛¡õö ≤Ãò¢≈ö ܲˆÏÓ˚ öy– xÓü®ˆÏöÓ˚
üyey ~Ó˚ ˆÌˆÏܲ ˆÓ!¢ •ˆÏ°ñ ï˛yˆÏܲ x!ï˛ xÓü®ö Ó°y •ˆÏÓ ~ÓÇ ˆ§ˆÏ«˛ˆÏeÄ Ü˛¡õö ˆòáy ÎyˆÏÓ öyñ Î!òÄ
ܲ¡õö«˛ü Ó› xˆÏ˛õ«˛yÜ,˛ï˛ ˆÓ!¢ §üÎ˚ !öˆÏÎ˚ §yüƒyÓfliyˆÏö ~ˆÏ§ !fliÓ˚ •ˆÏÎ˚ ÎyˆÏÓ– xyÓ˚ xÓü®ˆÏöÓ˚ üyey e´yhsˇ#Î˚
xÓü®ö ˆÌˆÏܲ ܲü •ˆÏ°ñ §yü!@ˇÃܲ ÓƒÓfliy!ê˛ ~ܲÓyÓ˚ §yüƒyÓfliy ˆÌˆÏܲ !Óã%˛ƒï˛ •ˆÏ° ܲ!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ–
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ñ ~•z e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ ˆÜ˛yöÄ !ӈϢ£Ï ï˛yͲõÎ≈ xyˆÏåÈ !ܲ öy– ÓƒÓ•y!Ó˚ܲ
69
ò,!‹TˆÏܲyî ˆÌˆÏܲ ~•z xÓü®ˆÏöÓ˚ !ܲå%È !ӈϢ£Ï ≤ÈÏÎ˚yà Ó˚ˆÏÎ˚ˆÏåÈ– xy˛õ!ö ˆòˆÏá ÌyܲˆÏÓöñ òÓ˚çy ˆë˛ˆÏ° ˆáy°yÓ˚
˛õÓ˚ ï˛y !öˆÏç ˆÌˆÏܲ•z Ó¶˛ •ˆÏÎ˚ ÎyÄÎ˚yÓ˚ çöƒ ~ܲ!ê˛ !ӈϢ£Ï ôÓ˚ˆÏöÓ˚ òÓ˚çy Ó¶˛ ܲÓ˚yÓ˚ ÓƒÓfliy (door closer)
òÓ˚çyÎ˚ °yàyˆÏöy ÌyˆÏܲ– ~•z ÓƒÓfliy!ê˛ˆÏï˛ òÓ˚çy ӈ϶˛Ó˚ çöƒ e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ ≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~ˆÏï˛
òÓ˚çy!ê˛ ï˛yÓ˚ !Óã%˛ƒï˛ xÓfliyö ˆÌˆÏܲ xˆÏ˛õ«˛yÜ,˛ï˛ o&ï˛ §yüƒyÓfliyˆÏö ˆú˛ˆÏÓ˚ ~ÓÇ ï˛y ܲ!¡õï˛ •ÄÎ˚yÓ˚ ˆã˛‹Ty öy
ܲˆÏÓ˚ !fliÓ˚ •ˆÏÎ˚ òÑy!í˛¸ˆÏÎ˚ ÎyÎ˚– xÓü®ˆÏöÓ˚ üyey ˆÓˆÏí˛¸ ˆàˆÏ° òÓ˚çy!ê˛ ò#â≈ï˛Ó˚ §üÎ˚ !öˆÏÎ˚ Ó¶˛ •ˆÏÓ xyÓ˚ xÓü®ö
ܲˆÏü ˆàˆÏ° ˆ§!ê˛ §yüƒyÓfliyˆÏöÓ˚ ò%Û˛õyˆÏ¢ ò%°ˆÏï˛ ÌyܲˆÏÓ xÌÓyñ ï˛yÓ˚ §%ˆÏÎyà öy ÌyܲˆÏ° §yüƒyÓfliyˆÏö ~ˆÏ§
§ˆÏçyˆÏÓ˚ Ä §¢ˆÏ∑ ˆã˛ÔܲyˆÏë˛ xyâyï˛ Ü˛Ó˚ˆÏÓ– Ó°y Óy‡°ƒñ ~Ó˚ ˆÜ˛yöÄ!ê˛•z ÓƒÓ•y!Ó˚ܲ ò,!‹TˆÏܲyî ˆÌˆÏܲ ܲyüƒ
öÎ˚– x!ï˛ xÓü®ö Óy e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ xyÓ˚ ~ܲ!ê˛ í˛zòy•Ó˚î ˆòÄÎ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– xyüÓ˚y ˛õÓ˚#«˛yàyˆÏÓ˚
ˆÎ §Ó ˆû˛yŒê˛!üê˛yÓ˚ñ xƒy!üê˛yÓ˚ Ä ˆê˛Ó° àƒy°û˛ƒyˆÏöy!üê˛yÓ˚ ÓƒÓ•yÓ˚ ܲ!Ó˚ xÌÓy ˆê˛˛õ ˆÓ˚ܲí≈˛yˆÏÓ˚ ˆÎ ¢∑hflÏÓ˚
!öˆÏò≈¢Ü˛ (level indicator) ÌyˆÏܲñ ˆ§=!°ˆÏï˛ §)ã˛Ü˛ÈÙÈÜÑ˛yê˛y ÓƒÓ•*ï˛ •Î˚– ~•z §)ã˛Ü˛ ÜÑ˛yê˛yÓ˚ à!ï˛ˆÏï˛ e´yhsˇ#Î˚
xÓü®ˆÏöÓ˚ ÓƒÓfliy ÌyˆÏܲ ÎyˆÏï˛ ˆ§!ê˛ §yüƒyÓfliyÓ˚ ò%ÛôyˆÏÓ˚ ܲ!¡õï˛ öy •Î˚–
ày!î!ï˛Ü˛ ò,!‹TˆÏܲyî ˆÌˆÏܲ Ó°y ÎyÎ˚ ˆÎñ e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe b2– ω0
2 = 0 ~ÓÇ ˆ§ˆÏ«˛ˆÏe §ü#ܲÓ˚î
3.9 ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yÎ˚ñ
x(t) = (a1 + a
2) exp (–bt) = a exp (–bt) ...3.12
~áyˆÏö a = a1 + a
2
°«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ 3.12 §ü#ܲÓ˚ˆÏî ~ܲ!ê˛üye ˆfl∫FåÈ xã˛Ó˚ Ó˚ˆÏÎ˚ˆÏåÈñ xÌã˛ xy˛õ!ö çyˆÏöö ˆÎñ ~ܲ!ê˛ !müyeyÓ˚
xÓܲ° §ü#ܲÓ˚ˆÏî ò%•z!ê˛ ˆfl∫FåÈ xã˛Ó˚ ÌyܲˆÏÓñ ÎyˆÏòÓ˚ üyö ≤ÃyÌ!üܲ ¢ˆÏï≈˛Ó˚ IJõÓ˚ !û˛!_ ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyˆÏÓ–
3.12 §ü#ܲÓ˚î !ܲv ï˛y•ˆÏ° ü)° xÓܲ° §ü#ܲÓ˚î 3.3 ~Ó˚ §üyôyö öÎ˚–
§yôyÓ˚î §üyôyö!ê˛ ˛õyÄÎ˚yÓ˚ çöƒ xyüÓ˚y 3.10 §üyôyö!ê˛ˆÏܲ ~û˛yˆÏÓ !°áˆÏï˛ ˛õy!Ó˚ ≠
x(t) = exp (– bt) [a1 exp (βt) + a
2 exp (–βt)]
= 2 2 2 2
1 21 1exp( ) 1 .... 1 ...2 2bt a t t a t t⎡ ⎤⎛ ⎞ ⎛ ⎞− + + + + − + +⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
β β β β
= ( ) ( ) ( ) 2 21 2 1 2 1 2
1exp( ) ...2
bt a a a a t a a t⎡ ⎤− + + − + + +⎢ ⎥⎣ ⎦β β
Îáö b → ω0ñ ï˛áö 2 20b ω− Óy β → 0–
~áö β → 0 §#üyÎ˚ xyüÓ˚y β 2 ~ÓÇ β ~Ó˚ ï˛ò)ôù≈ âyï˛=!°ˆÏܲ í˛zˆÏ˛õ«˛y ܲÓ˚ˆÏï˛ ˛õy!Ó˚– Î!ò (a1 + a
2)
= p ~ÓÇ (a1 – a2)β = q ˆ°áy •Î˚ ï˛ˆÏÓ §üyôyö!ê˛ •ˆÏÓ ≠
70
x(t) = (p+qt) exp (–bt) ...3.13
~áyˆÏö p ~ÓÇ q ò%!ê˛ ˆfl∫FåÈ xã˛Ó˚ ÎyˆÏòÓ˚ üyö ≤ÃyÌ!üܲ ¢ˆÏï≈˛Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– xÓܲ°ˆÏöÓ˚ §y•yˆÏ΃
xy˛õ!ö ˆòˆÏá !öˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ 3.13 §üyôyö!ê˛ ≤ÃÜ,˛ï˛•z 3.3 xÓܲ° §ü#ܲÓ˚îˆÏܲ !§k˛ ܲˆÏÓ˚–
3.4 !ã˛e!ê˛ˆÏï˛ ~ܲ•z ܲ¡õö«˛ü ÓƒÓfliyÎ˚ x!ï˛ñ e´yhsˇ#Î˚ Ä °â% ~•z !ï˛ö ˆ◊î#Ó˚ xÓü®ˆÏöÓ˚ çöƒ §üÎ˚ÈÙÈò)Ó˚c
ˆ°á!ã˛e ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˆ°á!ã˛ˆÏe SܲV x!ï˛ xÓü®öñ SáV e´yhsˇ#Î˚ xÓü®ö ~ÓÇ SàV °â% xÓü®ö
§)!ã˛ï˛ ܲÓ˚ˆÏåÈ– ~•z °â% xÓü®ö Ä ï˛yÓ˚ ˜Ó!¢ˆÏ‹TƒÓ˚ !Ó£ÏÎ˚ ~ÓyÓ˚ xyüÓ˚y !ÓhflÏy!Ó˚ï˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
!ã˛e !ã˛e !ã˛e !ã˛e !ã˛e 3.4 §yüƒyÓfliy ˆÌˆÏܲ ~ܲ!ê˛ Ü˛¡õö«˛ü ÓƒÓfliyˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ ˆåȈÏí˛¸ !òˆÏ° !Ó!û˛ß¨ üyeyÓ˚ xÓü®ˆÏöÓ˚
ˆ«˛ˆÏe ï˛yÓ˚ §üÎ˚ ò)Ó˚c ˆ°á!ã˛e ˆÜ˛üö •ˆÏÓñ ï˛y ~áyˆÏö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– !ï˛ö!ê˛ ˆ«˛ˆÏe ÎÌye´ˆÏü x!ï˛
xÓü®ö SܲVñ e´yhsˇ#Î˚ xÓü®ö SáV ~ÓÇ °â% xÓü®ö SàV §)!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ–
3.3.3 °â% xÓü®ö°â% xÓü®ö°â% xÓü®ö°â% xÓü®ö°â% xÓü®ö
ày!î!ï˛Ü˛ ò,!‹TˆÏܲyî ˆÌˆÏܲ °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe b2 < ω02 •Î˚– ~Ó˚ ú˛ˆÏ° 3.7(a) Ä (b) §ü#ܲÓ˚ˆÏî
a1 ~ÓÇ a
2 ~Ó˚ üyö ò%!ê˛ ç!ê˛° Ó˚y!¢ •ˆÏÎ˚ ˛õˆÏí˛¸–
~áö xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
12 2 2
0b ω⎡ ⎤−⎣ ⎦ =
()() 220 1b −− ω
= jω'
ˆÎáyˆÏö j = 1− ~ÓÇ ω' =
220b ω−
°«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ ω' ~ܲ!ê˛ ÓyhflÏÓ §Çáƒy ~ÓÇ Îáö ~•z °â% xÓü®ˆÏöÓ˚ üyey xï˛ƒhsˇ ܲü •ˆÏÎ˚ ÎyÎ˚
xÌ≈yÍ Îáö b → 0 •Î˚ ï˛áö ω'=ω0 xÌ≈yÍ ω' ܲ¡õö«˛ü ÓƒÓfliyÓ˚ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ §üyö •ˆÏÎ˚ ÎyÎ˚–
3.9 §ü#ܲÓ˚ˆÏî ω' ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y ˆÎ §üÎ˚ÈÙÈò)Ó˚c §ü#ܲÓ˚î ˛õy•zñ ï˛y ~•zÓ˚ܲü ≠
SܲV
SáVSàV
x
t
71
x(t) = exp (– bt) [a1 exp (jω't) + a2 exp (–jω't)] ...3.14
exp (±jω't) = cos ωt ±j sin ω't §)e ÓƒÓ•yÓ˚ ܲˆÏÓ˚ §ü#ܲÓ˚î 3.14 ˆÜ˛ ˆ°áy ÎyÎ˚ ≠
x(t) = exp (–bt) [(a1 + a2) cos ω't + j (a1 – a2) sin ω't] ...3.15
~áyˆÏö a1 ~ÓÇ a
2 §yôyÓ˚îû˛yˆÏÓ ò%!ê˛ ç!ê˛° Ó˚y!¢ üˆÏö ܲÓ˚y ÎyÎ˚ ~ÓÇ §ü#ܲÓ˚î 3.15 ÈÙÈ~Ó˚ ÓyhflÏÓ Óy
ܲy“!öܲ ˆÎ ˆÜ˛yö xÇ¢•z ü)° §üyôyö §)!ã˛ï˛ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– ~•z ò%!ê˛ xLjϢÓ˚ ˆÎ ˆÜ˛yö!ê˛ˆÏܲ•z ˆ°áy ÎyÎ˚ñ
Óy x(t) = exp (–bt) [A' cos ω't + B' sin ω't] ...3.16
ˆÎáyˆÏö A' = a1 + a2, B' = a1 – a2
~áyˆÏö A' ~ÓÇ B' í˛zû˛ˆÏÎ˚•z ÓyhflÏÓ Ó˚y!¢ ~ÓÇ ≤ÃyÌ!üܲ ¢ˆÏï≈˛Ó˚ §y•yˆÏ΃ !öî≈Î˚ˆÏÎyàƒ– xyüÓ˚y xÓ¢ƒ ~áö
~•z Ó˚y!¢=!° !öî≈ˆÏÎ˚Ó˚ ˆã˛‹Ty ܲÓ˚Ó öy– xyüÓ˚y 3.16 §ü#ܲÓ˚îˆÏܲ ü%_´ ܲ¡õˆÏöÓ˚ §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy
ܲÓ˚yÓ˚ í˛z˛õˆÏÎyà# ˆã˛•yÓ˚y ˆòÄÎ˚yÓ˚ ˆã˛‹Ty ܲÓ˚Ó– Î!ò xyüÓ˚y ô!Ó˚ ˆÎñ
A' = A0 cos φ, B' = –A0 sin φ
ˆÎáyˆÏö A0 =
22 '' AB +
~ÓÇ tan φ =
''
BA
−~=!° 3.16 §ü#ܲÓ˚ˆÏî Ó!§ˆÏÎ˚ ˛õy•zñ
x(t) = A0 exp (–bt) [cos φ cos ω't – sin φ sin ω't]
= A0 exp (–bt) cos (ω't + φ ) ...3.17(a)
3.17 §ü#ܲÓ˚î!ê˛ °â% xÓü®ˆÏöÓ˚ (b2 < ω0
2) ˆ«˛ˆÏe 3.3 xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §yôyÓ˚î §üyôyö– ~áyˆÏö
3,16 §ü#ܲÓ˚ˆÏîÓ˚ A' ~ÓÇ B' ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ A0 ~ÓÇ φ ~•z ò%•z!ê˛ ˆfl∫FåÈ xã˛Ó˚ ˛õyÄÎ˚y ˆàˆÏåÈ–
xy˛õöyÓ˚y ~áö §•ˆÏç•z 3.17 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ≤ÃÌü ~ܲˆÏܲ §Ó˚° ˆòy°à!ï˛Ó˚ ˆÎ §üÎ˚ ò)Ó˚c §ü#ܲÓ˚î
ˆ˛õˆÏÎ˚!åȈϰö ï˛yÓ˚ §ˆÏD ï%˛°öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~•z ï%˛°öy ܲÓ˚ˆÏï˛ ˆàˆÏ° ˆÎ !Ó£ÏÎ˚=!° xy˛õöyÓ˚ ò,!‹T xyܲ£Ï≈î
ܲÓ˚yÓ˚ ˆ§=!° •°ñ
(i) ~•z xÓü!®ï˛ ܲ¡õˆÏöÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω0 öÎ˚ñ ~•z ܲ¡õyB˛
() 2200b ωω =−
xÌ≈yÍ xÓü®ˆÏöÓ˚
çöƒ ܲ¡õyˆÏB˛Ó˚ •…y§ âˆÏê˛ˆÏåÈ– xÓü®ö Îï˛ °â% •ˆÏÓ ~•z ˛õ!Ó˚Óï≈˛ö •ˆÏÓ ï˛ï˛•z §yüyöƒ–
(ii) 3.17 §ü#ܲÓ˚ˆÏî ˆòáy ÎyˆÏFåÈ ˆÎñ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ !•ˆÏ§ˆÏÓ xyüÓ˚y ˛õy•z
A(t) = A0e–bt ...3.17(b)
72
~Ó˚ xÌ≈ñ ~ˆÏ«˛ˆÏe !ÓhflÏyÓ˚ §üˆÏÎ˚Ó˚ §ˆÏD e´ü•…y§üyö– ˆòy°ˆÏöÓ˚ ≤ÃyÌ!üܲ !ÓhflÏyˆÏÓ˚Ó˚ §üˆÏÎ˚Ó˚ §ˆÏD e–bt •yˆÏÓ˚
•…y§≤Ãy!Æ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ §ÓˆÏã˛ˆÏÎ˚ Óí˛¸ ˜Ó!¢‹Tƒ– Ó›ï˛ñ ~•z !ÓhflÏyÓ˚ e´ü¢ ܲˆÏü ÎyÄÎ˚yÓ˚ ú˛ˆÏ° ˆòy°ö¢#°
ÓƒÓfliy!ê˛Ó˚ ¢!_´Ä e´ü¢ •…y§ ˛õyÎ˚– xÓü®öܲyÓ˚# ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ ܲyÎ≈ ܲÓ˚y•z ~•z •…yˆÏ§Ó˚ ܲyÓ˚î–
ï˛y•ˆÏ° Ó°y ÎyÎ˚ ˆÎñ °â% xÓü®ˆÏöÓ˚ ú˛ˆÏ° ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyˆÏB˛ e´ü•…y§üyö !ÓhflÏyˆÏÓ˚ ˆòy°ö §¡õy!òï˛
•Î˚– 3.17a §ü#ܲÓ˚ˆÏî §üˆÏÎ˚Ó˚ àîöy Î!ò ~üöû˛yˆÏÓ Ü˛Ó˚y •Î˚ ÎyˆÏï˛ ≤ÃyÓ˚!Ω˛Ü˛ ò¢yˆÏܲyî φ = 0 •Î˚ ï˛ˆÏÓ
~•z §ü#ܲÓ˚ˆÏîÓ˚ Ó˚*˛õ •ˆÏÓñ
x(t) = A0 exp (–bt) cos ω't
3.5 !ã˛ˆÏe ~•z §ü#ܲÓ˚ˆÏîÓ˚ ˆ°á!ã˛e!ê˛ ˆòáˆÏï˛ ˛õyˆÏÓö– ˆ°á!ã˛e!ê˛Ó˚ üˆÏôƒ í˛ƒy¢ ˆòÄÎ˚y ˆÓ˚áy myÓ˚y §üˆÏÎ˚Ó˚
§ˆÏD !ÓhflÏyˆÏÓ˚Ó˚ §)ã˛Ü˛ (exponential) •yˆÏÓ˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ü)° ˆ°á!ã˛e!ê˛ e´ü•…y§üyö !ÓhflÏyˆÏÓ˚Ó˚
ˆòy°öà!ï˛ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ˆòy°öܲy° T' •ˆÏ° ˆ°áy ÎyÎ˚ ˆÎñ
T' = 2'
πω =
() 220
2
b −π
ω
= 2
2
2
4km m
−
πγ ...3.18
3.18 ÈÙÈ~ ˆòáy ÎyˆÏFåÈ ˆÎñ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ˆòy°öܲy° xÓü!®ï˛ ˆòy°öܲyˆÏ°Ó˚ ˆÌˆÏܲ ˆÓ!¢ ܲyÓ˚î
b > 0 ~ÓÇ È
2200 b ωω −<
– xyÓyÓ˚ °«˛ƒ ܲÓ˚&ö ˆÎ b = 0 •ˆÏ° í˛zû˛Î˚ ˆ«˛ˆÏe T ~Ó˚ üyö xöÓü!®ï˛
!ã˛e 3.5 °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe §üÎ˚ ò)Ó˚c ˆ°á!ã˛e–
x§hsˇï˛ (broken) ˆÓ˚áy §üˆÏÎ˚Ó˚ §ˆÏD
!ÓhflÏyˆÏÓ˚Ó˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏFåÈ–
A0
73
ˆòy°öܲyˆÏ°Ó˚ §üyö •Î˚– ܲyÓ˚î b = 0ÈÙÈÓ˚ xÌ≈ ˆòy°ö xÓü®ö•#ö– xï˛~Ó ï˛y §Dï˛ Ü˛yÓ˚ˆÏî•z xöÓü!®ï˛
(undamped) ˆòy°öܲyˆÏ°Ó˚ §üyö– xyÓyÓ˚ xÓü®ˆÏöÓ˚ çöƒ ˆòy°öܲy° ò#â≈!Î˚ï˛ •ÄÎ˚yÓ˚ !Ó£ÏÎ˚!ê˛Ä
≤Ãï˛ƒy!¢ï˛ñ ˆÜ˛ö öy ÓyôyÓ˚ !ÓÓ˚&ˆÏk˛ ~ܲ•z ˛õÌ x!ï˛e´ü ܲÓ˚ˆÏï˛ !àˆÏÎ˚ §üÎ˚ ˆÓ!¢ °yàˆÏåÈ–
xyüÓ˚y ~áö ˛õÎ≈hsˇ Îy xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ ï˛y xyÓ˚Ä û˛y° ܲˆÏÓ˚ ˆÓyé˛ÓyÓ˚ çöƒ ~ܲ!ê˛ ày!î!ï˛Ü˛ ≤ß¿ í˛zay˛õö
ܲÓ˚y Îyܲ– xy˛õöyÓ˚y ~•z xö%¢#°ö#!ê˛ Ü˛Ó˚yÓ˚ ˆã˛‹Ty ܲÓ˚&ö– ≤ÈÏÎ˚yçˆÏö xyˆÏàÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˛õ,¤˛yÓ˚ !Ó£ÏÎ˚Ó› xyÓ˚Ä
~ܲÓyÓ˚ ˆòˆÏá !öö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -1 :
°â% xÓü!®ï˛ ~ܲ!ê˛ !fl±ÇÈÙÈ˛û˛Ó˚ ÓƒÓfliyÓ˚ !ÓhflÏyÓ˚ .08m ˆÌˆÏܲ .02m ˆöˆÏü xy§ˆÏï˛ 160s §üÎ˚ ˆöÎ˚–
~•z §üˆÏÎ˚Ó˚ üˆÏôƒ Î!ò ÓƒÓfliy!ê˛ 80!ê˛ Ü˛¡õö §¡õߨ ܲˆÏÓ˚ ÌyˆÏܲ ï˛y•ˆÏ° ~•z ˆòy°ˆÏöÓ˚ ˆòy°öܲy° ܲï˛⁄
~•z ˆòy°öܲy° xöÓü!®ï˛ ˆòy°öܲyˆÏ°Ó˚ ˆÌˆÏܲ ܲï˛ê˛y !û˛ß¨⁄ xyüÓ˚y !ܲ ~•z ˛õyÌ≈ܲƒˆÏܲ í˛zˆÏ˛õ«˛y ܲÓ˚ˆÏï˛
˛õy!Ó˚⁄
3.4 °â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´°â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´°â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´°â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´°â% xÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆüyê˛ ¢!_´
°â% xÓü®ˆÏöÓ˚ çöƒ Îáö ܲ¡õö¢#° ÓƒÓfliyÓ˚ !ÓhflÏyÓ˚ e´üyàï˛ •…y§ ˛õyÎ˚ñ ï˛áö ÓƒÓfliy!ê˛Ó˚ ¢!_´Ó˚Ä «˛Î˚
•ˆÏï˛ ÌyˆÏܲ– xÓ¢ƒ ~•z ¢!_´ Ü˛ï˛ o&ï˛ «˛!Î˚ï˛ •ˆÏÓ ï˛y !öû≈˛Ó˚ ܲˆÏÓ˚ ÓƒÓfliy!ê˛ˆÏï˛ í˛z˛õ!fliï˛ xÓü®ˆÏöÓ˚ í˛z˛õÓ˚–
~•z ˛õÎ≈yˆÏÎ˚Ó˚ ≤ÃÌü ~ܲˆÏܲ xy˛õöyÓ˚y ˆòˆÏáˆÏåÈö ˆÎ ˆÜ˛yö ü%_´ Óy xöÓü!®ï˛ ˆòy°à!ï˛Ó˚ ˆ«˛ˆÏe ˆüyê˛ ¢!_´
E ˆÜ˛ ˆ°áy ÎyÎ˚ñ
E =
2 12
kA
...3.19
~áyˆÏö ω02 =
km
Óy k = mω02
~ÓÇ k ˆÜ˛ ܲ¡õö¢#°ÓƒÓfliy!ê˛Ó˚ !fl±ÇÈÙÈ=îyB˛ (Spring factor) Ó°y •Î˚– ~ˆÏ«˛ˆÏe ˆÜ˛yö xÓü®ö í˛z˛õ!fliï˛
öy ÌyܲyÎ˚ A §üˆÏÎ˚Ó˚ §ˆÏD ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ öyñ xÌ≈yÍ A ~ܲ!ê˛ ô &Óܲ–
!ܲv xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ˆ«˛ˆÏe A ô &Óܲ öÎ˚ ~ÓÇ xyüÓ˚y çy!ö ˆÎñ ˆÜ˛yö §üˆÏÎ˚Ó˚ !ÓhflÏyÓ˚ A ˆòy°ˆÏöÓ˚
≤ÃyÓ˚!Ω˛Ü˛ !ÓhflÏyÓ˚ A0ÈÙÈÓ˚ ˆÌˆÏܲ §)ã˛Ü˛ •yˆÏÓ˚ •…y§ ˛õyÎ˚ xÌ≈yÍ
A = A0e–bt
ˆÎˆÏ•ï%˛ ˆòy°ˆÏöÓ˚ ˆüyê˛ ¢!_´ E !ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õy!ï˛Ü˛ñ ï˛y•z ˆ°áy ÎyÎ˚ ˆÎñ
E ∝ A2
Óy E ∝ [A0e–bt]2
Óy E = C A0
2e–2bt = E0e–2bt ...3.20
74
~áyˆÏö C ~ܲ!ê˛ ô &Óܲ Ó˚y!¢ Sˆû˛ò#Î˚ ô &ÓܲV ~ÓÇ CA2 = E0 ˆÎáyˆÏö E0 ≤ÃyÓ˚!Ω˛Ü˛ ¢!_´ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
3.20 §ü#ܲÓ˚ˆÏî ˆòáy ÎyˆÏFåÈ ˆÎñ ˆüyê˛
¢!_´ E °â% xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ˆ«˛ˆÏe
§üˆÏÎ˚Ó˚ §ˆÏD §)ã˛Ü˛ •yˆÏÓ˚ •…y§ ˛õyÎ˚– ï˛ˆÏÓ
~•z •…y§ !ÓhflÏyˆÏÓ˚Ó˚ •…yˆÏ§Ó˚ ï%˛°öyÎ˚ o&ï˛ï˛Ó˚
ܲyÓ˚î !ÓhflÏyˆÏÓ˚Ó˚ •…yˆÏ§Ó˚ •yÓ˚ e-bt !ܲv ˆüyê˛
¢!_´Ó˚ •…yˆÏ§Ó˚ •yÓ˚ e–2bt–
3.6 öÇ !ã˛ˆÏe §üˆÏÎ˚Ó˚ §ˆÏD ˆüyê˛ ¢!_´
E ~Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˆ°á!ã˛e ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲÓ˚&ö ˆÎñ ~•z ˆ°á!ã˛e!ê˛
§üˆÏÎ˚Ó˚ §ˆÏD !ÓhflÏyˆÏÓ˚Ó˚ ˛õ!Ó˚Óï˛≈ˆÏöÓ˚
ˆ°á!ã˛ˆÏeÓ˚ S!ã˛e 3.5 ~Ó˚ í˛ƒy¢ ˆòÄÎ˚y
ˆÓ˚áyV ï%˛°öyÎ˚ ˆÓ!¢ ö!ï˛!Ó!¢‹T– ÎyÓ˚ xÌ≈ñ ¢!_´Ó˚ •…y§ âê˛ˆÏåÈ o&ï˛ï˛Ó˚ •yˆÏÓ˚–
≤ÃÌü ~ܲˆÏܲÓ˚ 1.9 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆÎñ ˆüyê˛ ¢!_´ E =
22 12
mA ω= 2 2 2
012
btm A eω −
§%ï˛Ó˚yÇñ 3.20 §ü#ܲÓ˚ˆÏîÓ˚ C ô &Óܲ!ê˛ xy§ˆÏ° 21
2mω – ~ÓyÓ˚ xy˛õ!ö xyÓ˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òˆÏÎ˚
!öˆÏï˛ ˛õyˆÏÓ˚ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -2 :
~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliyÎ˚ !fl±ÇÈÙÈô &Óܲ 2Nm–1 Î%_´ û˛Ó˚!ê˛ 2kg ~ÓÇ !e´Î˚y¢#° xÓü®ö Ó°
0.4 Nsm–1
(i) û˛Ó˚!ê˛Ó˚ à!ï˛ !ܲ ܲ¡õöÎ%_´ (oscillatory)?
(ii) ~ܲܲ à!ï˛ˆÏÓˆÏà Ü˛ï˛ xÓü®ö ӈϰÓ˚ çöƒ à!ï˛!ê˛ e´yhsˇ#Î˚û˛yˆÏÓ xÓü!®ï˛ •ˆÏÓ⁄
(iii) !fl±Ç ô &Óܲ Ä xÓü®ö Ó° ≤Ãò_ üyˆÏö ÌyܲˆÏ° ˆÜ˛yö û˛ˆÏÓ˚Ó˚ çöƒ e´yhsˇ#Î˚ xÓü®ö ˛õyÄÎ˚y ÎyˆÏÓ⁄
(iv) ≤ÃyÌ!üܲ ¢!_´ «˛!Î˚ï˛ •ˆÏÎ˚ ~ܲ ï,˛ï˛#Î˚yÇ¢ üyˆÏö ˆ˛õÑÔåȈÏï˛ Ü˛ï˛ §üÎ˚ °yàˆÏÓ⁄
3.5 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹TƒxÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹TƒxÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹TƒxÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹TƒxÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˜Ó!¢‹Tƒ
xy˛õ!ö ~áö çyˆÏöö ˆÎñ ˆÜ˛Ó°üye °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe•z ܲ¡õö ˛õyÄÎ˚y §Ω˛Ó ~ÓÇ xÓü®ˆÏöÓ˚ ú˛ˆÏ°
ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ §üˆÏÎ˚Ó˚ §ˆÏD e´ü¢ •…y§ ˛õyÎ˚– ÓyhflÏÓ çàˆÏï˛Ó˚ ܲ¡õö¢#° ÓƒÓfliyÎ˚ ˆÜ˛Ó°üye °â% xÓü®ö
E0exp(–2bt)
0.37E0
75
í˛z˛õ!fliï˛ ÌyܲˆÏ°Äñ ˆ§•z xÓü®ˆÏöÓ˚ ˆÜ˛Ó°üye =îàï˛ ˛õ!Ó˚ã˛Î˚ öÎ˚ñ ï˛yÓ˚ ˛õ!Ó˚üyîàï˛ ˛õ!Ó˚ã˛Î˚Ä xï˛ƒhsˇ
≤ÈÏÎ˚yçö#Î˚– ï˛y•z xÓü®öˆÏܲ ≤ÃyÌ!üܲû˛yˆÏÓ °â% !•§yˆÏÓ !ã˛!•´ï˛ ܲÓ˚yÓ˚ ˛õˆÏÓ˚ñ ï˛yÓ˚ xyÓ˚Ä ˛õ!Ó˚ã˛Î˚ ˆòÄÎ˚yÓ˚
çöƒ xyüÓ˚y °â% xÓü®ˆÏöÓ˚ ܲï˛=!° ˜Ó!¢ˆÏ‹TƒÓ˚ !òˆÏܲ öçÓ˚ !ò•z–
§yôyÓ˚îï˛ °â% xÓü®ˆÏöÓ˚ ˜Ó!¢‹Tƒ !ã˛!•´ï˛ ܲÓ˚yÓ˚ çöƒ xÓü®ˆÏöÓ˚ §ˆÏD §Ç!Ÿ’‹T !ï˛ö!ê˛ !ӣψÏÎ˚Ó˚ àîöy
ܲÓ˚y •ˆÏÎ˚ ÌyˆÏܲ– ~=!°Ó˚ üyö xyüyˆÏòÓ˚ °â% xÓü®ˆÏöÓ˚ ˛õ!Ó˚üyî §¡õˆÏÜ≈˛ ôyÓ˚îy ˆòÎ˚ ~ÓÇ ò%!ê˛ !û˛ß¨ ˆ«˛ˆÏe
í˛z˛õ!fliï˛ °â% xÓü®ˆÏöÓ˚ ˛õ!Ó˚üyîàï˛ ï%˛°öy ܲÓ˚yÓ˚ §%ˆÏÎyà ˆòÎ˚– ˆÎ !ï˛ö!ê˛ Ó˚y!¢ ~çöƒ !ӈϢ£Ïû˛yˆÏÓ àîöy
ܲÓ˚y •Î˚ ˆ§=!° •°ñ
SܲV °à#Î˚ •…y§ (Log decrement) (λ)
SáV Ÿ’Ìö ܲy° (Relaxation time) (τ)
SàV Q S!ܲí˛zV =îyB˛ SQ - factor) (Q)
~•z Ó˚y!¢=!°Ó˚ ~ܲ!ê˛Ó˚ Óy ~ܲy!ôˆÏܲÓ˚ àîöyÓ˚ üyôƒˆÏü xyüÓ˚y xÓü®ˆÏöÓ˚ ˜Ó!¢‹Tƒ Ó%é˛ˆÏï˛ ˛õy!Ó˚– ܲáöñ
ˆÜ˛yö Ó˚y!¢!ê˛ àîöy ܲÓ˚y §%!Óôyçöܲñ ï˛y ܲ¡õö«˛ü ÓƒÓfliy!ê˛Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ~áö ~•z !ï˛ö!ê˛ Ó˚y!¢Ó˚
àîöyÓ˚ ≤Ã!e´Î˚y !öˆÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
3.5.1 °à#Î˚ •…y§°à#Î˚ •…y§°à#Î˚ •…y§°à#Î˚ •…y§°à#Î˚ •…y§
°â% xÓü®ˆÏöÓ˚ üyey ˛õ!Ó˚üyˆÏ˛õÓ˚ §ÓˆÏã˛ˆÏÎ˚ §%!Óôyçöܲ í˛z˛õyÎ˚ •ˆÏFåÈñ ˆÜ˛yöÄ ÓyhflÏÓ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚
§üˆÏÎ˚Ó˚ §ˆÏD ܲ# •yˆÏÓ˚ •…y§ ˛õyÎ˚ñ ï˛y !öî≈Î˚ ܲÓ˚y– °à#Î˚ •…yˆÏ§Ó˚ àîöyÓ˚ üyôƒˆÏü ~ê˛y ܲÓ˚y §Ω˛Ó– °à#Î˚ •…y§
SλV Ó°ˆÏï˛ ~ܲ!ê˛ ˛õ)î≈ ˆòy°ˆÏöÓ˚ ≤ÃyÓ˚!Ω˛Ü˛ Ä x!hsˇü !ÓhflÏyˆÏÓ˚Ó˚ xÌ≈yÍñ ÎÌye´ˆÏü ˆòy°ö!ê˛ ÷Ó˚& Ä ˆ¢£Ï •ÄÎ˚yÓ˚
˛õÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ xö%˛õyˆÏï˛Ó˚ ≤ÃyÜ,˛ï˛ °ày!Ó˚òü‰ˆÏܲ (natural logarithm) Óyé˛yÎ˚– ~ܲ!ê˛ õ)î≈ òy°ˆÏöÓ˚ !ë˛Ü˛ ÷Ó˚&Ó˚
Ä ˆ¢ˆÏ£ÏÓ˚ !ÓhflÏyÓ˚ xÓü®ˆÏöÓ˚ çöƒ•z !û˛ß¨ •Î˚ñ ï˛y•z ~•z xö%˛õyˆÏï˛Ó˚ °ày!Ó˚òü‰ xÌ≈yÍñ °à#Î˚ •…y§ ˆÌˆÏܲ
xÓü®ö §¡õˆÏÜ≈˛ ~ܲ!ê˛ ã˛üÍܲyÓ˚ ˛õ!Ó˚üyîàï˛ ôyÓ˚îy ˛õyÄÎ˚y ÎyÎ˚–
A2
76
3.7 !ã˛ˆÏe xyÓyÓ˚ °â% xÓü®öÎ%_´ ܲ¡õˆÏöÓ˚ §üÎ˚ ò)Ó˚c ˆ°á!ã˛e!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲÓ˚&ö ˆÎñ
˛õÓ˚˛õÓ˚ !ï˛ö!ê˛ ˆòy°ˆÏöÓ˚ ˆ«˛ˆÏe ~ܲ•z !òˆÏܲ !ÓhflÏyÓ˚ ÎÌye´ˆÏü A0, A
1 ~ÓÇ A
2– ˆÎ ˆòy°ˆÏöÓ˚ ≤ÃyÓ˚!Ω˛Ü˛ !ÓhflÏyÓ˚
!åÈ° A0ñ ~ܲ!ê˛ ˆòy°ö ˛õ)î≈ •ÄÎ˚yÓ˚ §ˆÏD ï˛y ܲˆÏü •ˆÏÎ˚ˆÏåÈ A1 ~ÓÇ A1 !ÓhflÏyÓ˚ !öˆÏÎ˚ ˆÎ ˆòy°ö ÷Ó˚& •ˆÏÎ˚ˆÏåÈ
~ܲ!ê˛ ˛õ)î≈ ˆòy°ˆÏöÓ˚ ˆ¢ˆÏ£Ï ï˛y •ˆÏÎ˚ ˆàˆÏåÈ A2– ~áö xyüÓ˚y çy!ö ˆÎñ °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe
A(t) = A0e–bt S§ü#ܲÓ˚î 3.17 (a))
~ˆÏ«˛ˆÏe A0 ˆÌˆÏܲ !ÓhflÏyÓ˚ A1 ~ xy§ˆÏï˛ ~ܲ!ê˛ ˛õ)î≈ ˆòy°ˆÏöÓ˚ §üÎ˚ !öˆÏFåÈ– ï˛y•z ~ˆÏ«˛ˆÏe §üÎ˚ t ˙
ܲ¡õˆÏöÓ˚ ˆòy°öܲyˆÏ°Ó˚ (T) §üyö– xï˛~Ó 3.17(a) §ü#ܲÓ˚î ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆ°áy ÎyÎ˚ ˆÎñ
1
0
exp() =− AbT
A
Óy0
1
exp( )=AbT
A
~áö xyüÓ˚y §ü#ܲÓ˚î 3.3 ˆÌˆÏܲ çy!ö ˆÎñ
2b = mγ ∴ b =
2mγ§%ï˛Ó˚yÇ ˆ°áy ÎyÎ˚ ˆÎñ
0
1
AA
= exp
2Tmγ ⎛⎞
⎜⎟ ⎝⎠...3.21
~•z xö%˛õyï˛ˆÏܲ •…y§ (decrement) d Ó°y •Î˚ ~ÓÇ ~•z xö%˛õyˆÏï˛Ó˚ ≤ÃyÜ,˛ï˛ °ày!Ó˚òü‰ˆÏܲ xyüÓ˚y °à#Î˚
•…y§ (logarithmic decrement) Ó!°–
§%ï˛Ó˚yÇñ °à#Î˚ •…y§ λ ˆÜ˛ ˆ°áy ÎyÎ˚–
λ =
0
1
loge
AA
⎛⎞⎜⎟ ⎝⎠
= bt =
2Tmγ
...3.22
ˆÎˆÏ•ï%˛ A0 > Añ °à#Î˚ •…y§ λ ~ܲ!ê˛ ôöydܲ Ó˚y!¢– ~áö xyüÓ˚y Ó°ˆÏï˛ ˛õy!Ó˚ ˆÎñ λ Ó˚y!¢!ê˛ xÓü®ö
b ~ÓÇ ÓƒÓfliy!ê˛Ó˚ ˆòy°öܲy° T ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°–
xy˛õ!ö !ܲ °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ •…y§ Óy ~ܲ•z !òˆÏܲ ˆöÄÎ˚y ~ܲ!ê˛ ˛õ)î≈ ˆòy°öܲyˆÏ°Ó˚ ÓƒÓôyˆÏö ò%!ê˛ !ÓhflÏyˆÏÓ˚Ó˚
xö%˛õyï˛ §Ó≈òy•z §üyö⁄ xyüÓ˚y ~áyˆÏö •…y§ !•ˆÏ§ˆÏÓ
0
1
AA
xö%˛õyï˛ˆÏܲ !ã˛!•´ï˛ ܲÓ˚ˆÏ°Ä
1
2
AA
Óy ~ܲ•zû˛yˆÏÓ
77
2
3
AA
,
3
4
AA
≤Ãû,˛!ï˛ xö%˛õyï˛ Ä •…yˆÏ§Ó˚ (d) §üyö ~ÓÇ ˛≤Ã!ï˛ˆÏ«˛ˆÏe•z loged ~Ó˚ üyö °à#Î˚ •…y§ §)!ã˛ï˛ ܲˆÏÓ˚–
ˆÎüöñ
1
2
AA
=
0
0
exp()exp(2)
AbtAbt
−−
= exp (bt) = d
∴ λ = 1
2
ln AA = ln d = bt ...3.23
S°à#Î˚ •…yˆÏ§Ó˚ xyˆÏÓ˚ܲ!ê˛ !Óܲ“ Ä ÓƒÓ•y!Ó˚ܲ !òܲ ˆÌˆÏܲ §%!Óôyçöܲ §ÇK˛y Ó˚ˆÏÎ˚ˆÏåÈ– ~•z ~ܲˆÏܲÓ˚ xhsˇû≈%˛_´
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#Ó˚ 4 öÇ ≤ß¿!ê˛Ó˚ í˛z_Ó˚ ܲÓ˚yÓ˚ ˆã˛‹Ty ܲÓ˚&öñ ˆ§áyˆÏö !Ó£ÏÎ˚!ê˛ xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ–V
3.5.2 Ÿ’Ìö ܲy°Ÿ’Ìö ܲy°Ÿ’Ìö ܲy°Ÿ’Ìö ܲy°Ÿ’Ìö ܲy°
°â% xÓü®ö §ÇÓ!°ï˛ ˆòy°ˆÏö §üˆÏÎ˚Ó˚ §ˆÏD ˆÎ !ÓhflÏyˆÏÓ˚Ó˚ §)ã˛Ü˛ •…y§ •Î˚ñ ï˛y xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈ–
~ܲÌyÄ xyüyˆÏòÓ˚ çyöy xyˆÏåÈ ˆÎñ ˆÜ˛yöÄ §)ã˛Ü˛ •…yˆÏ§Ó˚ ˆ«˛ˆÏe e´ü•…y§üyö Ó˚y!¢!ê˛ ˆÜ˛yöÄ ˛õ!Ó˚üy˛õˆÏÎyàƒ
§§#ü §üˆÏÎ˚ ¢)öƒ üyˆÏö ˆ˛õÑÔåÈÎ˚ öy– xÓü®ˆÏöÓ˚ ˛õ!Ó˚üy˛õ ܲÓ˚yÓ˚ çöƒ ï˛y•z ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ ˆÜ˛yöÄ !ö!ò≈‹T
xö%˛õyˆÏï˛ •…y§ ˆ˛õˆÏï˛ Ü˛ï˛ê˛y §üÎ˚ °yˆÏà ˆ§!ê˛ !öî≈Î˚ ܲÓ˚y •Î˚–
§yôyÓ˚îï˛ ˆÜ˛yöÄ ˆû˛Ôï˛ Ó˚y!¢ Îáö §)ã˛Ü˛ •yˆÏÓ˚ •…y§ ˛õyÎ˚ñ ï˛áö ˆû˛Ôï˛ Ó˚y!¢!ê˛ ï˛yÓ˚ ≤ÃyÌ!üܲ Óy §ˆÏÓ≈yFã˛
üyˆÏöÓ˚ e–1 xLjϢ xÌ≈yÍñ 36.8 ¢ï˛yLjϢ ˆ˛õÑÔåȈÏï˛ ˆÎ §üÎ˚ ˆöÎ˚ñ ï˛yÓ˚ !•§yÓ Ü˛Ó˚y •Î˚– ~•z §üÎ˚!ê˛ˆÏܲ Ó°y
•Î˚ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ Ÿ’Ìöܲy° (relaxation time)– 3.6 !ã˛ˆÏe ~•z Ÿ’Ìöܲy°!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~ˆÏܲ
ˆ°áy •Î˚ τ Sê˛yí˛zV !•ˆÏ§ˆÏÓ–
ˆÎ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe Ÿ’Ìöܲy° ï%˛°öyÎ˚ ò#â≈ï˛Ó˚ xÌ≈yÍñ ˆÎˆÏ«˛ˆÏe ≤ÃyÌ!üܲ Óy §ˆÏÓ≈yFã˛ !ÓhflÏyÓ˚ ï˛yÓ˚ 36.8
¢ï˛yLjϢ ˛õ!Ó˚îï˛ •ˆÏï˛ xˆÏ˛õ«˛yÜ,˛ï˛ ˆÓ!¢ §üÎ˚ ˆöÎ˚ñ ˆÓyé˛y ÎyÎ˚ ˆÎ ˆ§áyˆÏö xÓü®ˆÏöÓ˚ üyey ܲü– Ÿ’Ìöܲy°
Îï˛ ˆåÈyê˛ •Î˚ñ xÓü®ˆÏöÓ˚ üyeyÄ •Î˚ ï˛ï˛ ˆÓ!¢– ï˛y•z Ÿ’Ìöܲy° !öî≈ˆÏÎ˚Ó˚ üˆÏôƒ !òˆÏÎ˚ xÓü®ˆÏöÓ˚ ˛õ!Ó˚üy˛õ
ܲÓ˚y ÎyÎ˚– xÓ¢ƒ ~!ê˛ üˆÏö Ó˚yáˆÏï˛ •ˆÏÓ ˆÎñ Ÿ’ÌöܲyˆÏ°Ó˚ ôyÓ˚îy ˆÜ˛Ó°üye °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe•z ˛≤ÈÏÎy烖
Ÿ’Ìöܲy° àîöyÓ˚ çöƒ ~ÓyÓ˚ 3.17(a) §ü#ܲÓ˚ˆÏî !ú˛ˆÏÓ˚ ÎyÄÎ˚y Îyܲ– ˙ §ü#ܲÓ˚î xö%ÎyÎ˚# t §üˆÏÎ˚ °â%
xÓü!®ï˛ ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ A(t) = A0e-bt
~áyö ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈñ Î!ò t = τ SŸ’Ìö ܲy°V •Î˚ ï˛ˆÏÓ A(τ) = A0e-bτ = A
0e-1
∴ bτ = 1
xÌ≈yÍñ τ =
1b
=
2mγ
...3.24
78
~áyˆÏö ˆòáy ÎyˆÏFåÈ ˆÎñ τ • ÏFåÈ b ~Ó˚ !Ó˛õÓ˚#ï˛ Ó˚y!¢– xï˛~Ó b ~Ó˚ üyö Îï˛ Óí˛¸ •ˆÏÓñ τ ~Ó˚ üyö
ï˛ï˛ ˆåÈyê˛ •ˆÏÓ– xyÓyÓ˚ b ~Ó˚ üyö Óí˛¸ •ÄÎ˚yÓ˚ xÌ≈ xÓü®ˆÏöÓ˚ üyey Ó,!k˛– ï˛y•z ˆåÈyê˛ Ÿ’Ìö ܲy° x!ôܲï˛Ó˚
xÓü®ö §)!ã˛ï˛ ܲˆÏÓ˚ ~ÓÇ ï˛y xyˆÏàÄ í˛zˆÏÕ‘á ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ Ÿ’Ìö ܲy° SτV ˆÜ˛Ó°üye
b ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– !ܲv °à#Î˚ •…y§ SτVb ~ÓÇ ˆòy°öܲy° T ~Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ï˛y•z xÓü®ˆÏöÓ˚
üyey ˆÓyé˛yˆÏöyÓ˚ çöƒ τ Óy λ Ó˚ ˆÜ˛yö!ê˛ ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÓ ï˛y ˆ«˛e!ӈϢ£Ï ~ÓÇ xyüyˆÏòÓ˚ ≤ÃyÆÓƒ ï˛ˆÏ̃Ó˚
!òˆÏܲ °«˛ƒ ˆÓ˚ˆÏá !öô≈y!Ó˚ï˛ Ü˛Ó˚ˆÏï˛ •ˆÏÓ–
3.5.3 !ܲí˛z!ܲí˛z!ܲí˛z!ܲí˛z!ܲí˛z (Q) ÈÙÈ =îyB˛ÙÈ =îyB˛ÙÈ =îyB˛ÙÈ =îyB˛ÙÈ =îyB˛
!ܲí˛z (Q) ÈÙ È=îyB˛ (Q - factor Óy quality factor) öyˆÏü ˛õ!Ó˚!ã˛ï˛ ~ܲ!ê˛ !ӈϢ£Ï Ó˚y!¢ àîöyÓ˚ üyôƒˆÏüÄ
xÓü®ˆÏöÓ˚ ˛õ!Ó˚üyî !öˆÏò≈¢ ܲÓ˚y ÎyÎ˚–
ôÓ˚y Îyܲñ ˆÜ˛yöÄ ~ܲ!ê˛ Ü˛¡õö¢#° ÓƒÓfliyÎ˚ E0 •ˆÏFåÈ xöÓü!®ï˛ xÓfliyÓ˚ ˆüyê˛ ¢!_´– ~áö ~•z
ÓƒÓfliy!ê˛ˆÏï˛ xÓü®ö í˛z˛õ!fliï˛ ÌyܲˆÏ° ~•z ¢!_´ e´üyàï˛ •…y§≤ÃyÆ •Î˚– ~•z ¢!_´ñ xyüÓ˚y çy!ö ˆÎñ §)ã˛Ü˛
•yˆÏÓ˚ ܲüˆÏï˛ ÌyˆÏܲ ~ÓÇ !ܲå%È §üÎ˚ ˛õˆÏÓ˚ ï˛y E0ÈÙÈ~Ó˚ üyˆÏöÓ˚ e–1 xÇ¢ Óy ≤ÃyÎ˚ 37 ¢ï˛yLjϢ ˛õ!Ó˚îï˛ •Î˚–
~•z §üˆÏÎ˚Ó˚ üˆÏôƒ ÓƒÓfliy!ê˛ Ü˛ˆÏÎ˚ܲ!ê˛ ˆòy°ö §¡õߨ ܲÓ˚ˆÏÓ– ~ܲ!ê˛ ˆòy°ö §¡õߨ ܲÓ˚yÓ˚ xÌ≈ ò¢yˆÏܲyˆÏîÓ˚
2π ˆÓ˚!í˛Î˚yö x!ï˛e´ü ܲÓ˚y– ~û˛yˆÏÓ E0 ï˛yÓ˚ 37 ¢ï˛yLjϢ ˆöˆÏü xy§yÓ˚ §üÎ˚ê%˛Ü%˛Ó˚ üˆÏôƒ SÎy ≤ÃÜ,˛ï˛˛õˆÏ«˛
ÓƒÓfliy!ê˛Ó˚ Ÿ’Ìö ܲy°ñ τ ~Ó˚ xˆÏô≈ˆÏܲÓ˚ §üyöV Ó›!ê˛Ó˚ ܲ¡õˆÏöÓ˚ ò¢yˆÏܲyî Îï˛ ˆÓ˚!í˛Î˚yö x!ï˛e´ü ܲˆÏÓ˚ˆÏåÈñ
ï˛y ˙ ÓƒÓfliy!ê˛Ó˚ !ܲí˛z ÈÙÈ=îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õܲ–
3.20 §ü#ܲÓ˚î ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ E0 ï˛yÓ˚ 37 ¢ï˛yÇ¢ Óy E0e–1 üyˆÏö ˆ˛õÑÔåȈÏï˛ t §üÎ˚ !öˆÏ°ñ
t =
12b
=
mγ
...3.25
ˆÎˆÏ•ï%˛ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω', t §üˆÏÎ˚ ܲ¡õö¢#° ÓƒÓfliy!ê˛ Îï˛ ˆÓ˚!í˛Î˚yö x!ï˛e´ü
ܲˆÏÓ˚ˆÏåÈ ï˛yÓ˚ ˛õ!Ó˚üyî òÑyí˛¸yÎ˚ ω' × t
ˆÎˆÏ•ï%˛ ~•z §Çáƒy!ê˛•z !ܲí˛zÈÙÈ=îyB˛ !•§yˆÏÓ §ÇK˛y!Î˚ï˛ •ˆÏÎ˚ˆÏåÈñ ï˛y•z Ó°y ÎyÎ˚ ˆÎñ
!ܲí˛z ÈÙÈ =îyB˛ Q = ω't =
'2bω
=
'2
ωτ
=
'm ωγ
...3.26
3.26 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ !ܲí˛z ÈÙÈ =îyB˛ˆÏܲ Ÿ’Ìö ܲyˆÏ°Ó˚ §y•yˆÏ΃ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚ xÌ≈yÍñ
ò%!ê˛ Ó˚y!¢ §¡õÜ≈˛Î%_´– xyÓ˚Ä °«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ ~•z ˆÎñ !ܲí˛zÈÙ =îyˆÏB˛ Q ~ܲ!ê˛ ~ܲܲ•#ö !Ó÷k˛ §Çáƒy–
xy˛õ!ö Î!ò 3.3 §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y !Ó£ÏÎ˚ܲ xyˆÏ°yã˛öyÎ˚ ~ܲê%˛ !ú˛ˆÏÓ˚ Îyöñ ï˛y•ˆÏ° ˆòáˆÏï˛ ˛õyˆÏÓö ˆÎ
γ •ˆÏFåÈ xÓü®ö =îyB˛– Èγ ÙÈ~Ó˚ Ó,•_Ó˚ üyö x!ôܲï˛Ó˚ xÓü®ö §)!ã˛ï˛ ܲˆÏÓ˚– xyÓyÓ˚ xï˛ƒhsˇ °â% xÓü®ˆÏöÓ˚
ˆ«˛ˆÏe Èγ ÙÈ~Ó˚ üyö á%Ó•z ܲü •Î˚ ~ÓÇ ˆ§ˆÏ«˛ˆÏe 3.2 §ü#ܲÓ˚ˆÏîÓ˚ Q ÈÙÈ~Ó˚ üyö á%Ó•z Óí˛¸ •Î˚– §%Ó˚¢°yܲyÓ˚
ܲ¡õˆÏöÓ˚ ˆ«˛ˆÏe Q =îyˆÏB˛Ó˚ üyö ˆüyê˛yü%!ê˛û˛yˆÏÓ Ü˛ˆÏÎ˚ܲ •yçyˆÏÓ˚Ó˚ üï˛ ~ÓÇ ˆ§áyˆÏö xÓü®ö á%Ó•z ܲü–
79
Ó›ï˛ 3.26 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈñ xöÓü!®ï˛ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏeñ ˆÎáyˆÏö xÓü®ö xö%˛õ!fliï˛ ÌyܲyÎ˚
γ = 0, Q =îyˆÏB˛Ó˚ üyö •ˆÏÓ x§#ü– ï˛ˆÏÓ ≤ÃÜ,˛ï˛˛õˆÏ«˛ ˆÜ˛yöÄ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe•z xÓöü®ö =îyˆÏB˛Ó˚ üyö
¢)öƒ •Î˚ öy– ú˛ˆÏ° Q ~Ó˚ üyöÄ §§#ü ÌyˆÏܲ–
xÓü®ˆÏöÓ˚ üyey á%Ó•z ܲü •ˆÏ° Îáö ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚ ˆÎ ω0
2 >> b2ñ ï˛áö xÓü®ˆÏöÓ˚ ú˛ˆÏ° ˛õ!Ó˚Ó!ï≈˛ï˛
ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω' ˆÜ˛ ˆ°áy ÎyÎ˚ È
ω' ≈ ω0 =
km
~Ó˚ ú˛ˆÏ° Q ÈÙÈ ~Ó˚ üyö òÑyí˛¸yÎ˚
Q =
0m ωγ
=
km
.
mγ
=
kmγ
=
0
2ωτ
...3.27
xÌ≈yÍñ ~ˆÏ«˛ˆÏe Q ÈÙÈ ~Ó˚ üyö !fl±ÇÈÙÈô &ÓˆÏܲÓ˚ Óà≈ü)ˆÏ°Ó˚ §ˆÏD §üyö%˛õy!ï˛Ü˛ ~ÓÇ γ ~Ó˚ ÓƒhflÏyö%˛õy!ï˛Ü˛ •Î˚–
≤ÈÏîy!òï˛ ˆòy°ö (forced vibration) Ä xö%öyò (resonance) ~Ó˚ ˆ«˛ˆÏe Q ÈÙÈ =îyˆÏB˛Ó˚ !ӈϢ£Ï ï˛yͲõÎ≈
Ä =Ó˚&c xyˆÏåÈ– ~ !ӣψÏÎ˚ xy˛õ!ö ˛õÓ˚Óï≈˛# ã˛ï%˛Ì≈ ~ܲˆÏܲ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö– Q ÈÙÈ =îyˆÏB˛Ó˚ !Óܲ“ §ÇK˛y §¡∫ˆÏ¶˛
ˆ§áyˆÏö xyˆÏ°yã˛öy •ˆÏÓ– ~áyˆÏö ˆÓ¢ ܲˆÏÎ˚ܲ!ê˛ !Ó£ÏÎ˚ xyüÓ˚y xyˆÏ°yã˛öy ܲÓ˚°yü– xy§%ö ~ÓyÓ˚ ˆÓyô•Î˚
ò%ÛÈÙÈ~ܲ!ê˛ xö%¢#°ö#Ó˚ ü%ˆÏáyü%!á •ˆÏï˛ xy˛õöyÓ˚ û˛yˆÏ°y•z °yàˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -3 :
~ܲ!ê˛ û˛Ó˚•#ö !fl±ÇÈÙÈ~Ó˚ §ˆÏD Î%_´ 200g û˛ˆÏÓ˚Ó˚ ˆòy°öܲy° 0.25– û˛Ó˚!ê˛Ó˚ ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ ï˛yÓ˚ ≤ÃyÌ!üܲ
üyˆÏöÓ˚ ~ܲÈÙÈã˛ï%˛Ì≈yLjϢ ˆöˆÏü xy§ˆÏï˛ 100s §üÎ˚ ˆöÎ˚– ÓƒÓfliy!ê˛Ó˚ xÓü®ö =îyB˛ γ ñ Ÿ’Ìö ܲy°ñ Q - =îyB˛
~ÓÇ !fl±Ç!ê˛Ó˚ !fl±ÇÈÙÈô &Óܲ !öî≈Î˚ ܲÓ˚&ö– ~ˆÏ«˛ˆÏe °à#Î˚ •…yˆÏ§Ó˚ üyö ܲï˛⁄
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 4 :
341 Hz ܲ¡õyB˛ !Ó!¢‹T ~ܲ!ê˛ §%Ó˚¢°yܲyÓ˚ Q ÈÙÈ =îyB˛ 1800– §%Ó˚¢°yܲy!ê˛Ó˚ ≤ÃyÓ˚!Ω˛Ü˛ ¢!_´ •…y§ ˆ˛õˆÏÎ˚
20 ¢ï˛yÇ¢ •ˆÏï˛ Ü˛ï˛ §üÎ˚ °yàˆÏÓ⁄
3.6 xÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î ≠ ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ñ ôyÓ˚ܲ ÄxÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î ≠ ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ñ ôyÓ˚ܲ ÄxÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î ≠ ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ñ ôyÓ˚ܲ ÄxÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î ≠ ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ñ ôyÓ˚ܲ ÄxÓü!®ï˛ ˆòy°ˆÏöÓ˚ í˛zòy•Ó˚î ≠ ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ñ ôyÓ˚ܲ Ä
ˆÓ˚yô§• Óï≈˛ö#ˆÓ˚yô§• Óï≈˛ö#ˆÓ˚yô§• Óï≈˛ö#ˆÓ˚yô§• Óï≈˛ö#ˆÓ˚yô§• Óï≈˛ö#
xy˛õöyÓ˚y ~•z ˛õÎ≈yˆÏÎ˚Ó˚ ≤ÃÌü ~ܲˆÏܲ ˆòˆÏáˆÏåÈö ˆÎñ ~ܲ!ê˛ ˆÓ˚yô•#ö Óï≈˛ö#ˆÏï˛ Îáö ~ܲ!ê˛ xyˆÏÓ¢ (L)
~ÓÇ ôyÓ˚ܲ (C) ˆ◊î# §üÓyˆÏÎ˚ Î%_´ ÌyˆÏܲ ~ÓÇ Óï≈˛ö#ˆÏï˛ ~ܲ!ê˛ Óƒyê˛y!Ó˚ ˆÎyà ܲˆÏÓ˚ ôyÓ˚ܲˆÏܲ §¡õ)î≈ xy!•ï˛
ܲˆÏÓ˚ §!Ó˚ˆÏÎ˚ ˆöÄÎ˚y •Î˚ñ ï˛áö ôyÓ˚ˆÏܲÓ˚ ˙ xyôyö ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛ ≤Ãò¢≈ö ܲˆÏÓ˚– !ܲv ˆû˛ˆÏÓ ˆòá%öñ
ÓyhflÏÓ ˆ«˛ˆÏe !ܲ ˆÓ˚yô•#ö Óï≈˛ö# ˛õyÄÎ˚y §Ω˛Ó⁄ xyüÓ˚y Î!ò ˆ§áyˆÏö §ˆÏã˛ï˛öû˛yˆÏÓ ˆÜ˛yöÄ ˆÓ˚yô (R) Î%_´
öyÄ Ü˛!Ó˚ñ ï˛y•ˆÏ°Ä xyˆÏÓ¢!ê˛ ˆÎ ˛õ!Ó˚Óy•# myÓ˚y ˜ï˛!Ó˚ñ ï˛yÓ˚ !öçfl∫ ˆÓ˚yô ~ÓÇ §ÇˆÏÎyà ï˛yˆÏÓ˚Ó˚ (connecting
80
wire) ˆÓ˚yô Óï≈˛ö#ˆÏï˛ xhsˇû≈%˛_´ •ˆÏÓ– ï˛y•z L - C - R Óï≈˛ö#•z ≤ÃÜ,˛ï˛˛õˆÏ«˛ ~ܲ!ê˛ ÓyhflÏÓ Óï≈˛ö# ~ÓÇ xyüÓ˚y
~áyˆÏö ï˛yÓ˚ xyã˛Ó˚î !ӈϟ’£Ïî ܲˆÏÓ˚ ˆòáÓ ˆÎñ
§ü≤ÃÓyˆÏ•Ó˚ í˛zͧÎ%_´ •ˆÏÎ˚ ôyÓ˚ܲˆÏܲ §¡õ)î≈û˛yˆÏÓ
xy!•ï˛ ܲÓ˚yÓ˚ ˛õˆÏÓ˚ í˛zͧ!ê˛ˆÏܲ §!Ó˚ˆÏÎ˚ !öˆÏ° ˙
Óï≈˛ö#ˆÏï˛ §üˆÏÎ˚Ó˚ §ˆÏD xyôyö ܲ# Ó˚ܲü ˛õ!Ó˚Óï≈˛ö
ˆòáyÎ˚– !ã˛e 3.8 ~ ~ܲ!ê˛ L - C - R Óï≈˛ö# ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– ≤ÃÌü ~ܲˆÏܲÓ˚ 1.4.6 xLjϢÓ˚ §ü#ܲÓ˚î
qdI LCdt
+
= 0 ˛õ!Ó ˚Ó!ï ≈ ˛ï˛ •ˆ ÏÎ ˚ ò Ñ yí˛ ¸ yˆ ÏÓ
q dIL RIC dt
+ + = 0
ˆÎˆÏ•ï%˛ñ I = + dqdt
ñ ~•z §ü#ܲÓ˚î!ê˛ §y!çˆÏÎ˚ !öˆÏÎ˚
ˆ°áy ÎyÎ˚ñ
2
2dqdqq LR
dtC dt++
= 0 ...3.28
í˛zû˛Î˚ ˛õ«˛ˆÏܲ L !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ˛õy•zñ
2
2dqdqq R
LdtLC dt++
= 0 ...3.29
~•z §ü#ܲÓ˚î!ê˛ 3.3 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ– xyüÓ˚y 3.3 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˛õy•zñ
ω02 = 1
LC ~ÓÇ b =
2R
L
xÌ≈yÍñ Óï≈˛ö#Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öô≈yÓ˚î ܲˆÏÓ˚ L ~ÓÇ C, ~ÓÇ xÓü®ö !öÎ˚!sfï˛ •Î˚ R ~ÓÇ LÈÙÈ~Ó˚
myÓ˚y–
1LC
ÈÙÈÓ˚ ~ܲܲ •ˆÏÓ s–2 ~ÓÇ
RL
~Ó˚ ~ܲܲ •ˆÏÓ s–1ñ Óy Èω0 ÙÈÓ˚ xö%Ó˚)˛õ–
~ܲ•zû˛yˆÏÓ 3.3 §ü#ܲÓ˚ˆÏîÓ˚ °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe §üyôyˆÏöÓ˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ Ó°y ÎyÎ˚ ˆÎñ Îáö ~•z
Óï≈˛ö#ˆÏï˛ b << ω0 xÌ≈yÍ R <<
2Lc
ñ ï˛áö §üˆÏÎ˚Ó˚ §ˆÏD ôyÓ˚ˆÏܲÓ˚ xyôyˆÏöÓ˚ ˛õ!Ó˚Óï≈˛ö !ö¡¨!°!áï §ü#ܲÓ˚î
xö%ÎyÎ˚# •ˆÏÓñ
q(t) =
01 expcos()2
R qttL
ωφ ⎛⎞ −+ ⎜⎟ ⎝⎠
...3.30
ˆÎáyˆÏö ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω1 •ˆÏFåÈñ
ω1 =
220b ω−
=
2
21
4R
LCL−
...3.31
81
xï˛~Óñ ~áyˆÏöÄ ˆòáy ÎyˆÏFåÈ ˆÎñ q §üˆÏÎ˚Ó˚ §ˆÏD §)ã˛Ü˛ •yˆÏÓ˚ •…y§ ˛õyÎ˚ñ xyÓyÓ˚ cos (ω1t+φ) ÈÙÈ~Ó˚
çöƒ xyˆÏ®y!°ï˛ •Î˚– L - C - R Óï≈˛ö#ˆÏï˛ñ R Ó,!k˛ ܲÓ˚ˆÏ° xÓü®ö Ó,!k˛ ˛õyÎ˚ ~ÓÇ R ~áyˆÏö xÓü®ö
=îyB˛ γ ~Ó˚ û)˛!üܲy ˛õy°ö ܲˆÏÓ˚– L ~Ó˚ û)˛!üܲy 3.3 §ü#ܲÓ˚ˆÏîÓ˚ û˛ˆÏÓ˚Ó˚ §üï%˛°ñ ܲyÓ˚î 3.3 §ü#ܲÓ˚ˆÏî 2b
= mγ ~ÓÇ 3.30 §ü#ܲÓ˚ˆÏî 2b =
RL
–
3.30 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ ˆüy«˛ˆÏîÓ˚ (discharge) §üÎ˚ xyôyˆÏöÓ˚ üyˆÏöÓ˚ !ÓhflÏyÓ˚ §)ã˛Ü˛ •yˆÏÓ˚
•…y§ ˛õyˆÏÓ–
°«˛ƒ ܲÓ˚&öñ xyôyˆÏöÓ˚ ܲ¡õö §Ω˛Ó •ÄÎ˚yÓ˚ çöƒ R <
2LC
•ÄÎ˚y ≤ÈÏÎ˚yçö– !ܲv R Î!ò
2LC
ÈÙÈ~Ó˚
ï%˛°öyÎ˚ í˛zˆÏ˛õ«˛î#Î˚ •ˆÏÎ˚ ÎyÎ˚ñ ï˛y•ˆÏ° ˆüy«˛ˆÏîÓ˚ ܲ¡õyB˛ ω1
~ÓÇ ω0
≤ÃyÎ˚ §üyö •ˆÏÎ˚ ÎyÎ˚ñ xÌ≈yÍ ï˛áö
ω12 ≅ ω0
2 =
1LC
Ó›ï˛ñ xyüÓ˚y ~áyˆÏö Óï≈˛ö#Ó˚ Q ÈÙÈ =îyB˛ !•§yˆÏÓ L Ä C ÈÙÈÓ˚ !•§yˆÏÓ QÙÈ=îyˆÏB˛Ó˚ ~ܲ!ê˛ Ó˚y!¢üy°y
ˆ˛õˆÏï˛ ˛õy!Ó˚– xy˛õ!ö ~!ê˛ !öˆÏç ܲˆÏÓ˚ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö–
L - C - R Óï≈˛ö#ˆÏï˛ ˆÓ˚yô R ÈÙÈ~Ó˚ !òˆÏܲ !ӈϢ£Ï ò,!‹T ˆòÄÎ˚yÓ˚ ≤ÈÏÎ˚yçö Ó˚ˆÏÎ˚ˆÏåÈ– ˆÓ˚yô•z Óï≈˛ö#ˆÏï˛
xÓü®ˆÏöÓ˚ §MÈ˛yÓ˚ ܲˆÏÓ˚– ˆÓ˚yˆÏôÓ˚ üyö Ó,!k˛Ó˚ §ˆÏD §ˆÏD xÓü®ö Ó,!k˛ ˛õyÎ˚– ˆÓ˚yˆÏôÓ˚ üyö
2LC
x!ï˛e´ü
ܲÓ˚ˆÏ° Óï≈˛ö#!ê˛ x!ï˛ xÓü!®ï˛ •Î˚ñ ˆ§!ê˛ˆÏï˛ xyÓ˚ ܲ¡õö âˆÏê˛ öy– ~ÓyˆÏÓ˚Ó˚ xö%¢#°ö#!ê˛ L – C – R Óï≈˛ö#
§Çe´yhsˇ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 5 :
~ܲ!ê˛ L – C – R Óï≈˛ö#ˆÏï˛ L = 5mH ~ÓÇ C = 1μF– Î!ò ò%!ê˛ ˆ«˛ˆÏe R ÎÌye´ Ïü 2Ω ~ÓÇ 100Ωñ
•Î˚ ï˛ˆÏÓ xyôyˆÏöÓ˚ ܲ¡õö à!ï˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ Ä =îyB˛ í˛zû˛Î˚ ˆ«˛ˆÏe Ü˛ï˛ •ˆÏÓ⁄
3.7 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~•z ~ܲˆÏܲ ˆÎ !Ó£ÏÎ˚=!° §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚y •°– ˆ§=!° •°ÈÙÙÙÈ
1. ÓyhflÏÓ çàˆÏï˛Ó˚ ˆòy°ˆÏö xÓü®ö í˛z˛õ!fliï˛ ÌyˆÏܲ ~ÓÇ xÓü®ˆÏöÓ˚ ú˛ˆÏ° ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ Ä ¢!_´
§üˆÏÎ˚Ó˚ §ˆÏD •…y§ ˛õyÎ˚–
2. xÓü!®ï˛ ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î !•§yˆÏÓ
22
0 22 dxdx bxdt dt
ω ++
= 0 §ü#ܲÓ˚î!ê˛ ˛õyÄÎ˚y ÎyÎ˚–
82
~áyˆÏö 2b = mγ ,
20
km
ω=
– γ = ~ܲܲ ˆÓà !˛õå%È xÓü®öܲyÓ˚# ӈϰÓ˚ ˛õ!Ó˚üyîñ k = ~ܲܲ §Ó˚î
!˛õå%È ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ ˛õ!Ó˚üyî–
3. !Ó!û˛ß¨ üyeyÓ˚ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyˆÏöÓ˚ Ó˚*˛õ!ê˛ !û˛ß¨ •Î˚– ˆÜ˛Ó°üye °â%
xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe ˆòy°à!ï˛ °«˛ƒ ܲÓ˚y ÎyÎ˚ ~ÓÇ ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ ~•z ˆ◊î#Ó˚ xÓü®ö
§Ó≈yˆÏ˛õ«˛y =Ó˚&c˛õ)î≈– °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe §üyôyö!ê˛ ˆ°áy ÎyÎ˚ñ
x(t) = A0e–bt cos (ω't + Q)
ˆÎáyˆÏöñ ω' =
220b ω−
x!ï˛ Óy e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe ˆòy°à!ï˛ §Ω˛Ó˛õÓ˚ •Î˚ öy–
4. xÓü!®ï˛ ˆòy°à!ï˛Ó˚ ˆ«˛ˆÏe !ÓhflÏyÓ˚ Ä ¢!_´ §üˆÏÎ˚Ó˚ §ˆÏD §)ã˛Ü˛ •yˆÏÓ˚ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– ˆÎüö
A = A0e–bt
~ÓÇ E = E0e–2bt
~áyˆÏöñ E ˆüyê˛ ¢!_´ ~ÓÇ A0 Ä E
0 ÎÌye´ˆÏü ˆòy°ˆÏöÓ˚ ≤ÃyÓ˚!Ω˛Ü˛ !ÓhflÏyÓ˚ Ä ˆüyê˛ ¢!_´–
5. °â% xÓü®ˆÏöÓ˚ ú˛ˆÏ° ˆòy°öܲy°Ä ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– Î!òÄ b << ω0 •ˆÏ° ~•z ˛õ!Ó˚Óï≈˛ö x!ï˛ §yüyöƒ
•Î˚ ~ÓÇ ≤ÃyÎ˚¢•z ï˛y x@ˇÃy•ƒ ܲÓ˚y •Î˚– °â% xÓü®ˆÏöÓ˚ ú˛ˆÏ° ˆÎ ˆòy°öܲy° ˛õyÄÎ˚y ÎyÎ˚ ï˛y •°ñ
T =
2'πω
=
220
2
b
πω−
6. °â% xÓü®ˆÏöÓ˚ üyey ˛õ!Ó˚üyîàï˛û˛yˆÏÓ !öˆÏò≈¢ ܲÓ˚yÓ˚ í˛zˆÏjˆÏ¢ƒ ܲˆÏÎ˚ܲ!ê˛ Ó˚y!¢ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ~=!°
•°ñ °à#Î˚ •…y§ SλVñ Ÿ’Ìö ܲy° SτV ~ÓÇ !ܲí˛z SQV ÈÙÈ=îyB˛– ~•z Ó˚y!¢=!° b, t, ω ≤Ãû,˛!ï˛Ó˚ §y•yˆÏ΃
≤Ãܲy¢ ܲÓ˚y ÎyÎ˚–
7. xyˆÏÓ¢ SLVñ ôyÓ˚ܲ ScV ~ÓÇ ˆÓ˚yô SRV ˆ◊î# §üÓyˆÏÎ˚ Î%_´ ~üö Óï≈˛ö#ˆÏï˛ Óƒyê˛y!Ó˚ Óy ˆÜ˛yöÄ §ü
!Óû˛Ó í˛zͧ Î%_´ ܲˆÏÓ˚ ôyÓ˚ܲ!ê˛ˆÏܲ ˛õ)î≈ xy!•ï˛ ܲÓ˚yÓ˚ ˛õÓ˚ Óƒyê˛y!Ó˚ Óy í˛zͧ!ê˛ˆÏܲ !Ó!FåÈߨ ܲÓ˚ˆÏ° ôyÓ˚ˆÏܲÓ˚ xyôyö
§üˆÏÎ˚Ó˚ §ˆÏD •…y§ ˛õyÎ˚ ~ÓÇ xÓü!®ï˛ ˆòy°à!ï˛ ≤Ãò¢≈ö ܲˆÏÓ˚– ~•z xÓü®ö Óï≈˛ö#Ó˚ ˆÓ˚yˆÏôÓ˚ IJõÓ˚ !öû˛≈Ó˚¢#°–
3.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. û˛Ó˚•#ö ~ÓÇ 8Nm–1 !fl±Ç ô &Óܲ !Ó!¢‹T ~ܲ!ê˛ !fl±ÇˆÏܲ ~ܲ!ê˛ ò,ì˛¸ !Ó®% ˆÌˆÏܲ é%˛!°ˆÏÎ˚ ï˛yÓ˚ !öˆÏã˛
0.2 kg û˛Ó˚ Î%_´ ܲÓ˚y •°– ˆòáy ˆà° ˆÎñ !fl±ÇÈÙÈ~Ó˚ §ˆÏD Î%_´ û˛Ó˚!ê˛ xÓü!®ï˛ ˆòy°ˆÏö xyˆÏ®y!°ï˛ •ˆÏFåÈ
83
~ÓÇ ï˛yÓ˚ ¢!_´ ≤ÃyÌ!üܲ ¢!_´Ó˚ 37 ¢ï˛yÇ¢ ˆöˆÏü xy§ˆÏï˛ 40s §üÎ˚ !öˆÏFåÈ– xÓü®ö =îyB˛ñ °à#Î˚ •…y§
~ÓÇ Ÿ’Ìö ܲy° ~ˆÏ«˛ˆÏe ܲï˛⁄ ÓƒÓfliy!ê˛Ó˚ QÈÙÈ=îyB˛ àîöy ܲÓ˚%ö–
2. ~ܲ!ê˛ §Ó˚° ˆòy°ˆÏܲÓ˚ !ÓhflÏyÓ˚ 4º ˆÌˆÏܲ 3º ˆï˛ •…y§ ˛õyÄÎ˚yÓ˚ §üÎ˚ 30!ê˛ ˛õ)î≈ ˆòy°ö §¡õߨ •Î˚–
ˆòy°ˆÏܲÓ˚ ˆòy°öܲy° 2.2s •ˆÏ° Ÿ’Ìö ܲy° Ä °à#Î˚ •…y§ àîöy ܲÓ˚&ö–
3. L– C – R ˆ◊î# §üÓyˆÏÎ˚ Î%_´ ~ܲ!ê˛ Óï≈˛ö#ˆÏï˛ L = 15mH, C = 3μF ~ÓÇ R = 10Ω– ~•z Óï≈˛ö#ˆÏï˛
~ܲ!ê˛ Óƒyê˛y!Ó˚ Î%_´ ܲˆÏÓ˚ ôyÓ˚ܲ!ê˛ˆÏܲ ˛õ)î≈ xy!•ï˛ ܲÓ˚yÓ˚ ˛õÓ˚ Óƒyê˛y!Ó˚!ê˛ˆÏܲ !Ó!FåÈߨ ܲÓ˚y •°– ~áö ôyÓ˚ˆÏܲÓ˚
xyôyˆÏöÓ˚ •…y§ ܲ# ܲ¡õö à!ï˛ ≤Ãò¢≈ö ܲÓ˚ˆÏÓ⁄ xyôyˆÏöÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ˛õ!Ó˚üyî Ü˛ï˛ §üˆÏÎ˚ xˆÏô≈ܲ •ˆÏÓ⁄ R ~Ó˚
ö)ƒöï˛ü ˆÜ˛yö üyˆÏöÓ˚ çöƒ xyôyˆÏöÓ˚ •…y§ ܲ¡õö à!ï˛ˆÏï˛ §Ω˛Ó˛õÓ˚ •ˆÏÓ öy⁄
4. 3.5.1 xö%ˆÏFåȈÏò xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ °à#Î˚ •…y§ λ !öî≈ˆÏÎ˚Ó˚ çöƒ ܲ#û˛yˆÏÓ ˛õÓ˚ ˛õÓ˚ ò%!ê˛ Ü˛¡õˆÏöÓ˚
!ÓhflÏyÓ˚ Ä ï˛yˆÏòÓ˚ xö%˛õyï˛ˆÏܲ ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– !ܲv ÓƒÓ•y!Ó˚ܲ ˆ«˛ˆÏe Îáö ˛õÓ˚#«˛yü%°Ü˛ û˛yˆÏÓ Èλ ÙÈÓ˚
üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ •Î˚ñ ï˛áö xyÓ˚Ä !öû≈˛Ó˚ˆÏÎyàƒ üyö ˛õyÄÎ˚yÓ˚ çöƒ §yôyÓ˚îï˛ ~üö ò%!ê˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ˆÏܲ
!Óã˛yÓ˚ ܲÓ˚y •Î˚ñ ÎyˆÏòÓ˚ üˆÏôƒ ‘n’ (n > 1) §ÇáƒÜ˛ ˛õ)î≈ ܲ¡õö §¡õߨ •ˆÏÎ˚ˆÏåÈ– Î!ò a0 ~ÓÇ an ~ˆÏ«˛ˆÏeÓ˚
≤ÃÌü Ä (n+1) ï˛ü ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ •Î˚ñ ï˛y•ˆÏ° ˆòáyö ˆÎñ Èλ ÙÈÓ˚ !Óܲ“ Ó˚y!¢üy°y •° λ =
0 1lnn
ana
⎛⎞⎜⎟ ⎝⎠
í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#
1. ˆÎˆÏ•ï%˛ A = A0e–bt ~ÓÇ ~áyˆÏö
A0 = .08m, A = .02m, t = 160s, xyüÓ˚y ˛õy•zñ
0
AA
=
.02
.08
= .25 = e–160b
Óyñ e160b = 4
∴ 160b = ln 4 = 1.386
~áyö ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ b = .00866s–1
xÓü!®ï˛ ˆòy°öܲy° T =
16080
= 2s ∴ ω' =
2Tπ
= πs–1
!ܲv ω' =
220b ω−
ñ xÌ≈yÍ ω0 =
22 'b ω+
=
22 (.00866) π+
84
∴ xÓü!®ï˛ ˆòy°öܲy° = 0
2πω =
22
2
(.00866)
ππ+
= 1.999992s Óy 2s
§%ï˛Ó˚yÇñ ò%•z ˆòy°öܲyˆÏ°Ó˚ üˆÏôƒ ˛õyÌ≈ܲƒ í˛zˆÏ˛õ«˛î#Î˚–
2. ˆòÄÎ˚y xyˆÏåÈñ û˛Ó˚ m = 2kg ~ÓÇ !fl±Ç ô &Óܲ k = 2Nm–1 ~ÓÇ ~ܲܲ à!ï˛ˆÏÓˆÏà xÓü®ö Ó°
γ = 0.4Nsm–1
∴ b = 2mγ =
0.422 ×
= 0.1s–1
xÓü!®ï˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω0 =
km
= 1s–1
(i) ˆòáy ÎyˆÏFåÈ ˆÎñ ω02 > b2ñ xï˛~Ó à!ï˛!ê˛ Ü˛¡õö Î%_´ •ˆÏÓ– ܲyÓ˚îñ ω0
2 > b2 •ÄÎ˚yÓ˚ xÌ≈ ~•z
ˆÎñ ÓƒÓfliy!ê˛ˆÏï˛ xÓü®ö ÌyܲˆÏ°Ä ï˛y °â% xÓü®ö–
(ii) e´yhsˇ#Î˚ xÓü®ˆÏöÓ˚ çöƒ ω0 = b xÌ≈yÍñ b = 1s–1
§%ï˛Ó˚yÇñ γ = 2bm = 2 × 1 × 2 = 4NSm–1 xÌ≈yÍñ ~ܲܲ à!ï˛ˆÏÓˆÏà 4N ӈϰÓ˚ çöƒ à!ï˛!ê˛Ó˚ e´yhsˇ#Î˚
xÓü®ö âê˛ˆÏÓ–
(iii) û˛Ó˚ mÈÙÈ~Ó˚ ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ω0 = b §¡õܲ≈!ê˛ ˛õy°ö ܲÓ˚y •ˆÏ° e´yhsˇ#Î˚ xÓü®ö §,!‹T •ˆÏÓ– ~ˆÏ«˛ˆÏe
ω02 = b2 xÌÓyñ
km
=
2 2
2mγ ⎛⎞
⎜⎟ ⎝⎠
, Óyñ m =
2
4kγ
– γ Ä k ~Ó˚ üyö Ó!§ˆÏÎ˚ñ
m =
2 (0.4)2
= .02 kg–
(iv) xyüÓ˚y çy!ö ˆÎñ
ˆüyê˛ ¢!_´ E(t) = E0e–2bt
~ˆÏ«˛ˆÏe b = 0.1s–1 ~ÓÇ
0
() EtE
=
13
∴
13
= e–2bt ln 3 = 2bt, Óyñ t =
ln32b
=
1.102.1 ×
= 5.5s
85
3. ˆòy°öܲy° T = 0.2s, ∴ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω' =
22.0.2 T
ππ
= 10π
!ÓhflÏyÓ˚ •…yˆÏ§Ó˚ ˆ«˛ˆÏe
A(t) = A0e–bt ∴
0
4A
= A0e–b.100
∴ ln 4 = b.100 ∴ b = .01386s–1
ˆÎˆÏ•ï%˛ m = 200g = 0.2kgñ ˆ°áy ÎyÎ˚ ˆÎñ xÓü®ö =îyB˛
γ = 2bm = 2.0.2 × .01386 = .0055 NSm–1
Ÿ’Ìö ܲy° τ =
1b
=
1.01386
= 72s
Q =îyB˛ =
'2bω
=
102.01386
π×
= ≤ÃyÎ˚ 1100
ˆÎˆÏ•ï%˛ b << ω', ω0 ≈ ω'
~áö K = mω0
2 ≈ mω'2
= 0.2 ×(10π)2
= 197Nm–1
°à#Î˚ •…y§ λ = bt = .01386 × .2 ≈ 2.8 × 10–3– °«˛ƒ ܲÓ˚&öñ ~•z ày!î!ï˛Ü˛ ≤ß¿!ê˛Ó˚ í˛z_ˆÏÓ˚Ó˚ çöƒ
xyüÓ˚y xyüyˆÏòÓ˚ §%!Óôyüï˛ !Ó!û˛ß¨ §¡õÜ≈˛=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚!åÈ– !ܲv !Óܲ“ §¡õÜ≈˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚Ä í˛z_Ó˚=!°
˛õyÄÎ˚y §Ω˛Ó– ˆÎüöñ λ =
Tτ
, Q =
πτ
– ~=!° xy˛õ!ö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö–
4. Q = 1800 ~ÓÇ Ü˛¡õyB˛ υ =
0
2ω
π
= 341Hz
Q ~Ó˚ í˛zFã˛ üyö x!ï˛ °â% xÓü®ö §)!ã˛ï˛ ܲÓ˚ˆÏåÈ
∴ Q =
0
2ωτ
∴τ =
0
2Qω
=
218003412π
××
= 1.68s
ˆÎˆÏ•ï%˛ τ =
1b
∴ b =
1 11.68
s−
= .595s–1
E = E0b–2bt
86
~ˆÏ«˛ˆÏe
0
EE
= 20% = 0.2 =
15
∴ ln 5 = 2bt ∴ t =
ln52b
= 1.35s
5. ≤ÃÌü ˆ«˛ˆÏe L = 5mH, C = 1μF, R = 2Ω
∴ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω1 =
2
21
4R
LCL−
= 8 42 10 4 10× − × = 1.4 ×104s–1
°«˛ƒ ܲÓ˚&öñ ˆÎˆÏ•ï%˛ 1
LC>>
2
2 4R
L
ñ xï˛~Ó ~áyˆÏö ôyÓ˚ˆÏܲÓ˚ ܲ!¡õï˛ ˆüy«˛î (oscillatory discharge)
âê˛ˆÏÓ ~ÓÇ ω1
≅ ω0 =
1LC
•ˆÏÓ– !mï˛#Î˚ ˆ«˛ˆÏe ˆÜ˛Ô!îܲ ܲ¡õyB˛
ω2 =
88 210110 ×−×
= 1.0 × 104s–1– ~•z ˆ«˛ˆÏe ˆÎˆÏ•ï%˛ R = 100Ω xÓü®ˆÏöÓ˚ üyey ~áyˆÏö
xˆÏöܲ ˆÓ!¢ ~ÓÇ Óï≈˛ö#Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ω0(=141.2 × 102s–1)– ~•z ˆÜ˛Ô!îܲ ܲ¡õyB˛ Èω2 ÙÈ~Ó˚ ˆÌˆÏܲ
xˆÏöܲáy!ö !û˛ß¨–
≤ÃÌü ˆ«˛ˆÏe QÈÙÈ=îyB˛ Q1 = 1
2bω =
1LRω
=
43 1.4105102
− ×××
= 35
!mï˛#Î˚ ˆ«˛ˆÏe Q2 = 4 31.0 10 5 10
100
−× × × = 0.5
°«˛ƒ ܲÓ˚&öñ ≤ÃÌü ˆ«˛ˆÏe xÓü®ö ܲü ÌyܲyÎ˚ Q ~Ó˚ üyö ˆÓ!¢–
3.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. K = 8Nm–1– xyüÓ˚y çy!ö E = E0e–2bt– 37 ¢ï˛yÇ¢ ˛õÎ≈hsˇ •…yˆÏ§Ó˚ xÌ≈ ≤ÃyÌ!üܲ ¢!_´ E
0 ï˛yÓ˚
0.37 xLjϢ ˆöˆÏü xy§ˆÏï˛ 40s §üÎ˚ !öˆÏFåÈ
∴ e–2bt = e–2b.40= 0.37
∴ 80b = – ln 0.37 = 1.0 xÌ≈yÍñ b = 1180
s−
87
∴ xÓü®ö =îyB˛ γ = 2bm =
1 20.2[0.2]80
mkg ××= ∵
= .005 NSm–1
xöÓü!®ï˛ ˆòy°öܲy° T = 2 mk
π = 0.228
π ≈ 1s
∴ ω0 = 2
Tπ = 2πs–1
°à#Î˚ •…y§ λ = bt = bT =
1180
×
= 0.0125
Ÿ’Ìö ܲy° τ =
1b
= 80s
Q =îyB˛ Q =
'2bω
=
220
2b
bω−
=
221 (2)
801 280
π⎛⎞ −⎜⎟ ⎝⎠×
= 2π ×40 ≈ 250
°«˛ƒ ܲÓ˚&öñ xÓü®ö =îyB˛ xï˛ƒhsˇ ܲü •ÄÎ˚yÎ˚ ~áyˆÏö ω' ≅ ω0
2. ˆòy°öܲy° T = 2.2s ∴ 30T = 66s
ˆÎˆÏ•ï%˛ A = A0e–bt
∴ 34
= e–66b ∴ ln
43
= 66b
xÌ≈yÍñ b =
4 ln366
= 4.4 × 10–3s–1
Ÿ’Ìöܲy° τ =
1b
= 230 s.
°à#Î˚ •…y§ λ = bt = .0096
88
3. ~áyˆÏö
2LC
=
3
61510 2.
310
−
−××
= 141Ω, ˆÎˆÏ•ï%˛ñ R (1.0Ω) <
2LC
– ôyÓ˚ˆÏܲÓ˚ xyôyö
ܲ¡õöà!ï˛ˆÏï˛ •…y§ ˛õyˆÏÓ–
xyôyˆÏöÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ˛õ!Ó˚üyî q =
0
2q
=
.2
0
RtL qe
−
∴ t = 2 ln2LR
= 32.15 10 .0.693
1.0
−× = 0.0207s
xyôyˆÏöÓ˚ ˆüy«˛î (discharge) ܲ¡õöà!ï˛ˆÏï˛ öy •ÄÎ˚yÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ö)ƒöï˛ü ˆÓ˚yô 2 LC
Óy
141Ω–
4. xyüÓ˚y •…y§ Óy d-Ó˚ ˆÎ §ÇK˛yÓ˚ §ˆÏD 3.5.1 xö%ˆÏFåȈÏò ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚!åÈñ ï˛yÓ˚ !û˛!_ˆÏï˛ ˆ°áy ÎyÎ˚ ˆÎñ
1 n
n
aa
−
= d = eλ
~áy Ïö an – 1 ~ÓÇ anÎÌye´ˆÏü n ï˛ü ~ÓÇ (n + 1) ï˛ü !ÓhflÏyÓ˚–
∴ °à#Î˚ •…y§ λ = ln d = 1
lnn
n
aa
− ⎛⎞⎜⎟ ⎝⎠
!ܲv a0 Î!ò ≤ÃÌü ~ÓÇ a
n Î!ò (n + 1) ï˛ü !ÓhflÏyÓ˚ •Î˚ ï˛ˆÏÓ ˆ°áy ÎyÎ˚ ˆÎñ
0
n
aa
=
021 12
1231.....nn
nn
aaa aaaaaaa
−−
−
⎛⎞⎛⎞⎛⎞⎛⎞ ⎛⎞ ×× ⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ ⎝⎠ ⎝⎠⎝⎠⎝⎠⎝⎠
= d × d × d ......d × d
= dn
ˆÜ˛ööyñ ≤Ã!ï˛!ê˛ Ó¶˛ö#û%˛_´ xö%˛õyï˛ dÈÙÈÓ˚ §üyö
xÌ≈yÍñ
012
123
......aaa
daaa
===
§%ï˛Ó˚yÇ ñ ln
0
n
aa
= lndn = nlnd = nλ
∴ λ =
0 1lnn
ana
89
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 4 ≤ÈÏîy!òï˛≤ÈÏîy!òï˛≤ÈÏîy!òï˛≤ÈÏîy!òï˛≤ÈÏîy!òï˛ Ü˛¡õöܲ¡õöܲ¡õöܲ¡õöܲ¡õö ÄÄÄÄÄ xö%öyòxö%öyòxö%öyòxö%öyòxö%öyò (Forced vibration and Resonance)
àë˛öàë˛öàë˛öàë˛öàë˛ö
4.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
4.2 ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®öÈÙȧ• ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚
≤Ã!ï˛¤˛y–
4.3 ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
4.3.1 fl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyö
4.3.2 fliyÎ˚# xÓfliyÓ˚ §üyôyö
4.4 ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy
4.5 xö%öyò
4.5.1 !ÓhflÏyˆÏÓ˚Ó˚ xö%öyò
4.5.2 à!ï˛ˆÏÓà xö%öyò
4.6 ≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚
4.7 xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y Ä Q =îyB˛
4.8 ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ˆ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢Ü˛ (L), ôyÓ˚ܲ (C) ~ÓÇ ˆÓ˚yôܲ (R) §¡∫!°ï˛
Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–
4.9 §yÓ˚yÇ¢
4.10 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
4.11 í˛z_Ó˚üy°y
4.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
˛õ)Ó≈Óï≈˛# ~ܲˆÏܲ S~ܲܲ 3) xyüÓ˚y ˆòˆÏá!åÈñ xÓü®ˆÏöÓ˚ ú˛ˆÏ° ܲ#û˛yˆÏÓ ~ܲ!ê˛ ˆòy°ˆÏöÓ˚ !ÓhflÏyÓ˚ Ä ¢!_´
e´ü¢ •…y§ ˆ˛õˆÏï˛ ˆ˛õˆÏï˛ §¡õ)î≈ !Ó°#ö •ˆÏÎ˚ ÎyÎ˚– ï˛y•ˆÏ° !ܲ ÓyhflÏÓ çàˆÏï˛ ˆÜ˛yöÄ ˆòy°ö•z ò#â≈fliyÎ˚# •ÄÎ˚y
§Ω˛Ó öÎ˚⁄ xÌã˛ ˆÜ˛yöÄ ˆÜ˛yöÄ ˆ«˛ˆÏeñ ˆÎüö â!í˛¸Ó˚ ˆ˛õu%˛°yˆÏüÓ˚ ˆòy°ö Óy !Óò%ƒÍ ã˛y!°ï˛ §%Ó˚¢°yܲyÓ˚
ܲ¡õˆÏöÓ˚ §üÎ˚ xyüyˆÏòÓ˚ üˆÏö •Î˚ ˙ ˆòy°ö Óy ܲ¡õö ˆÎö xyÓ•üyö ܲy° ôˆÏÓ˚ ã˛°ˆÏï˛ ˛õyˆÏÓ˚– Ó›ï˛ ~•z
ˆ«˛e=!°ˆÏï˛Ä !ܲå%È xÓü®ö í˛z˛õ!fliï˛ ÌyˆÏܲ– ï˛y §ˆÏ_¥Ä ~•z§Ó ˆòy°ö ÓçyÎ˚ Ó˚yáy §Ω˛Ó •Î˚ ~ܲ!ê˛ !ӈϢ£Ï
ܲyÓ˚ˆÏî– ~•zÓ˚ܲü ≤Ã!ï˛!ê˛ ˆ«˛ˆÏe ܲ¡õö¢#° ÓƒÓfliy!ê˛ˆÏï˛ Óy•zˆÏÓ˚ ˆÌˆÏܲ ¢!_´ §Ó˚ÓÓ˚y• ܲÓ˚y •Î˚ ~ÓÇ ~•z ¢!_´
xÓü®ˆÏöÓ˚ í˛z˛õ!fli!ï˛ §ˆÏ_¥Ä ܲ¡õö ÓçyÎ˚ Ó˚yáˆÏï˛ §«˛ü •Î˚–
ˆÜ˛yöÄ Ü˛¡õö«˛ü ï˛ˆÏsf (system) Óy•zˆÏÓ˚ ˆÌˆÏܲ ¢!_´ §Ó˚ÓÓ˚y• ܲÓ˚yÓ˚ çöƒ §yôyÓ˚îï˛ §üˆÏÎ˚Ó˚ §ˆÏD
˛õ!Ó˚Óï≈˛ö¢#° ~ܲ!ê˛ Ó° ≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~•z ˛õ!Ó˚Óï≈˛ö¢#° Ó°!ê˛ ≤ÃyÎ˚¢•z ˛õÎ≈yÓ,_û˛yˆÏÓ (harmonically)•…y§ÈÙÈÓ,!k˛ ˛õyÎ˚ ~ÓÇ ï˛yÓ˚ ܲ¡õyB˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ˆÌˆÏܲ §ã˛Ó˚yã˛Ó˚ !û˛ß¨ •ˆÏÎ˚ ÌyˆÏܲ– Ó›ï˛ ~•z ܲ¡õyB˛
~ÓÇ ï˛sf!ê˛Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ (natural frequency) Îáö ~ܲ •ˆÏÎ˚ ÎyÎ˚ñ ï˛áö ~ܲ!ê˛ !ӈϢ£Ï xÓfliyÓ˚
§,!‹T •Î˚– ˛õòyÌ≈!ÓòƒyÓ˚ ˛õ!Ó˚û˛y£ÏyÎ˚ñ Óy•zˆÏÓ˚ ˆÌˆÏܲ xyˆÏÓ˚y!˛õï˛ ˛õÎ≈yÓ,_ ӈϰÓ˚ xôy#ˆÏö ܲ¡õöˆÏܲ ≤ÈÏîy!òï˛
90
ܲ¡õö (forced vibration) ~ÓÇ ò%•z ܲ¡õyˆÏB˛Ó˚ §üï˛yÓ˚ ú˛ˆÏ° !ӈϢ£Ï xÓfliy!ê˛ˆÏܲ xö%öyò (resonance)Ó°y •Î˚– ~•z ~ܲˆÏܲ ò%!ê˛ !Ó£ÏÎ˚•z !ÓhflÏy!Ó˚ï˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÓ–
xy§%öñ xyüÓ˚y ÓÓ˚Ç ~áyˆÏö xyÓ˚Ä Ü˛ˆÏÎ˚ܲ!ê˛ §Ç!Ÿ’‹T ˛õ!Ó˚û˛y£ÏyÓ˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ ••z– Óy•zˆÏÓ˚ ˆÌˆÏܲ ˆÎ
Ó° ≤ÃÎ%_´ •ˆÏÎ˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ §)ã˛öy ܲˆÏÓ˚ ~ÓÇ xÓü®ö §ˆÏ_¥Ä ܲ¡õˆÏö Ó˚yˆÏáñ ï˛yˆÏܲ xyüÓ˚y ã˛y°Ü˛
Ó° Ó!° ~ÓÇ ˙ Ó° ˆÎ ÓƒÓfliyÓ˚ IJõÓ˚ ≤ÃÎ%_´ •Î˚ñ ï˛yˆÏܲ xyüÓ˚y ã˛y!°ï˛ ÓƒÓfliy Ó!°– ≤ÈÏîy!òï˛ ˆòy°ö
Óy Ü¡õˆÏöÓ˚ ˆ«˛ˆÏe §yü!@ˇÃܲû˛yˆÏÓ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ x!û˛ü%á ÌyˆÏܲ ã˛y°Ü˛ ˆÌˆÏܲ ã˛y!°ˆÏï˛Ó˚ !òˆÏܲ– ã˛y!°ï˛
ÓƒÓfliy ã˛y°ˆÏܲÓ˚ ˜Ó!¢ˆÏ‹TƒÓ˚ xÌ≈yÍñ ã˛y°Ü˛ ӈϰÓ˚ üyö Óy ܲ¡õyˆÏB˛Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï˛≈ö âê˛yÎ˚ öy– ≤ÈÏîy!òï˛
ˆòy°ˆÏöÓ˚ ~!ê˛ xöƒï˛ü ˜Ó!¢‹Tƒ–
xöƒ!òˆÏܲñ !ܲå%È !ӈϢ£Ï ôÓ˚ˆÏöÓ˚ ÓƒÓfliyÎ˚ !ܲå%È ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ x!û˛ü%á ˛õÎ≈yÎ˚e´ˆÏü ˛õyˆÏŒê˛ ÎyÎ˚ ~ÓÇ ã˛y°Ü˛
ï˛sf Ä ã˛y!°ï˛ ï˛ˆÏsfÓ˚ û)˛!üܲy e´üyß∫ˆÏÎ˚ •hflÏyhsˇ!Ó˚ï˛ •ˆÏï˛ ÌyˆÏܲ– ~•z ôÓ˚ˆÏöÓ˚ ò%!ê˛ ï˛ˆÏsf ˆÎ ܲ¡õö ˆòáy ÎyÎ˚ñ
ï˛y ≤ÈÏîy!òï˛ Ü˛¡õö ˆÌˆÏܲ !û˛ß¨– ~•z ܲ¡õöˆÏܲ Ó°y •Î˚ Î%!@¬ï˛ ܲ¡õö (coupled vibration)– ò%!ê˛
ܲ¡õö«˛ü ï˛ˆÏsfÓ˚ üˆÏôƒ ~ܲ!ê˛ !ӈϢ£Ï §ÇˆÏÎyçöñ ÎyˆÏܲ ˛õ!Ó˚û˛y£ÏyÎ˚ Ó°y •Î˚ Î%@¬ö (coupling) í˛z˛õ!fliï˛
ÌyܲˆÏ°ñ ~!ê˛ °«˛ƒ ܲÓ˚y ÎyÎ˚– xyüÓ˚y ˛õÓ˚Óï≈˛# ~ܲˆÏܲ S~ܲܲ - 5) !Ó£ÏÎ˚!ê˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
˛õòyÌ≈!ÓòƒyÓ˚ !Ó!û˛ß¨ ¢yáyÎ˚ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyˆÏòÓ˚ Óƒy˛õܲ ≤ÈÏÎ˚yà Ó˚ˆÏÎ˚ˆÏåÈ– =Ó˚&c˛õ)î≈ ~•z
≤ÃyˆÏÎ˚y!àܲ !Ó£ÏÎ˚=!° !öˆÏÎ˚ xyüÓ˚y ~•z ~ܲˆÏܲ ˛õˆÏÓ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
í˛ zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒ
~•z ~ܲˆÏܲ xyˆÏ°y!ã˛ï˛ !Ó£ÏÎ˚Ó› ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õˆÏÓ˚ xy˛õ!ö ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyˆÏòÓ˚ §¡∫ˆÏ¶˛ xˆÏöܲê˛y
çyöˆÏï˛ ˛õyÓ˚ˆÏÓö– ~ !ӣψÏÎ˚ xy˛õ!ö ˆÎ ܲyç=!° ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓöñ ˆ§=!° •°ÈÙÙÙÈ
°â% xÓü®ö §• ܲ¡õö¢#° ï˛ˆÏsf ˛õÎ≈yÓ,_ Ó° ≤ÈÏÎ˚yˆÏàÓ˚ ú˛ˆÏ° §,‹T xÓfliyÓ˚ ày!î!ï˛Ü˛ !ÓÓÓ˚ˆÏîÓ˚ çöƒ
≤ÈÏÎ˚yçö#Î˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y Ä ï˛yÓ˚ §üyôyö ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
≤ÃÎ%_´ ˛õÎ≈yÓ,_ ӈϰÓ˚ ܲ¡õyB˛ñ !ÓhflÏyÓ˚ñ ò¢yÓ˚ §ˆÏD fliyÎ˚# xÓfliyÓ˚ (steady state) ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚
!Ó!û˛ß¨ ˜Ó!¢ˆÏ‹TƒÓ˚ §¡õÜ≈˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
xö%öyò (resonance) ~ÓÇ ï˛yÓ˚ !Ó!û˛ß¨ ˜Ó!¢‹Tƒ=!° !ӈϟ’£Ïî ܲÓ˚ˆÏï˛ ~ÓÇ ≤ÈÏîy!òï˛ Ü˛¡õˆÏö ~ܲ!ê˛
˛õ)î≈ ã˛ˆÏe´ ã˛y°Ü˛ ˆÌˆÏܲ ã˛y!°ˆÏï˛Ó˚ !òˆÏܲ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ àí˛¸ •yÓ˚ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe !ܲí˛z =îyB˛ (Q-factor) àîöy ~ÓÇ ï˛yÓ˚ =Ó˚&c Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
≤Ãï˛ƒyÓï≈˛# !Óû˛Ó í˛zͧ §ˆÏüï˛ ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢ÈÙÈôyÓ˚ܲÈÙȈÓ˚yô Óï≈˛ö#Ó˚ üï˛ !ӈϢ£Ï ˆ«˛ˆÏe ≤ÈÏîy!òï˛
ˆòy°ˆÏöÓ˚ !ӈϟ’£Ïî ≤ÈÏÎ˚yà ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓöñ ˙ ôÓ˚ˆÏöÓ˚ ï˛ˆÏsfÓ˚ §y!Ó≈ܲ xyã˛Ó˚ˆÏîÓ˚ ˛õÎ≈yˆÏ°yã˛öy ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
4.2 ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®ö §• ≤ÈÏîy!òï˛≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®ö §• ≤ÈÏîy!òï˛≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®ö §• ≤ÈÏîy!òï˛≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®ö §• ≤ÈÏîy!òï˛≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏîÈÙÙÙÈ°â% xÓü®ö §• ≤ÈÏîy!òï˛
ܲ¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yܲ¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yܲ¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yܲ¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yܲ¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y
˛õ)Ó≈Óï≈˛# ~ܲˆÏܲÓ˚ S~ܲܲ -3) 3.3 xö%ˆÏFåȈÏò xy˛õ!ö ~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliyÓ˚ ܲÌy ˛õˆÏí˛¸ˆÏåÈö– ~áyˆÏö
4.1 !ã˛ˆÏe !ë˛Ü˛ ˙Ó˚ܲü ~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– !fl±ÇÈÙÈ~Ó˚ §ˆÏD Î%_´ m û˛Ó˚!ê˛ °â% xÓü®ö
91
§• ܲ!¡õï˛ •ˆÏï˛ ˛õyˆÏÓ˚ñ ~ê˛y xyüÓ˚y xyˆÏà ˆòˆÏá!åÈ– ~áö ~•z ÓƒÓfliyÓ˚ IJõÓ˚ 4.1 !ã˛ˆÏe ˆÎüö ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈñ ˆï˛üö ~ܲ!ê˛ §üÎ˚ !öû≈˛Ó˚ ˛õÎ≈yÓ,_ Ó° F(t) ≤ÈÏÎ˚yà ܲÓ˚y
•°– ~áö xyüÓ˚y !ӈϟ’£Ïî ܲÓ˚yÓ˚ ˆã˛‹Ty ܲÓ˚Óñ Óy•zˆÏÓ˚ ˆÌˆÏܲ
≤ÃÎ%_´ ~•z ӈϰÓ˚ ≤Ãû˛yˆÏÓ m û˛Ó˚!ê˛ !ë˛Ü˛ ܲ#û˛yˆÏÓ Ü˛!¡õï˛ •ˆÏÓ–
xyüÓ˚y ӈϰ!åÈ F(t) ~ܲ!ê˛ ˛õÎ≈yÓ,_ Ó°ñ ï˛y•z ˆ°áy ÎyÎ˚ ˆÎñ
F(t) = F0 cos ωt ...4.1
~áyˆÏö F0 •ˆÏFåÈ Ó°!ê˛Ó˚ §ˆÏÓ≈yFã˛ üyö Óy !ÓhflÏyÓ˚ ~ÓÇ ω •ˆÏFåÈ
ï˛yÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛– ~áyˆÏö xyüÓ˚y F(t) = F0 cos (ωt + φ) !°áˆÏï˛ ˛õyÓ˚ï˛yüñ ï˛ˆÏÓ Îáö F = F
0
ˆ§•z ü%•)ï≈˛ ˆÌˆÏܲ §üˆÏÎ˚Ó˚ àîöy ÷Ó˚& ܲÓ˚ˆÏ° φ ~Ó˚ üyö ¢)öƒ ˆöÄÎ˚y ÎyˆÏÓ–
~ÓyÓ˚ xyüÓ˚y Ó!•≤Ã≈Î%_´ ~•z ӈϰÓ˚ ≤Ãû˛yˆÏÓ !fl±ÇÈÙÈ~Ó˚ §ˆÏD Î%_´ û˛Ó˚!ê˛Ó˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xyˆÏ°yã˛öyÓ˚
çöƒ ˛õ)Ó≈Óï≈˛# ~ܲˆÏܲÓ˚ (3.2) §ü#ܲÓ˚î!ê˛ ˛õ%öÓ˚&k,˛ï˛ ܲÓ˚!åÈ– ˛õò=!°ˆÏܲ §y!çˆÏÎ˚ !öˆÏÎ˚ñ
2
2dxdx mkx
dt dtγ ++
= 0 ...4.2
ï˛ˆÏsfÓ˚ IJõÓ˚ F(t) Ó°!ê˛ ≤ÃÎ%_´ •ÄÎ˚yÓ˚ ˛õˆÏÓ˚ (4.2) §ü#ܲÓ˚î!ê˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yˆÏÓñ
2
2dxdx mkx
dt dtγ ++
= F0 cos ωt
í˛zû˛Î˚ ˛õ«˛ˆÏܲ m !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ˛õy•zñ
22
0 22 ++ dxdx bxdt dt
ω
= f0 cos ωt ...4.3
~áyˆÏöñ 2b = mγ , ω
02 =
km
~ÓÇñ f0 =
0 Fm
°«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ñ (4.3) §ü#ܲÓ˚î!ê˛Ó˚ í˛yö!òˆÏܲ ¢)öƒ öÎ˚– ˆ§áyˆÏö í˛z˛õ!fliï˛ Ó˚ˆÏÎ˚ˆÏåÈ f0 cos ωt– ~Ó˚ ú˛ˆÏ°
~•z §ü#ܲÓ˚î!ê˛ 4.2 ˆÌˆÏܲ !û˛ß¨ ~ÓÇ ~•z xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö ܲÓ˚ˆÏï˛ •ˆÏÓ !ܲå%Èê˛y !û˛ß¨ ˛õk˛!ï˛ˆÏï˛–
4.3 xÓܲ° §ü#ܲÓ˚î!ê˛ !mï˛#Î˚ üyeyÓ˚ ˜Ó˚!áܲ §ü#ܲÓ˚î– !ܲv §ü#ܲÓ˚î!ê˛Ó˚ í˛yö!òܲ ¢)öƒ öy •ÄÎ˚yÎ˚ñ ~!ê˛
~ܲ!ê˛ x§ü§_¥ (inhomogeneous) §ü#ܲÓ˚î– ~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyˆÏöÓ˚ üyôƒˆÏü ≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚
§üÎ˚ÈÙÈò)Ó˚c §¡õÜ≈˛ !öî≈Î˚ ܲÓ˚y ÎyˆÏÓ ~ÓÇ §¡õÜ≈˛!ê˛ !ӈϟ’£ÏˆÏîÓ˚ üyôƒˆÏü ≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚ ˜Ó!¢‹Tƒ=!°
˛õÎ≈yˆÏ°yã˛öy ܲÓ˚y §Ω˛Ó •ˆÏÓ– ˛õˆÏÓ˚Ó˚ xö%ˆÏFåȈÏò xyüÓ˚y ~•z ܲyç!ê˛ Ü˛Ó˚Ó–
92
4.3 ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
4.3 §ü#ܲÓ˚î!ê˛Ó˚ §üyôyˆÏöÓ˚ xyˆÏà (4.2) §ü#ܲÓ˚î!ê˛Ó˚ §ˆÏD ï˛yÓ˚ ~ܲê%˛ ï%˛°öy ܲÓ˚y Îyܲ– Ó›ï˛ ~Ó˚ ú˛ˆÏ°
xyüyˆÏòÓ˚ Ó%é˛ˆÏï˛ §%!Óôy •ˆÏÓ ˆÎñ !ë˛Ü˛ ܲ# ôÓ˚ˆÏöÓ˚ §üyôyö xyüÓ˚y ˆ˛õˆÏï˛ ˛õy!Ó˚– °«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ (4.2)
§ü#ܲÓ˚î!ê˛ ~ܲ!ê˛ xÓü!®ï˛ ˆòy°ö §)!ã˛ï˛ ܲÓ˚ˆÏåÈ ~ÓÇ °â% xÓü®ˆÏöÓ˚ ˆ«˛ˆÏe ï˛sf!ê˛ ï˛yÓ˚ fl∫yû˛y!Óܲ ˆÜ˛Ô!îܲ
ܲ¡õyB˛ ω0 ˆÌˆÏܲ £ÏÍ ˛õ!Ó˚Ó!ï≈˛ï˛ ~ܲ!ê˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛
() 220 'b ωω =−
!öˆÏÎ˚ xyˆÏ®y!°ï˛ •Î˚– xyüÓ˚y
~ܲܲ-3ÈÙÈ~ ~ê˛yÄ ˆòˆÏá!åÈ ˆÎñ °â% xÓü®ˆÏöÓ˚ çöƒ ω0ÈÙÈÓ˚ ~•z ˛õ!Ó˚Óï≈˛ö x“ ~ÓÇ b << ω0 •ˆÏ° ω0 ≅
ω' Ó°y ÎyÎ˚–
~ÓyÓ˚ ~•z ܲ¡õö¢#° ï˛ˆÏsf xyˆÏÓ˚y!˛õï˛ •ˆÏÎ˚ˆÏåÈ ~ܲ!ê˛ Óy•zˆÏÓ˚Ó˚ ã˛y°Ü˛ Ó°ñ ÎyÓ˚ !öçfl∫ ˆÜ˛Ô!îܲ ܲ¡õyB˛
ω– ã˛y°Ü˛ Ó° ˆã˛‹Ty ܲÓ˚ˆÏÓ ï˛sf!ê˛ˆÏܲ ï˛yÓ˚ !öçfl∫ ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ ܲÓ˚ˆÏï˛ñ ú˛ˆÏ° xyüÓ˚y ò%!ê˛ !û˛ß¨
ܲ¡õyˆÏB˛Ó˚ xyˆÏ®y°ˆÏöÓ˚ í˛z˛õ!Ó˚˛õyï˛ ≤Ãï˛ƒ«˛ ܲÓ˚Ó– ~Ó˚ ≤ÃÌü!ê˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛ xyˆÏ®y°ö ~ÓÇ
!mï˛#Î˚!ê˛ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛ xyˆÏ®y°ö– xy˛õ!ö û˛yÓˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ Î!ò ~•z ò%!ê˛ Ü˛¡õyB˛ §üyö •ˆÏÎ˚
ÎyÎ˚ñ ï˛y•ˆÏ° xyüÓ˚y ò%!ê˛ !û˛ß¨ ܲ¡õyˆÏB˛Ó˚ xyˆÏ®y°ö ˛õyÓ öy ~ÓÇ ˆ§ˆÏ«˛ˆÏe ܲ# âê˛ˆÏÓ⁄ ~•z âê˛öy ~ܲ!ê˛ !ӈϢ£Ï
xÓfliyÓ˚ §,!‹T ܲÓ˚ˆÏÓñ ÎyÓ˚ xyˆÏ°yã˛öy xyüÓ˚y ÎÌy§üˆÏÎ˚ ܲÓ˚Ó– xy˛õyï˛ï˛ xyüÓ˚y ω ≠ ω0 ôˆÏÓ˚ !öˆÏÎ˚ 4.3
§ü#ܲÓ˚ˆÏîÓ˚ §üyôyˆÏöÓ˚ ˆã˛‹Ty ܲÓ˚Ó–
xy˛õ!ö •Î˚ï˛ xy®yç ܲÓ˚ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈö ˆÎñ (4.3) §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö x(t) à!ë˛ï˛ •ˆÏÓ ò%!ê˛ xLjϢÓ˚
§üß∫ˆÏÎ˚ ܲyÓ˚îñ ~áyˆÏö ò%!ê˛ ˛õ,Ìܲ ܲ¡õyˆÏB˛Ó˚ ܲ¡õö í˛z˛õ!fliï˛ ÌyܲˆÏÓ– ï˛y•z ˆ°áy ÎyÎ˚ ˆÎñ
x(t) = x1(t) + x2(t) ...4.4
~áyˆÏö x1(t) •ˆÏFåÈ 4.2 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyöñ ˆÎ!ê˛ ã˛y°Ü˛ ӈϰÓ˚ xö%˛õ!fli!ï˛ˆÏï˛ xyüÓ˚y °â% xÓü®öÈÙȧ•
ܲ¡õˆÏöÓ˚ ˆ«˛ˆÏe ˆ˛õˆÏÎ˚!åÈ°yü–
§%ï˛Ó˚yÇ
221 1
0 12 2d x dx
b xdtdt
ω+ + = 0 ...4.5
xyÓ˚ x2(t) •ˆÏFåÈ 4.3 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö xÌ≈yÍñ ã˛y°Ü˛ Ó° ï˛yÓ˚ ≤Ãû˛yˆÏÓ ï˛sfˆÏܲ ˆÎû˛yˆÏÓ ã˛y°öy ܲÓ˚ˆÏï˛
ã˛y•zˆÏåÈñ ï˛y §)!ã˛ï˛ ܲÓ˚ˆÏÓ x2–
222 2
0 22 2d x dx
b xdtdt
ω+ + = f0 cos ωt ...4.6
xï˛~Ó ˛õ)î≈ §üyôyö (complete solution) x(t) ˆÜ˛ x1(t) ~ÓÇ x
2(t) ÈÙÈÓ˚ ˆÎyàú˛° !•§yˆÏÓ ˛õyÄÎ˚y ÎyÎ˚
S§ü#ܲÓ˚î 4.4) xÓܲ° §ü#ܲÓ˚î !Ó£ÏÎ˚ܲ ˛õyë˛ƒe´ˆÏü xy˛õöyÓ˚y °«˛ƒ ܲÓ˚ˆÏÓö ˆÎñ x1(t) ~ÓÇ x2(t) ˆÜ˛ ày!î!ï˛Ü˛
˛õ!Ó˚û˛y£ÏyÎ˚ ÎÌye´ˆÏü ˛õ)Ó˚ܲ xˆÏ˛õ«˛Ü˛ (complementary function) ~ÓÇ !ӈϢ£Ï §üyܲ° (particular
integral) öyˆÏü x!û˛!•ï˛ ܲÓ˚y •Î˚– 4.5 ~ÓÇ 4.6 §ü#ܲÓ˚î ò%!ê˛ˆÏܲ ˆÎyà ܲˆÏÓ˚ ˛õy•zñ
93
22
1 2 1 2 0 1 22 ( ) 2 ( ) ( )d dx x b x x x xdtdt
ω+ + + + + = f0 cos ωt ...4.7
4.7 §ü#ܲÓ˚î ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ 4.4 §ü#ܲÓ˚ˆÏî ˆÜ˛ö x ˆÜ˛ x1 ~ÓÇ x2 ÈÙÈÓ˚ §ü!‹T !•§yˆÏÓ ˆ°áy •ˆÏÎ˚ˆÏåÈ–
xyüÓ˚y ~ÓyÓ˚ x1
Ä x2ÈÙÈÓ˚ myÓ˚y §)!ã˛ï˛ ˛õ)î≈ §üyôyˆÏöÓ˚ ò%!ê˛ xÇ¢ ~ÓÇ ï˛yˆÏòÓ˚ ˜Ó!¢‹Tƒ=!° xyˆÏ°yã˛öy ܲÓ˚Ó–
4.3.1 fl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyöfl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyöfl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyöfl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyöfl∫“fliyÎ˚# xÓfliyÓ˚ §üyôyö (Transient solution)
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ xyüÓ˚y •z!ï˛üˆÏôƒ•z ˛õ)Ó≈Óï˛≈# ~ܲˆÏܲÓ˚ S~ܲܲ 3) 3.4.3 xö%ˆÏFåȈÏÓ˚ 4.5
§ü#ܲÓ˚î!ê˛Ó˚ §üyôyö ܲˆÏÓ˚!åÈ– ~•z §üyôyö xÌ≈yÍñ 3.17 §ü#ܲÓ˚î xö%§Ó˚î ܲˆÏÓ˚ 4.5 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö
ˆ°áy ÎyÎ˚–
x1(t) = A0 exp (–bt)[cos (ω't + φ)] ...4.8
~áy Ïö ω' = 2 20 bω − ~ÓÇ xöƒyöƒ !ã˛•´=!° ˛õ!Ó˚!ã˛ï˛ xÌ≈ Ó•ö ܲÓ˚ˆÏåÈ–
~ܲܲ 3ÈÙÈ~Ó˚ xyˆÏ°yã˛öyÎ˚ xy˛õ!ö ~Ä ˆòˆÏáˆÏåÈö ˆÎñ 4.8 §üyôyˆÏö x1(t) ~Ó˚ üyö xÓü®ˆÏöÓ˚ ú˛ˆÏ° §üˆÏÎ˚Ó˚
§ˆÏD •…y§ ˛õyˆÏÓ ~ÓÇ e´ü¢ ï˛y ≤ÃyÎ˚ ¢)öƒ •ˆÏÎ˚ ÎyˆÏÓ– ~•zçöƒ ~•z §üyôyö!ê˛ˆÏܲ Ó°y •Î˚ fl∫“fliyÎ˚# xÓfliyÓ˚
§üyôyö (Transient solution)– ≤ß¿ í˛zë˛ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ~•z fl∫“fliyÎ˚# §üyôyˆÏöÓ˚ ≤ÃÜ,˛ï˛ xyÎ˚% Ü˛ï˛«˛î⁄ fl∫“fliyÎ˚#
Ó°ˆÏï˛ §yôyÓ˚îï˛ Ü˛ï˛ê˛y §üÎ˚ ˆÓyé˛yÎ˚⁄
§ˆÏ®• ˆö•zñ ~•z fl∫“ fliy!Î˚ˆÏcÓ˚ §üÎ˚ xyÓyÓ˚ xÓü®ˆÏöÓ˚ üyeyÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°– §yôyÓ˚îï˛ Ó°y •Î˚
ˆÎñ x!ï˛e´yhsˇ §üÎ˚ t Îáö Ÿ’Ìö ܲy° τ ~Ó˚ ˆÌˆÏܲ ˆÓ¢ xˆÏöܲáy!ö Óí˛¸ (t >> τ Óy t > 5τ) •ˆÏÎ˚ ÎyÎ˚ñ
ï˛áö ~•z fl∫“fliyÎ˚# xÓfliy!ê˛ xyÓ˚ ôÓ˚y ˛õˆÏí˛¸ öy– xyüÓ˚y ~ܲܲ 3ÈÙÈ~ ˆòˆÏá!åÈ ˆÎñ ~Ó˚ܲü ˆ«˛ˆÏe ˆòy°ö ≤ÃyÎ˚
Ó¶˛ •ˆÏÎ˚ ÎyÎ˚– ~áyˆÏö !ܲv ã˛y°Ü˛ ӈϰÓ˚ í˛z˛õ!fli!ï˛Ó˚ çöƒ ˛õ!Ó˚!fli!ï˛ !û˛ß¨– ï˛y•z §yôyÓ˚îï˛˛ õ)î≈ §üyôyˆÏöÓ˚
(complete solution) ò%!ê˛ xLjϢÓ˚ ~ܲ!ê˛ [x1] ¢)öƒ •ˆÏÎ˚ ˆàˆÏ°Äñ x !ܲv x
2 Ó˚ í˛z˛õ!fli!ï˛Ó˚ çöƒ ¢)öƒ •Î˚
öyÈÙÙÙÈÓ›ï˛ ï˛sf!ê˛ ï˛áö ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛ ~ܲ•zû˛yˆÏÓ xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyˆÏܲ– ˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏò xyüÓ˚y
~•z fliyÎ˚# xÓfliyÓ˚ §üyôyö!ê˛ !öî≈Î˚ ܲÓ˚Ó–
4.3.2 fliyÎ˚# xÓfliyÓ˚ §üyôyöfliyÎ˚# xÓfliyÓ˚ §üyôyöfliyÎ˚# xÓfliyÓ˚ §üyôyöfliyÎ˚# xÓfliyÓ˚ §üyôyöfliyÎ˚# xÓfliyÓ˚ §üyôyö (Steady state solution)
xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈñ ≤ÃyÌ!üܲ !ܲå%Èê˛y §üÎ˚ x!ï˛e´yhsˇ •ÄÎ˚yÓ˚ ˛õˆÏÓ˚ x1(t) ~Ó˚ üyö x!ï˛ öàîƒ •ˆÏÎ˚
ÎyÎ˚ ~ÓÇ x(t) e´ü¢ 4.6 §ü#ܲÓ˚ˆÏîÓ˚ !ӈϢ£Ï §üyܲ° x2(t) ~Ó˚ §üyö •ˆÏï˛ ÌyˆÏܲ– x2(t) §üyôyö!ê˛ §¡õ)î≈û˛yˆÏÓ
ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#° x1(t) !Ó°#ö •ˆÏÎ˚ ÎyÄÎ˚yÓ˚ ˛õˆÏÓ˚ Îáö x(t) §¡õ)î≈û˛yˆÏÓ x2(t) myÓ˚y
!öô≈y!Ó˚ï˛ •Î˚ (transfer) ñ ï˛áö xyüÓ˚y ˆÎ §üyôyö!ê˛ ˛õy•zñ ï˛yˆÏܲ Ó°y •Î˚ !fliÓ˚yÓfliyÓ˚ §üyôyö (Steady
state solution)– ~•z xÓfliyÓ˚ ܲ¡õöñ Îï˛«˛î ã˛y°Ü˛ Ó° í˛z˛õ!fliï˛ ÌyܲˆÏÓñ ï˛ï˛«˛î ã˛°ˆÏï˛ ÌyܲˆÏÓ– xyüÓ˚y
ôˆÏÓ˚ ˆöÓ ˆÎñ ~•z ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ §üyö •ˆÏÓ– ~•z ò%!ê˛ ˛õÎ≈yÎ˚ xyÓ˚Ä ~ܲê%˛ !ӈϟ’£Ïî
94
ܲÓ˚y ≤ÈÏÎ˚yçö– ôÓ˚y Îyܲñ °â% xÓü®ö §• ~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy ≤ÃyÌ!üܲ û˛yˆÏÓ §yüƒyÓfliyˆÏö !fliÓ˚ Ó˚ˆÏÎ˚ˆÏåÈ–
~ÓyÓ˚ ~Ó˚ IJõÓ˚ ~ܲ!ê˛ §üO§ Ó° ≤ÃÎ%_´ •°ñ ˆÎ ӈϰÓ˚ ܲ¡õyB˛ ~•z !fl±ÈÙÈû˛Ó˚ ÓƒÓfliyÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛
xˆÏ˛õ«˛y !û˛ß¨– ~ÓyÓ˚ ܲ# âê˛ˆÏÓ⁄ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy!ê˛ ï˛yÓ˚ !öçfl∫ ܲ¡õyˆÏB˛ SÎy °â% xÓü®ˆÏöÓ˚ ú˛ˆÏ° x!ï˛
§yüyöƒ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ˆÏåÈ–V xyˆÏ®y!°ï˛ •ˆÏï˛ ˆã˛‹Ty ܲÓ˚ˆÏÓ– !ܲv Ó!•≤Ã≈Î%_´ §üO§ Ó° ˆã˛‹Ty ܲÓ˚ˆÏÓ
ÓƒÓfliy!ê˛ˆÏܲ ӈϰÓ˚ ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ ܲÓ˚ˆÏï˛– ~•z !mü%á# ≤ÃÎ˚y§ á%Ó ˆÓ!¢«˛î xÓ¢ƒ ã˛°ˆÏÓ öyñ ܲyÓ˚î
xÓü®ˆÏöÓ˚ ú˛ˆÏ° !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliy ï˛yÓ˚ ¢!_´ •yÓ˚yˆÏÓ ~ÓÇ ï˛yÓ˚ !ÓhflÏyÓ˚ •…y§ ˆ˛õˆÏï˛ ˆ˛õˆÏï˛ ¢)öƒ •ˆÏï˛ ã˛y•zˆÏÓ–
xöƒ!òˆÏܲ §üO§ Ó° ˛õÎ≈yÓ,_ •yˆÏÓ˚ ÓƒÓfliy!ê˛Ó˚ IJõÓ˚ í˛z˛õ!fliï˛ ˆÌˆÏܲ e´üyàï˛ ï˛yÓ˚ ܲ¡õyB˛ ˆ§áyˆÏö xyˆÏÓ˚y˛õ
ܲÓ˚ˆÏÓ– ~•z ܲyç!ê˛ §¡õ)î≈ •ÄÎ˚yÓ˚ xyˆÏàÓ˚ §üÎ˚ê%˛Ü%˛ ˛õÎ≈hsˇ x“fliyÎ˚# ܲ¡õö ˆòáy ÎyˆÏÓ– ï˛yÓ˚˛õÓ˚ ÷Ó˚& •ˆÏÓ
fliyÎ˚# xÓfliyÓ˚ ܲ¡õöñ ˆÎ ܲ¡õˆÏöÓ˚ §üÎ˚ÈÙÈò)Ó˚c §ü#ܲÓ˚î xyüÓ˚y ~ÓyÓ˚ !öî≈ˆÏÎ˚Ó˚ ˆã˛‹Ty ܲÓ˚Ó–
4.6 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyˆÏöÓ˚ çöƒ xyüÓ˚y ôˆÏÓ˚ !ö•z ˆÎñ ˙ §ü#ܲÓ˚ˆÏîÓ˚ !ӈϢ£Ï §üyôyö x2(t) ˆÜ˛ ˆ°áy
ÎyÎ˚ñ
x2(t) = B cos (ωt – θ) ...4.9
~áyˆÏö B ~ÓÇ θ ò%!ê˛ xK˛yï˛ ô &Óܲ–
xy˛õöyÓ˚ üˆÏö •ÄÎ˚y fl∫yû˛y!Óܲñ ˆÜ˛ö xyüÓ˚y x2(t) ˆÜ˛ §ü#ܲÓ˚î 4.9 ~Ó˚ üï˛ !°á°yü⁄ ~Ó˚ fl∫˛õˆÏ«˛ ò%!ê˛
Î%!_´ Ó˚ˆÏÎ˚ˆÏåÈ– ≤ÃÌüï˛ñ xyüÓ˚y ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ Ó%é˛ˆÏï˛ ˛õyÓ˚!åÈ ˆÎñ fliyÎ˚# xÓfliyÓ˚ §üyôyˆÏöÓ˚ ˆ«˛ˆÏe
§üÎ˚ÈÙÈò%Ó˚c §ü#ܲÓ˚ˆÏîÓ˚ ܲ¡õyB˛ !•§yˆÏÓ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ w í˛z˛õ!fliï˛ ÌyˆÏܲñ ܲyÓ˚î ~áö ˆòy°à!ï˛
§¡õ)î≈Ó˚*ˆÏ˛õ ã˛y°Ü˛ ӈϰÓ˚ myÓ˚y !öÎ˚!sfï˛– !mï˛#Î˚ï˛ñ xÓü®ˆÏöÓ˚ í˛z˛õ!fli!ï˛ˆÏï˛ ã˛y°Ü˛ Ó° ~ÓÇ §Ó˚ˆÏîÓ˚ üˆÏôƒ
~ܲê˛y ò¢yÓ˚ ˛õyÌ≈ܲƒ ˆÌˆÏܲ ÎyˆÏÓ– xÌ≈yÍñ ӈϰÓ˚ ò¢y xÇ¢!ê˛ ωt •ˆÏ°ñ §Ó˚ˆÏîÓ˚ ò¢y •ˆÏÓ (ωt – θ)–
4.9 §ü#ܲÓ˚ˆÏî í˛z˛õ!fliï˛ ò%!ê˛ ô &Óܲ B ~ÓÇ θ ˆÜ˛ !öî≈ˆÏÎ˚Ó˚ í˛zˆÏjˆÏ¢ƒ xyüÓ˚y 4.9 ~Ó˚ í˛zû˛Î˚ ˛õ«˛ˆÏܲ §üˆÏÎ˚Ó˚
§yˆÏ˛õˆÏ«˛ ò%ÛÓyÓ˚ xÓܲ°ö ܲˆÏÓ˚ ˛õy•z–
2 dxdt
= –Bω sin (ωt – θ)
222
dxdt
= –Bω2 cos (ωt – θ)
x2,
2 dxdt
~ÓÇ
222
dxdt
~Ó˚ üyö 4.6 §ü#ܲÓ˚ˆÏî Ó!§ˆÏÎ˚ ˛õy•zñ
(ω02 – ω2) B cos (ωt – θ) – 2B bω sin (ωt – θ) = f
0 cos ωt
cos (ωt – θ) ~ÓÇ sin (ωt – θ) ˆÜ˛ !Óhfl,Ïï˛ Ü˛ˆÏÓ˚ !°ˆÏá ~ÓÇ cos ωt ~ÓÇ sin ωt ~Ó˚ §•à=!°ˆÏܲ
~ܲe ܲˆÏÓ˚ ˛õy•z
[(ω02 – ω2) B cos θ + 2Bbω sin θ – f
0] cos ωt +
[(ω0
2 – ω2) B sin θ – 2Bbω cos θ ] sin ωt = 0 ...4.10
95
xyüÓ˚y çy!ö ˆÎñ sin ωt ~ÓÇ cos ωt ~ܲ•z §ˆÏD ¢)öƒ •ˆÏï˛ ˛õyˆÏÓ˚ öy– Ó›ï˛ Îáö ~ˆÏòÓ˚ ~ܲ!ê˛ ¢)öƒ
•Î˚ñ x˛õÓ˚!ê˛ ï˛áö ï˛yÓ˚ ã˛Ó˚ü üyˆÏö ˆ˛õÑÔåÈÎ˚– xï˛~Ó 4.10 §ü#ܲÓ˚ˆÏîÓ˚ ÓÑy!òˆÏܲ cos ωt ~ÓÇ sin ωtñ í˛zû˛ˆÏÎ˚Ó˚
§•à xy°yòyû˛yˆÏÓ ¢)öƒ •ÄÎ˚y ≤ÈÏÎ˚yçö– §%ï˛Ó˚yÇ ˆ°áy ÎyÎ˚ ˆÎñ
(ω02 – ω2) B cos θ + 2B bω sin θ = f
0...4.11
~ÓÇ (ω02 – ω2) B sin θ – 2Bbω cos θ = 0 ...4.12
(4.12) §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y §Ó˚y§!Ó˚ ˛õy•zñ
tan θ =
220
2b−ω
ωω
...4.13a
Óyñ θ =
122
0
2 tanb −
−ω
ωω
...4.13b
~•z (4.13) §ü#ܲÓ˚î ˆÌˆÏܲ ò%!ê˛ ô &ÓˆÏܲÓ˚ ~ܲ!ê˛ xÌ≈yÍñ θ ~Ó˚ üyö ˛õyÄÎ˚y ÎyˆÏFåÈ– ~áö xy˛õ!ö ˆòáyˆÏï˛
˛õyˆÏÓ˚ö ˆÎñ sin θ =
()1
22 22220
2
4
b
b ⎡⎤ −+ ⎢⎥ ⎣⎦
ω
ωωω
~ÓÇ cos θ =
( )
2 20
12 22 2 2 2
0 4b
−
⎡ ⎤− +⎢ ⎥⎣ ⎦
ω ω
ω ω ω
x˛õÓ˚ ô &Óܲ B Óy !ÓhflÏyˆÏÓ˚Ó˚ üyö ˛õyÄÎ˚yÓ˚ çöƒ sin θ Ä cos θ ~Ó˚ üyö 4.11 §ü#ܲÓ˚ˆÏî ÓƒÓ•yÓ˚ ܲˆÏÓ˚
xyüÓ˚y ˛õy•z
B = ( )0
2 20 cos 2 sin
fb− +ω ω θ ω θ
=
()0
122 2222
04
f
b ⎡⎤ −+ ⎢⎥ ⎣⎦ ωωω
= ( )
01
2 22 2 2 20 4
F
m b⎡ ⎤− +⎢ ⎥⎣ ⎦ω ω ω...4.14
fliyÎ˚# xÓfliyÓ˚ §üyôyˆÏöÓ˚ §¡õ)î≈ Ó˚y!¢üy°y!ê˛ ï˛y•ˆÏ° ~ÓyÓ˚ ˆ°áy Îyܲñ
x2(t) = ( )
01
2 22 2 2 20
cos( )
4
Ft
m b
−⎡ ⎤− +⎢ ⎥⎣ ⎦
ω θω ω ω
...4.15
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ~•z §üyôyˆÏö §Ó˚ˆÏîÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω ~ÓÇ ~!ê˛Ó˚ !ÓhflÏyÓ˚ ܲˆÏÎ˚ܲ!ê˛
!ӣψÏÎ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ~=!° •°ñ
96
(i) ã˛y°Ü˛ ӈϰÓ˚ !ÓhflÏyÓ˚ (F0)
(ii) ã˛y°Ü˛ ӈϰÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ (ω)
(iii) ܲ¡õö¢#° û˛Ó˚ (m)
(iv) ܲ¡õö¢#° ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ Sω0V ~ÓÇ
(v) xÓü®ö ô &Óܲ (b)
ï˛yåÈyí˛¸y 4.13 §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y θ ˆÜ˛yˆÏîÓ˚ üyö ˛õy•z– ~!ê˛ ≤ÃÎ%_´ ã˛y°Ü˛ Ó° §Ó˚ˆÏîÓ˚ ï%˛°öyÎ˚ ˆÎ
ò¢y ˆÜ˛yˆÏî x@ˇÃÓï≈˛# ÌyˆÏܲñ ˆ§!ê˛ˆÏܲ §)!ã˛ï˛ ܲˆÏÓ˚– °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆÎñ θ ˆÜ˛yîÄ Ü˛ˆÏÎ˚ܲ!ê˛ !ӣψÏÎ˚Ó˚
IJõÓ˚ !öû≈˛Ó˚¢#°– ~=!° •°ñ
(i) ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ SωV
(ii) ܲ¡õö¢#° ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ Sω0V
(iii) xÓü®ö ô &Óܲ (b)
˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏò Èθ ÙÈÓ˚ !Ó£ÏÎ˚!ê˛ xyÓyÓ˚ xyˆÏ°yã˛öyÎ˚ xy§ˆÏÓ– 4.3 §ü#ܲÓ˚ˆÏîÓ˚ §¡õ)î≈ §üyôyˆÏöÓ˚ Ó˚*˛õ!ê˛
~áö ˆ°áy ˆÎˆÏï˛ ˛õyˆÏÓ˚ ≠
x(t) = ( )
010
2 22 2 2 20
cos( )exp( )cos( ' )
4
F tA bt t
m b
−− + +⎡ ⎤− +⎢ ⎥⎣ ⎦
ω θω φω ω ω
...4.16
97
xyüÓ˚y ~ÓyÓ˚ §üˆÏÎ˚Ó˚ (t) § ÏD x1, x
2~ÓÇ x ~Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ !Ó£ÏÎ˚!ê˛ˆÏï˛ ò,!‹T !òˆÏï˛ ã˛y•z– 4.2 !ã˛ˆÏe §üˆÏÎ˚Ó˚
§ˆÏD ~=!°Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˆ°á!ã˛e ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~Ó˚ ≤ÃÌü!ê˛ xÌ≈yÍ 4.2a, (t – x1) xyüyˆÏòÓ˚ x!ï˛
˛õ!Ó˚!ã˛ï˛ xÓü!®ï˛ ü%_´ ˆòy°ˆÏöÓ˚ ˆ°á!ã˛e– ~ˆÏ«˛ˆÏe ~!ê˛ fl∫“fliyÎ˚# §üyôyö §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– §üˆÏÎ˚Ó˚ §ˆÏD !ÓhflÏyÓ˚
~áyˆÏö ≤Ãï˛ƒy!¢ï˛ û˛yˆÏÓ•z •…y§ ˛õyˆÏFåÈ– 4.2bñ (t – x2) ˆ°á!ã˛ˆÏe fliyÎ˚# xÓfliyÓ˚ §üyôyö!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
°«˛ƒ ܲÓ˚&öñ ~áyˆÏö !ܲv §üˆÏÎ˚Ó˚ §ˆÏD !ÓhflÏyˆÏÓ˚Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö ˆòáy ÎyˆÏFåÈ öy– !fliÓ˚yÓfliyÓ˚ ~!ê˛ xöƒï˛ü
˜Ó!¢‹Tƒ–
4.2c, (t – x ) ˆ°á!ã˛ˆÏe §üˆÏÎ˚Ó˚ §ˆÏD x1 + x
2) ~Ó˚ ˛õ!Ó˚Óï≈˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö ≤ÃÌü!òˆÏܲ
!ÓhflÏyÓ˚ ˛õ!Ó˚Óï≈˛ö¢#°– ˆÜ˛ööy ~•z ˛õÎ≈yˆÏÎ˚ ï˛sf!ê˛ ~ܲ!òˆÏܲ !öçfl∫ ω0 ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ •ˆÏï˛
ã˛yÎ˚ ~ÓÇ xÓü®ˆÏöÓ˚ çöƒ ï˛yÓ˚ !ÓhflÏyÓ˚ e´ü¢ ܲüˆÏï˛ ÌyˆÏܲ– !ܲv Óy•zˆÏÓ˚ ˆÌˆÏܲ ≤ÃÎ%_´ Ó° ï˛sf!ê˛ˆÏܲ ï˛yÓ˚
!öçfl∫ ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ SωV ã˛y°öy ܲÓ˚ˆÏï˛ ã˛yÎ˚– !û˛ß¨ ܲ¡õyˆÏB˛Ó˚ ò%•z xyˆÏ®y°ˆÏöÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ°
x“ §üˆÏÎ˚Ó˚ çöƒ ï˛ˆÏsf !ÓhflÏyˆÏÓ˚Ó˚ •…y§ÈÙÈÓ,!k˛ °«˛ƒ ܲÓ˚y ÎyÎ˚– ~•z xÓfliy!ê˛ˆÏܲ Ó°y •Î˚ x“fliyÎ˚# fl∫Ó˚ܲ¡õ
(transient beats)– xÓ¢ƒ ~•z xÓfliy!ê˛ x“ §üÎ˚ fliyÎ˚# •Î˚ ˆÜ˛ööyñ ï˛ˆÏsfÓ˚ !öçfl∫ ܲ¡õyˆÏB˛ xyˆÏ®y°ö
°Î˚≤ÃyÆ •ÄÎ˚yÓ˚ ˛õÓ˚ ã˛y°Ü˛ Ó° ï˛yÓ˚ !öçfl∫ ܲ¡õyˆÏB˛ ï˛sfˆÏܲ xyˆÏ®y!°ï˛ ܲˆÏÓ˚ ~ÓÇ ¢!_´Ó˚ §Ó˚ÓÓ˚yˆÏ•Ó˚ üyôƒˆÏü
!ÓhflÏyÓ˚ x«%˛] ÌyˆÏܲ– ï˛y•z §üˆÏÎ˚Ó˚ §ˆÏD §Ó˚ˆÏîÓ˚ ˛õ!Ó˚Óï≈˛ö x“ §üÎ˚ ˛õˆÏÓ˚•z x2 ~Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ xö%Ó˚*˛õ •ˆÏÎ˚
ÎyÎ˚– x“fliyÎ˚# fl∫Ó˚ܲˆÏ¡õÓ˚ xÓ§yö âê˛yÓ˚ ˛õÓ˚ fliyÎ˚# xÓfliyÓ˚ §üyôyö!ê˛ ~ܲy•z ï˛ˆÏsfÓ˚ xyã˛Ó˚î ≤Ãܲy¢ ܲˆÏÓ˚–
~•z xyˆÏ°yã˛öyÓ˚ ˛õˆÏÓ˚ !öˆÏã˛Ó˚ xö%¢#°ö#!ê˛ Ü˛Ó˚ˆÏï˛ xy˛õöyÓ˚ !öÿ˛Î˚•z û˛y° °yàˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -1 : 200Nm–1 !fl±Ç ô &Óܲ !Ó!¢‹T ~ܲ!ê˛ !fl±Ç ˆÌˆÏܲ 0.2 kg û˛Ó˚ é%˛°ˆÏåÈ– û˛Ó˚!ê˛Ó˚ IJõÓ˚
!e´Î˚y¢#° xÓü®ö Ó° – 2v Nm–1s ˆÎáyˆÏö v û˛Ó˚!ê˛Ó˚ ˆÓà– Î!ò ~•z û˛Ó˚!ê˛Ó˚ IJõÓ˚ 12 cos 50t !öí˛zê˛ö
Ó° !e´Î˚y ܲˆÏÓ˚ñ ï˛y•ˆÏ° fliyÎ˚# xÓfliyÎ˚ û˛Ó˚!ê˛Ó˚ !ÓhflÏyÓ˚ Ä ≤ÃÎ%_´ ӈϰÓ˚ §ˆÏD §Ó˚ˆÏîÓ˚ ò¢y ˛õyÌ≈ܲƒ !öî≈Î˚ ܲÓ˚&ö–
4.4 ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy–ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy–ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy–ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy–ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ û)˛!üܲy–
xyˆÏàÓ˚ xö%ˆÏFåȈÏòÓ˚ 4.15 §ü#ܲÓ˚î ˆÌˆÏܲ fliyÎ˚# xÓfliyÓ˚ ܲ¡õˆÏöÓ˚ ˆÎ §üyôyö ˛õyÄÎ˚y ˆàˆÏåÈñ ˆ§áyˆÏö
!öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ fliyÎ˚# xÓfliyÎ˚ ï˛ˆÏsfÓ˚ ܲ¡õyB˛ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyˆÏB˛Ó˚ §üyö– ï˛y•ˆÏ° !öÿ˛Î˚•z
Ó°y ÎyÎ˚ ˆÎñ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏ°ñ ï˛ˆÏsfÓ˚ xyˆÏ®y°ˆÏöÓ˚ ܲ¡õyB˛Ä ˛õyˆÏŒê˛ ÎyˆÏÓ– !Ó£ÏÎ˚!ê˛
!ܲv ~áyˆÏö•z ˆ¢£Ï öÎ˚– ܲyÓ˚îñ x2(t) ~Ó˚ !ÓhflÏyÓ˚ (B) ~ÓÇ ò¢yˆÏܲyî SθVñ ò%!ê˛ˆÏï˛•z ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛
ω ~ÓÇ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ω0 í˛zû˛ˆÏÎ˚•z í˛z˛õ!fliï˛ Ó˚ˆÏÎ˚ˆÏåÈ– ï˛y•z ~•z ≤ß¿ á%Ó•z §Dï˛ ˆÎñ ω ~ÓÇ ω0
ÈÙÈÓ˚ ï%˛°öyü)°Ü˛ üyˆÏöÓ˚ IJõÓ˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ=!° ܲ#û˛yˆÏÓ !öû≈˛Ó˚¢#°– xyüÓ˚y ~•z xö%ˆÏFåȈÏò
ω << ω0 ~ÓÇ ω >> ω
0ñ ~•z ò%•z ˆ«˛ˆÏeÓ˚ ˜Ó!¢‹Tƒ xyˆÏ°yã˛öy ܲÓ˚Ó– ï,˛ï˛#Î˚ ˆ«˛e!ê˛ñ Îáö ω = ω0ñ xyˆÏ°y!ã˛ï˛
•ˆÏÓ ˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏò–
Îáö ω << ω0ñ SxÌ≈yÍ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ ï%˛°öyÎ˚ á%Ó•z ˆåÈyê˛V
xyüÓ˚y çy!ö [(4.14) §ü#ܲÓ˚î ˆÌˆÏܲ ],
98
B = ( )
01
2 22 2 2 20 4
f
b⎡ ⎤− +⎢ ⎥⎣ ⎦ω ω ω
~áyö ˆÌˆÏܲ ˆ°áy ÎyÎ˚ ˆÎñ
B = 01
2 22 22 2
400 0
1 4
f
b⎡ ⎤⎧ ⎫⎪ ⎪⎛ ⎞⎢ ⎥− +⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦
ω ωω ω ω
ˆÎˆÏ•ï%˛ ω << ω0, 0
ωω <<1, §%ï˛Ó˚yÇ 1 ~Ó˚ ï%˛°öyÎ˚
2
0
⎛⎞⎜⎟ ⎝⎠
ωω
˛õòˆÏܲ öàîƒ ôˆÏÓ˚ xyüÓ˚y ˛õy•zñ
B0 = 020
fω =
02
0
Fmω
=
0 Fk
ˆÎáy Ïö ω02 =
km
~ÓÇ k = ≤Ãï˛ƒyöÎ˚ö =îyB˛ (stiffness const.), ~ˆÏ«˛ˆÏe !ÓhflÏyÓ˚ ≤Ãï˛ƒyöÎ˚ö =îyB˛
myÓ˚y !öÎ˚!sfï˛ •Î˚ ~ÓÇ ω0 Ó,!k˛Ó˚ §ˆÏD §ˆÏD •…y§ ˛õyÎ˚– ï˛y•z ~•z xÓfliyÎ˚ ï˛ˆÏsfÓ˚ ˆÎ à!ï˛ °«˛ƒ ܲÓ˚y ÎyÎ˚ñ
ï˛yˆÏܲ ≤Ãï˛ƒyöÎ˚ö =îyB˛ !öÎ˚!sfï˛ à!ï˛ (stiffness control motion) Ó°y •Î˚–
ò¢yÓ˚ !Ó£ÏÎ˚!ê˛ !ÓˆÏÓã˛öyÓ˚ çöƒ 4.13a §ü#ܲÓ˚î!ê˛ˆÏï˛ !ú˛ˆÏÓ˚ ÎyÄÎ˚y Îyܲ–
tan θ =
220
2bωωω−
= 2
220
0
2 01
b →⎛ ⎞−⎜ ⎟⎝ ⎠
ωωωω
Îáö 0
ωω << 1
§%ï˛Ó˚yÇ ò¢yˆÏܲyî θ → 0 xÌ≈yÍ ~ˆÏ«˛ˆÏe ã˛y°Ü˛ Ó° ~ÓÇ fliyÎ˚# xÓfliyÓ˚ §Ó˚ˆÏîÓ˚ üˆÏôƒ ò¢yÓ˚ ˛õyÌ≈ܲƒ ≤ÃyÎ˚
¢)öƒ •ˆÏÎ˚ ÎyÎ˚– 4.3 !ã˛ˆÏe Èω ÙÈÓ˚ §ˆÏD θ ÈÙÈÓ˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˆ°á!ã˛e!ê˛ ˆòáˆÏ° !Ó£ÏÎ˚!ê˛ Ó%é˛ˆÏï˛ xy˛õöyÓ˚ §%!Óôy
•ˆÏÓ–
Îáö ω >> ω0 SxÌ≈yÍ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ ï%˛°öyÎ˚ á%Ó•z ˆåÈyê˛Vñ
~ˆÏ«˛ˆÏe Îáö ω >> ω0,
0 ωω
<< 1
99
01
22 2220
224 1
fB
b
=⎡⎤ ⎛⎞ −+ ⎢⎥ ⎜⎟ ⎝⎠ ⎢⎥ ⎣⎦
ω ωωω
0 02 2
f Fmω ω
≅ =
~ˆÏ«˛ˆÏe ω Ó,!k˛Ó˚ §ˆÏD §ˆÏD !ÓhflÏyÓ˚ •…y§ ˛õyÎ˚– ~åÈyí˛¸y ܲ¡õö¢#° û˛Ó˚!ê˛Ó˚ üyö Ó,!k˛Ó˚ §ˆÏD §ˆÏD !ÓhflÏyÓ˚
•…y§ ˛õyÎ˚–
~ˆÏ«˛ˆÏeñ tan θ = 2 20
2bωω ω−
= 202
2
1
bωωω
−⎛ ⎞−⎜ ⎟⎝ ⎠
→ 0 ñÎáöñ
0 ωω
→ 0
xï˛~Óñ θ → π
ܲyÓ˚îñ ~ˆÏ«˛ˆÏe tan θ ÈÙÈÓ˚ üyö ω Ó,!k˛Ó˚ §ˆÏD §ˆÏD ¢)ˆÏöƒÓ˚
!öܲê˛Óï≈˛# •ˆÏ°Äñ ï˛y îydܲ üyˆÏöÓ˚ !òܲ ˆÌˆÏܲ ¢)ˆÏöƒÓ˚ !òˆÏܲ
x@ˇÃ§Ó˚ •ˆÏFåÈ– xï˛~Óñ θ fli(° ˆÜ˛yî §)!ã˛ï˛ ܲÓ˚ˆÏåÈ Ä Îáö ω →∞
•ˆÏFåÈ ï˛áö θ ˆÜ˛yö π (180º) ~Ó˚ !öܲê˛Óï≈˛# •ˆÏFåÈ– (4.3 !ã˛e
ˆòá%öV–
ï˛y•z Ó°y ÎyÎ˚ ˆÎñ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ
ܲ¡õyˆÏB˛Ó˚ ï%˛°öyÎ˚ Îï˛ Óí˛¸ •Î˚ñ ï˛ï˛ ã˛y°Ü˛ Ó° Ä §Ó˚ˆÏîÓ˚ üˆÏôƒ
ò¢yÓ˚ ˛õyÌ≈ܲƒ π Óy 180º Ó˚ !öܲê˛Óï≈˛# •Î˚ñ xÌ≈yÍ ï˛yÓ˚y !Ó˛õÓ˚#ï˛
ò¢yÎ˚ xÓfliyö ܲˆÏÓ˚– 4.3 !ã˛ˆÏe ~•z !Ó£ÏÎ˚!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏòÓ˚ 4.4 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ Ü˛¡õyB˛ ω ÈÙÈÓ˚ §ˆÏD !ÓhflÏyÓ˚ B ÈÙÈÓ˚ ˛õ!Ó˚Óï≈˛ö–
xy§%ö ~ÓyÓ˚ ÓÓ˚Ç !öˆÏã˛Ó˚ xö%¢#°ö#!ê˛ ˆã˛‹Ty ܲˆÏÓ˚ ˆòá%ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -2 : ~ܲ!ê˛ 0.1kg û˛Ó˚ 150Nm–1 !fl±Ç ô &Óܲ !Ó!¢‹T !fl±ÇÈÙÈ~ ˆé˛y°yˆÏöy Ó˚ˆÏÎ˚ˆÏåÈ– û˛Ó˚!ê˛Ó˚
IJõÓ˚ !e´Î˚y¢#° â£Ï≈î Ó° Î!ò Fd = 5v N •Î˚ Sv û˛Ó˚!ê˛Ó˚ ˆÓàVñ ï˛y•ˆÏ° xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚î
≤Ã!ï˛¤˛y ܲˆÏÓ˚ ˙ ˆòy°ˆÏöÓ˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚&ö– Î!ò ~ÓyÓ˚ ~•z ï˛sf!ê˛Ó˚ í˛z˛õÓ˚ F = 3 cos 20t N
üyˆÏöÓ˚ ˛õÎ≈yÓ,_ Ó° xyˆÏÓ˚y!˛õï˛ •Î˚ñ ï˛y•ˆÏ° fliyÎ˚# xÓfliyÎ˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ Ä ò¢y ˛õyÌ≈ܲƒ !öî≈Î˚
ܲÓ˚&ö–
ω0 ω
π
100
4.5 xö%öyòxö%öyòxö%öyòxö%öyòxö%öyò
4.5 xö%ˆÏFåȈÏòÓ˚ xyˆÏ°yã˛öyÎ˚ xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ fliyÎ˚# xÓfliyÓ˚ ܲ¡õˆÏöÓ˚ (Steady state vibration)
ˆ«˛ˆÏe ω ~ÓÇ ω0 ~Ó˚ ˛õyÌ≈ܲƒ Îï˛ ˆÓ!¢ •Î˚ñ ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ ï˛ï˛•z •…y§ ˛õyÎ˚– ï˛y•z ω >> ω0 Óy ω<< ω0 •ˆÏ° !ÓhflÏyÓ˚ í˛zû˛Î˚ ˆ«˛ˆÏe•z ˆÓ¢ ܲˆÏü ÎyÎ˚– fl∫yû˛y!Óܲû˛yˆÏÓ•z xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ω xyÓ˚
ω0 ÈÙÈ~Ó˚ ˛õyÌ≈ܲƒ ˆÓ!¢ •ˆÏ° !ÓhflÏyÓ˚ Îáö ܲˆÏü ÎyÎ˚ñ ï˛áö ˙ ò%!ê˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ܲyåÈyܲy!åÈ ~ˆÏ° !ÓhflÏyÓ˚
§Ω˛Óï˛ Ó,!k˛ ˆ˛õˆÏï˛ ˛õyˆÏÓ˚– ~ÓyÓ˚ xyüÓ˚y ω = ω0 •ˆÏ° xÌ≈yÍñ ã˛y°Ü˛ Ó° Ä ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ò%•z!ê˛
§üyö •ˆÏÎ˚ ˆàˆÏ° ܲ# •ˆÏÓ ˆ§•z !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚Ó– 4.14 §ü#ܲÓ˚ˆÏî !fliÓ˚yÓfliyÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ Ó˚y!¢üy°y!ê˛
•z!ï˛üˆÏôƒ•z ˛õyÄÎ˚y ˆàˆÏåÈ ≠
()0
122 2222
0
()
4
fB
b
ωωωω
=⎡⎤ −+ ⎢⎥ ⎣⎦
~áyˆÏö ω = ω0 Ó!§ˆÏÎ˚ ˛õy•z
000
00
()22
fFB
bmbωωω ==
...4.17
xÌ≈yÍñ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ω0 ~ÓÇ ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ SωV §üyö •ˆÏÎ˚ ˆàˆÏ° BÈÙÈ~Ó˚ Ó˚y!¢üy°yÓ˚
•Ó˚ (denominator) •…y§ ˛õyÎ˚ ~ÓÇ BÈÙÈ~Ó˚ üyö Ó,!k˛ ˛õyÎ˚– ~•z xÓfliy!ê˛ˆÏܲ xö%öyò Ó°y •Î˚– °«˛ƒ ܲÓ˚&ö
ˆÎñ B(ω0)ÈÙÈ~Ó˚ ~•z üyö 0
02f
bω ñ b Óy xÓü®ö ô &ÓˆÏܲÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°– Ó›ï˛ñ xÓü®ö ï˛Ìy b ~Ó˚
üyö Ó,!k˛Ó˚ §ˆÏD §ˆÏD B(ω0) ~Ó˚ üyö •…y§ ˛õyÎ˚– ˆÎ ˆÜ˛yöÄ ÓyhflÏÓ Ü˛¡õö¢#° ï˛ˆÏsf §Ó≈òy•z !ܲå%È xÓü®ö
í˛z˛õ!fliï˛ ÌyˆÏܲ xÌ≈yÍñ b ~Ó˚ üyö ¢)öƒ •Î˚ öy– ú˛ˆÏ° B(ω0) ~Ó˚ üyö x§#ˆÏüÓ˚ !òˆÏܲ ã˛ˆÏ° ÎyÎ˚ öy–
ω = ω0 •ÄÎ˚yÓ˚ §ˆÏD §ˆÏD ò¢y ˆÜ˛yî θ ܲ#Ó˚ܲü •ˆÏÓ⁄
4.13a §ü#ܲÓ˚î ˆÌˆÏܲ
tan θ =
220
2bωωω−
ú˛ˆÏ°ñ ω = ω0 •ˆÏ°ñ tan θ = ∞ ∴ θ = 2π
xï˛~Ó ~ˆÏ«˛ˆÏe ã˛y°Ü˛ ӈϰÓ˚ ï%˛°öyÎ˚ §Ó˚î
2π
ò¢yˆÏܲyî !˛õ!åȈÏÎ˚ ÌyܲˆÏÓ–
101
4.17 §ü#ܲÓ˚ˆÏî !ÓhflÏyˆÏÓ˚Ó˚ ˆÎ üyö B(ω0) ˛õyÄÎ˚y ˆàˆÏåÈñ ï˛y !ÓhflÏyˆÏÓ˚Ó˚ §ˆÏÓ≈yFã˛üyö öÎ˚– !ÓhflÏyˆÏÓ˚Ó˚ §ˆÏÓ≈yFã˛
üyö ˛õyÄÎ˚yÓ˚ çöƒ xyüyˆÏòÓ˚ !ܲå%Èê˛y !û˛ß¨ Ó˚yhflÏyÎ˚ ~ˆÏàyˆÏï˛ •ˆÏÓ– ˆÜ˛Ó° ï˛y•z öÎ˚ñ ≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚ ˆ«˛ˆÏe
xyüÓ˚y ˆòáˆÏï˛ ˛õyÓñ ï˛ˆÏsfÓ˚ à!ï˛ˆÏÓà !ӈϢ£Ï ¢ï≈˛ §yˆÏ˛õˆÏ«˛ §ˆÏÓ≈yFã˛ •Î˚ ~ÓÇ ~•z ¢ï≈˛ §ˆÏÓ≈yFã˛ !ÓhflÏyÓ˚ ˛õyÄÎ˚yÓ˚
¢ï≈˛ xˆÏ˛õ«˛y !û˛ß¨– ܲ¡õö¢#° Ó›Ó˚ !ÓhflÏyÓ˚ Îáö §ˆÏÓ≈yFã˛ •Î˚ñ ï˛áö ï˛yÓ˚ !fli!ï˛¢!_´ §ˆÏÓ≈yFã˛ üyˆÏö ˆ˛õÑÔåÈÎ˚
~ÓÇ Îáö ï˛yÓ˚ à!ï˛ˆÏÓà §Ó≈ˆÏ˛õ«˛y ˆÓ!¢ •Î˚ñ ï˛áö ï˛yÓ˚ à!ï˛¢!_´Ó˚ ˛õ!Ó˚üyî •Î˚ §ˆÏÓ≈yFã˛– ~•z ò%!ê˛ ˆ«˛eˆÏܲ
˛õòyÌ≈!ÓòƒyÓ˚ ˛õ!Ó˚û˛y£ÏyÎ˚ ÎÌye´ˆÏü !ÓhflÏyÓ˚ xö%öyò (amplitude resonance) ~ÓÇ à!ï˛ˆÏÓà xö%öyò
(velocity resonance) Ó°y •Î˚– ~áyˆÏö xyüÓ˚y !Ó£ÏÎ˚ ò%!ê˛ !öˆÏÎ˚ §ÇˆÏ«˛ˆÏ˛õ xyˆÏ°yã˛öy ܲÓ˚Ó–
4.5.1 !ÓhflÏyÓ˚ xö%öyò!ÓhflÏyÓ˚ xö%öyò!ÓhflÏyÓ˚ xö%öyò!ÓhflÏyÓ˚ xö%öyò!ÓhflÏyÓ˚ xö%öyò
~ÓyÓ˚ ˆòáy Îyܲñ ≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚ ˆ«˛ˆÏe !ÓhflÏyˆÏÓ˚Ó˚ §ˆÏÓ≈yFã˛ üyö ˛õyÄÎ˚yÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï˛≈!ê˛ Ü˛#⁄
4.14 §ü#ܲÓ˚ˆÏî xyüÓ˚y !ÓhflÏyÓ˚ B(ω) ~Ó˚ ˆÎ Ó˚y!¢üy°y!ê˛ ˆ˛õˆÏÎ˚!åÈ ˆ§!ê˛ •°ñ
B(ω) =
()0
122 2222
04
f
b ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
~Ó˚ üyö ω ~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° ~ÓÇ È
()2 222204b ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
Ó˚y!¢Ó˚ üyö Îáö §Ó≈!ö¡¨ñB(ω) ~Ó˚ üyö
ï˛áö §ˆÏÓ≈yFã˛ •ˆÏÓ– ~Ó˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï˛≈ ≠()2 222204 db
dωωω ω
⎡⎤ −+ ⎢⎥ ⎣⎦
= 0
Óyñ – 4ω (ω0
2 – ω2) + 8b2ω = 0
xÌ≈yÍñ ω =
()1
22202b ω ±−
xÌÓy 0
~Ó˚ üˆÏôƒ ω = 0 §üyôyö!ê˛ ˆÜ˛yöÄ Ó¶˛ö !öˆÏò≈¢ ܲˆÏÓ˚ öy– ï˛y•z xyüÓ˚y ˆ§!ê˛ Óyò ˆòÓ– ~åÈyí˛¸yñ ˆÎˆÏ•ï%˛
ω §Ó≈òy•z ôöydܲñ B(ω) ~Ó˚ üyö §ˆÏÓ≈yFã˛ •ÄÎ˚yÓ˚ ¢ï≈˛ ≠ ω = ωr = ( )
12 2 2
0 2bω − ... 4.18a
ωr ~áyˆÏö xö%öyò# ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˆÓyé˛yˆÏFåÈ– xy˛õ!ö •Î˚ï˛ °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ï,˛ï˛#Î˚ ~ܲˆÏܲ xyüÓ˚y
ˆÎ xÓü!®ï˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ 2 20 bω − ˆ˛õˆÏÎ˚!åÈñ ωr ~Ó˚ üyö ï˛yÓ˚ ˆã˛ˆÏÎ˚ ܲü– ≤çDï˛ Ó°y ÎyÎ˚ ˆÎñ
B(ω = ωr) ˆÎ B ~Ó˚ §ˆÏÓ≈yFã˛ üyöñ §Ó≈!ö¡¨ öÎ˚ñ ï˛y xy˛õ!ö
2
2
r
dBdωω ω=
⎛⎞⎜⎟ ⎝⎠
~Ó˚ üyö !öî≈Î˚ ܲˆÏÓ˚ ~ÓÇ ˆ§!ê˛Ó˚
îydܲ üyö °«˛ƒ ܲˆÏÓ˚ ≤Ã!ï˛˛õߨ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö–
~áö ω ~Ó˚ °∏˛üyö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y !ÓhflÏyÓ˚ BÈÙÈ~Ó˚ §ˆÏÓ≈yFã˛ üyö ˆ˛õˆÏï˛ ˛õy!Ó˚ ≠
102
Bmax
=
( ) ( )0
12 22 2 2 2 2 2
0 0 02 4 2
F
m b b bω ω ω⎡ ⎤− + + −⎢ ⎥⎣ ⎦
=
022
0 2
F
mbb ω−
... 4.18b
ï˛y•ˆÏ° ˆòáy ÎyˆÏFåÈñ ω = ω0 •ˆÏ° BÈÙÈ~Ó˚ üyö §ˆÏÓ≈yFã˛ •Î˚ öy– B ÈÙÈ~Ó˚ üyö §ˆÏÓ≈yFã˛ •Î˚ñ Îáö ω =
ωr =
()1
22202b ω−
– !ÓhflÏyÓ˚ BÈÙÈ~Ó˚ üyö §ˆÏÓ≈yFã˛ •ÄÎ˚yÓ˚ ~•z âê˛öy!ê˛ˆÏܲ Ó°y •Î˚ !ÓhflÏyÓ˚ xö%öyò (amplitude
resonance)– xyÓ˚ ωr ˆÜ˛ Ó°y •Î˚ !ÓhflÏyÓ˚ xö%öyˆÏòÓ˚
çöƒ xö%öyò# ˆÜ˛Ô!îܲ ܲ¡õyB˛ (resonant angular
frequency)–
4.4 !ã˛ˆÏe ω ~Ó˚ §ˆÏD ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚
B ~Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˆ°á!ã˛e ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– °«˛ƒ
ܲÓ˚ˆÏÓö xÓü®ö ô &Óܲ bÈÙÈ~Ó˚ Ó,!k˛Ó˚ §ˆÏD ˆÜ˛Ó° ωr
ÈÙÈ•z 4.18a §ü#ܲÓ˚î xö%ÎyÎ˚# ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ öyñ
ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ B-~Ó˚ üyöÄ •…y§ ˛õyÎ˚– Î!ò xyüÓ˚y
ˆÜ˛yöÄ xyò¢≈ ï˛ˆÏsfÓ˚ ܲÌy ܲ“öy ܲ!Ó˚ñ ˆÎáyˆÏö
xÓü®ö í˛z˛õ!fliï˛ ˆö•zñ xÌ≈yÍ b = 0, ï˛y•ˆÏ° ~ܲ!òˆÏܲ
ˆÎüö ωr = ω0 •ˆÏÎ˚ ÎyˆÏÓ Óy xö%öyò# ܲ¡õyB˛ ï˛ˆÏsfÓ˚
fl∫yû˛y!Óܲ ܲ¡õy ÏB˛Ó˚ §üyö • ÏÎ˚ Îy ÏÓñ ï˛üö•z xöƒ!ò Ïܲ
§ˆÏÓ≈yFã˛ !ÓhflÏyˆÏÓ˚Ó˚ üyö •ˆÏÎ˚ ÎyˆÏÓ x§#ü S§ü#ܲÓ˚î
4.17 ˆòá%öV– Ó°y Óy‡°ƒñ xyüÓ˚y ÓyhflÏÓ ˆ«˛ˆÏe ~Ó˚ܲü xÓfliy ˛õy•z öy–
!ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ xyÓ˚ ~ܲ!ê˛ =Ó˚&c˛õ)î≈ !òܲ Ó˚ˆÏÎ˚ˆÏåÈ– xy˛õ!ö çyˆÏöö ˆÎñ ܲ¡õö¢#° Ó›Ó˚ ˆ«˛ˆÏe ï˛yÓ˚
!fli!ï˛¢!_´ ≠
V = 212
kx ...4.19
~áyˆÏöñ k = ω02m = ≤Ãï˛ƒyöÎ˚ö =îyB˛ (spring constant)– ˆÎˆÏ•ï%˛ ~ܲ!ê˛ ï˛ˆÏsf k Ó˚ üyö ~ܲ!ê˛ ô &Óܲñ
ï˛y•z !fli!ï˛¢!_´Ó˚ üyö §ˆÏÓ≈yFã˛ •Î˚ Îáö x ~Ó˚ üyö §ˆÏÓ≈yFã˛ñ xÌ≈yÍñ Îáö x = Bmax
–
§%ï˛Ó˚yÇñ Vmax
=
2max
12
kB
ˆÎˆÏ•ï%˛ !ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ §üÎ˚ BÈÙÈ~Ó˚ üyö §ˆÏÓ≈yFã˛ •Î˚ñ ï˛y•z !ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ §üÎ˚•z ï˛ˆÏsfÓ˚ §ˆÏÓ≈yFã˛
!fli!ï˛¢!_´Ó˚ üyö §ˆÏÓ≈yFã˛ •Î˚– ~!ê˛ !ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ ~ܲ!ê˛ !Óܲ“ §ÇK˛y–
!ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ §üÎ˚ñ xÌ≈yÍ Îáö ω = ωr =
2202b ω−
103
§%ï˛Ó˚yÇñ §ˆÏÓ≈yFã˛ !fli!ï˛¢!_´ Vm =
2
02 2
0
12 2
Fk
mb b
⎛ ⎞⎜ ⎟
ω −⎝ ⎠ =
()2200
2220
18
Fmbb −
ωω
...4.20
4.5.2 à!ï˛ˆÏÓà xö%öyòà!ï˛ˆÏÓà xö%öyòà!ï˛ˆÏÓà xö%öyòà!ï˛ˆÏÓà xö%öyòà!ï˛ˆÏÓà xö%öyò
≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe fliyÎ˚# xÓfliyÓ˚ §üÎ˚ ÈÙÈ §Ó˚î §ü#ܲÓ˚ˆÏîÓ˚ S§ü#ܲÓ˚î 4.15) §ˆÏD xy˛õ!ö xyˆÏà•z
˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåöÈ– ~!ê˛ •° ≠
x = x2 = ( )
01
2 22 2 2 20
cos( )
4
F t
m b
−
⎡ ⎤− +⎢ ⎥⎣ ⎦
ω θ
ω ω ω
~!ê˛ ˆÌˆÏܲ xyüÓ˚y ܲ¡õö¢#° û˛ˆÏÓ˚Ó˚ à!ï˛ˆÏÓà ˆ˛õˆÏï˛ ˛õy!Ó˚ ≠
v = dxdt
=
()0
122 2222
0
sin()
4
Ft
mb
− −⎡⎤ −+ ⎢⎥ ⎣⎦
ωωθ
ωωω
îydܲ !ã˛•´ §Ó˚ˆÏîÓ˚ §yˆÏ˛õˆÏ«˛ à!ï˛ˆÏÓˆÏàÓ˚ x!û˛ü%á §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– ~•z à!ï˛ˆÏÓˆÏàÓ˚ §ˆÏÓ≈yFã˛ üyö Óy
à!ï˛ˆÏÓà !ÓhflÏyÓ˚ (velocity amplitude) v0 •ˆÏ° ˆ°áy ÎyÎ˚–
v0 =
( )0
12 22 2 2 2
0 4
F
m b⎡ ⎤− +⎢ ⎥⎣ ⎦
ω
ω ω ω...4.21
ˆòáy ÎyˆÏFåÈñ ˆÎ v0 ÈÙÈÓ˚ üyö F
0, m, b ~ÓÇ ω
0 åÈyí˛¸yÄ ω ÈÙÈ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– ~áö xy˛õ!ö §•ˆÏç•z
ÈÙÈ ω ~Ó˚ ˆÜ˛yö üyˆÏöÓ˚ çöƒ v0 §Ó≈y!ôܲ •ˆÏÓñ ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– 4.21 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ñ
0dvdω =
122222 02
0 [()4F
bm
− −+ ωωω
1
2 2 2 2 3 2 220 0( ) 4 (4 4 8 )]2 b b
−− − + − +ω ω ω ω ω ωω ω
=
( )
4 40 0
32 22 2 2 2
0 4
Fm
b
−
⎡ ⎤− +⎢ ⎥⎣ ⎦
ω ω
ω ω ω
ω ~Ó˚ ˆÎ @ˇÃ•îˆÏÎyàƒ üyˆÏöÓ˚ çöƒ 0dvdω = 0, ˆ§!ê˛ •° ω = ω
0– ˆòáyˆÏöy ÎyÎ˚ ˆÎñ Îáö ω = ω
0
ï˛áö
202
dvdω
~Ó˚ üyö îydܲ– ~Ó˚ xÌ≈ñ v0 ÈÙÈ~Ó˚ §ˆÏÓ≈yFã˛ üyˆÏöÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï˛≈ñ
ω = ω0
...4.22
104
~•z xÓfliy!ê˛ˆÏܲ Ó°y •Î˚ à!ï˛ˆÏÓà xö%öyò (velocity resonance)– 4.21 §ü#ܲÓ˚ˆÏî ω = ω0
Ó!§ ÏÎ˚
v0 ÈÙÈ~Ó˚ §ˆÏÓ≈yFã˛ üyö ˛õyÄÎ˚y ÎyÎ˚ ≠
v0m =
001
2220 04
F
mb
ω
ω ⎡⎤ + ⎣⎦
=
0
2Fmb
...4.23a
°«˛ƒ ܲÓ˚&öñ v0m
~Ó˚ Ó˚y!¢üy°yÓ˚ •ˆÏ° b
í˛z˛õ!fliï˛ Ó˚ˆÏÎ˚ˆÏåÈ xÌ≈yÍñ xÓü®ö Îï˛ Ó,!k˛ ˛õyˆÏÓ
à!ï˛ˆÏÓà xö%öyˆÏòÓ˚ §üÎ˚ ≤ÃyÆ v0 ~Ó˚ §ˆÏÓ≈yFã˛
üyö ï˛ï˛ ˆåÈyê˛ •ˆÏÓ–
4.5 !ã˛ˆÏe ω ~Ó˚ §ˆÏD v0 ~Ó˚ ˛õ!Ó˚Óï≈˛ö
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– §Óܲ!ê˛ ˆ«˛ˆÏe•z Îáö ω =
ω0ñ ˆÜ˛Ó° ï˛áö•z v0 ~Ó˚ üyö §ˆÏÓ≈yFã˛ •ˆÏï˛
ˆòáy ÎyˆÏFåÈ– ï˛ˆÏÓ ~•z ˆ°á!ã˛e ˆÌˆÏܲ ~!ê˛
˛õ!Ó˚‹ÒyÓ˚ ˆÎñ v0m
~Ó˚ üyö !öû≈˛Ó˚ ܲÓ˚ˆÏåÈ b ~Ó˚
IJõÓ˚– xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ
à!ï˛ˆÏÓà xö%öyˆÏòÓ˚ ˆ«˛ˆÏe ܲ¡õö¢#° û˛ˆÏÓ˚Ó˚
à!ï˛ˆÏÓà §ˆÏÓ≈yFã˛ •ÄÎ˚yÎ˚ ï˛yÓ˚ à!ï˛¢!_´Ä
(kinetic energy) ˙ xö%öyˆÏòÓ˚ §üÎ˚ §ˆÏÓ≈yFã˛
•ˆÏÓ– à!ï˛ˆÏÓà xö%öyˆÏòÓ˚ ~ܲ!ê˛ !Óܲ“ §ÇK˛y
!•§yˆÏÓ Ó°y ÎyÎ˚ñ ˆÎ xÓfliyÎ˚ ܲ¡õö¢#° Ó›Ó˚
à!ï˛¢!_´Ó˚ §ˆÏÓ≈yFã˛ üyö T0 §Ó≈!ôܲ •Î˚ñ ï˛yˆÏܲ Ó°y •Î˚ à!ï˛ˆÏÓà xö%öyò– ~•z xyˆÏ°yã˛öy ˆÌˆÏܲ xyüÓ˚y
à!ï˛ˆÏÓà xö%öyˆÏòÓ˚ §üÎ˚ §ˆÏÓ≈yFã˛ à!ï˛¢!_´Ó˚ T0m ÈÙÈ~Ó˚ Ó˚y!¢üy°y §•ˆÏç•z ˆ˛õˆÏï˛ ˛õy!Ó˚ ≠
∴ §ˆÏÓ≈yFã˛ à!ï˛¢!_´ T0m =
20
12m mv
=
2
0 122
Fm
mb⎛⎞⎜⎟ ⎝⎠
=
20
218
Fmb
...4.23b
xï˛~Ó ˆòáy ÎyˆÏFåÈ ˆÎñ !ÓhflÏyÓ˚ xö%öyò Ä à!ï˛ˆÏÓà xö%öyò ÎÌye´ˆÏü §ˆÏÓ≈yFã˛ !fli!ï˛¢!_´ Ä §ˆÏÓ≈yFã˛
à!ï˛¢!_´Ó˚ §Ó≈y!ôܲ üyö §)!ã˛ï˛ ܲˆÏÓ˚– xÓ¢ƒ Ó‡ ï˛ˆÏsf•z b xÌ≈yÍ xÓü®ˆÏöÓ˚ üyö á%Ó Ü˛ü ÌyˆÏܲ– ï˛áö xyüÓ˚y
!°áˆÏï˛ ˛õy!Ó˚ ≠
2b2 << ω02 ~ÓÇ ωr ≅ ω0 xÌ≈yÍ !ÓhflÏyÓ˚ xö%öyò Ä à!ï˛ˆÏÓà xö%öyò ï˛áö ≤ÃyÎ˚ ~ܲ•z ܲ¡õyˆÏB˛ âˆÏê˛–
˛õˆÏÓ˚Ó˚ xö%ˆÏFåȈÏò xyüÓ˚y ω = ω0 ¢ï≈˛!ê˛Ó˚ =Ó˚&c xyÓ˚Ä Ó%é˛ˆÏï˛ ˛õyÓ˚Ó Îáö ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ §üÎ˚ ï˛ˆÏsfÓ˚
¢!_´ @ˇÃ•ˆÏîÓ˚ •yˆÏÓ˚Ó˚ !Ó£ÏÎ˚!ê˛ xyˆÏ°y!ã˛ï˛ •ˆÏÓ–
~ï˛«˛î ˛õÎ≈hsˇ ˆÎ §Ó xyˆÏ°yã˛öy •° ï˛yÓ˚ IJõÓ˚ ò%Û~ܲ!ê˛ xö%¢#°ö#Ó˚ ã˛ã≈˛y ܲÓ˚ˆÏ° ˆÜ˛üö •Î˚⁄
105
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -3 : 200Nm–1 !fl±Ç ô &ÓˆÏܲÓ˚ !fl±ÇÈÙÈ~ 200g û˛Ó˚Î%_´ ܲˆÏÓ˚ ˆé˛y°yˆÏöy •°– !fl±Ç!ê˛ˆÏï˛ í˛z˛õ!fliï˛
xÓü®ö Ó° = –vNms–1– ~•z !fl±Ç!ê˛ˆÏܲ 5N !ÓhflÏyÓ˚ Ä 30 rads–1 ܲ¡õyB˛ !Ó!¢‹T Ó° ≤ÃÎ%_´ •°– !ÓhflÏyÓ˚ñ
xö%öyˆÏòÓ˚ ܲ¡õyB˛ñ §ˆÏÓ≈yFã˛ à!ï˛¢!_´ ~ÓÇ fliyÎ˚# xÓfliyÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ üyö ~•z ï˛ˆÏsfÓ˚ ˆ«˛ˆÏe Ü˛ï˛ •ˆÏÓ⁄
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -4 : §Ó˚°ˆÏÓ˚áyÎ˚ ܲ¡õö¢#° 70g û˛Ó˚!Ó!¢‹T ~ܲ!ê˛ Ó›Ü˛îyÓ˚ IJõÓ˚ 14Nm–1 ≤Ãï˛ƒyöÎ˚ܲ Ó°
!e´Î˚y ܲˆÏÓ˚– Î!ò ~•z ï˛ˆÏsf í˛z˛õ!fliï˛ xÓü®ö ӈϰÓ˚ ˛õ!Ó˚üyî ~ܲܲ à!ï˛ˆÏÓà 0.7Nsm–1 •Î˚ñ ï˛y•ˆÏ° ˆÜ˛yö
ܲ¡õyB˛ !Ó!¢‹T §üO§ Ó° Óy•zˆÏÓ˚ ˆÌˆÏܲ ≤ÈÏÎ˚yà ܲˆÏÓ˚ xö%öyò §,!‹T ܲÓ˚y ÎyˆÏÓ⁄ Î!ò ~•z ≤ÃÎ%_´ ӈϰÓ˚ !ÓhflÏyÓ˚
8N •Î˚ñ ï˛ˆÏÓ ï˛ˆÏsfÓ˚ §ˆÏÓ≈yFã˛ à!ï˛¢!_´ ܲï˛⁄
4.6 ≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚≤ÈÏîy!òï˛ Ü˛¡õö¢#° ï˛ˆÏsfÓ˚ ¢!_´ @ˇÃ•ˆÏîÓ˚ •yÓ˚
≤ÈÏîy!òï˛ Ü˛¡õˆÏö ≤Ãï˛ƒyöÎ˚ܲ Ó°ñ ã˛y°Ü˛ Ó° Ä xÓü®ö Ó° ~ܲ§ˆÏD ܲ¡õö¢#° Ó›Ó˚ í˛z˛õÓ˚ ܲyÎ≈ ܲˆÏÓ˚–
~ܲ!ê˛ §¡õ)î≈ ˛õÎ≈yˆÏÎ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° ˆüyˆÏê˛Ó˚ í˛z˛õÓ˚ ˆÜ˛yö ܲyÎ≈•z ܲˆÏÓ˚ öy– ã˛y°Ü˛ Ä xÓü®ö ӈϰÓ˚ ˆ«˛ˆÏe
~ܲ•z ܲÌy Ó°y ÎyÎ˚ öy– ܲ¡õö¢#° Ó›!ê˛ xÓü®ö ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ ܲyÎ≈ ܲÓ˚yÓ˚ ú˛ˆÏ° ï˛yÓ˚ ¢!_´Ó˚ «˛Î˚ •Î˚–
≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe ã˛y°Ü˛ Ó° Óy•zˆÏÓ˚ ˆÌˆÏܲ §ü•yˆÏÓ˚ ¢!_´ §Ó˚ÓÓ˚y• ܲˆÏÓ˚ fliyÎ˚# ܲ¡õö ÓçyÎ˚ Ó˚yˆÏá–
xyüÓ˚y ~áö ~•z ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yÓ˚ !öî≈Î˚ ܲÓ˚Ó– ~çöƒ xyüÓ˚y ≤Ã̈Ïü ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ ï˛yÍ«˛!îܲ üyö
~ÓÇ ï˛yÓ˚˛õÓ˚ ï˛yÓ˚ §üyܲ°ö ܲˆÏÓ˚ ~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yˆÏÎ˚ ~•z ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yÓ˚ !öô≈yÓ˚î ܲÓ˚Ó–
§ÇK˛yö%§yˆÏÓ˚ xyüÓ˚y çy!ö ˆÎñ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yÓ˚ Ó°ˆÏï˛ ~ܲܲ §üˆÏÎ˚ ¢!_´Ó˚ ˆÎyàyö Óy «˛üï˛y
(power) ˆÓyé˛yÎ˚– «˛üï˛yÓ˚ ï˛yÍ«˛!îܲ üyö P(tV •ˆÏ°ñ ˆ°áy ÎyÎ˚ ˆÎñ
P (t) = Ó° F (t)x à!ï˛ˆÏÓà v ...4.24(a)
4.6.2 xö%ˆÏFåÈò ˆÌˆÏܲ ˛õyÄÎ˚y v ~Ó˚ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆ°áy ÎyÎ˚ñ
v =
()
()0
122 2222
0
sin
4
Ft
mb
−−
⎡⎤ −+ ⎢⎥ ⎣⎦
ωωθ
ωωω
= 0 cos2
v t⎛ ⎞− +⎝ ⎠πω θ
= v0 cos (ωt – ψ) ...4.24(b)
~áyˆÏöñ v0 =
( )0
12 22 2 2 2
0 4
F
m b⎡ ⎤− +⎢ ⎥⎣ ⎦
ω
ω ω ω = ˆÓˆÏàÓ˚ !ÓhflÏyÓ˚ ~ÓÇñ ψ = 2
πθ⎛ ⎞−⎜ ⎟⎝ ⎠ = ˆÓà Ä ã˛y°Ü˛
ӈϰÓ˚ ò¢y ˛õyÌ≈ܲƒ v0 ~ÓÇ F(t) ~Ó˚ üyö (4.24a) §ü#ܲÓ˚ˆÏî ÓƒÓ•yÓ˚ ܲˆÏÓ˚ P(t) Ó˚ ˆÎ üyö ˛õyÄÎ˚y ÎyÎ˚
ï˛y •°ñ
106
P(t) = F0 cos ωt.v
0 cos (ωt – ψ)
= F0v0[cos2 ωt cos ψ + cos ωt sin ωt sin ψ]
= F0v0 cos ψ cos2 ωt +
12
F0v0 sin ψ sin 2ωt ...4.25
~ܲ!ê˛ õ)î≈ õÎ≈yˆÏÎ˚ •hflÏyhsˇ!Ó˚ï˛ ¢!_´Ó˚ àí˛¸ •yÓ˚ Óy àí˛¸ «˛üï˛y õyÄÎ˚yÓ˚ çöƒ (4.25) §ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚ õ«˛ˆÏܲ
0 ˆÌˆÏܲ T §üˆÏÎ˚Ó˚ üˆÏôƒ §üyܲ°ö ܲˆÏÓ˚ ~ÓÇ T !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ˛õy•zñ (T =
2πω
= ˆòy°öܲy°V
20000
00
sin 11 (cos)cossin22
TTFvPFvtdttdt
TT⎛⎞ =+⎜⎟ ⎝⎠ ∫∫ψ ψωω
= 0 01 cos 02
F v +ψ
Óyñ 0 01 sin2
P F v= θ ...4.26
ˆÎˆÏ•ï%˛ñ cos ψ = cos2
⎛⎞ − ⎝⎠π θ = cos
2⎛ ⎞−⎝ ⎠
π θ = sin θ
~áö tan θ ÈÙÈ~Ó˚ üyö (4.13a §)eV ˆÌˆÏܲ sin θ Ó˚ üyö !öî≈Î˚ ܲˆÏÓ˚ S§Ç!Ÿ’‹T ˛õ,¤˛yÓ˚ üy!ç≈ö ˆòá%öV ~ÓÇ
4.21b §ü#ܲÓ˚î ˆÌˆÏܲ v0 Ó˚ üyö 4.26 ÈÙÈ ~ Ó!§ˆÏÎ˚ ˛õy•zñ
()()0
11 022 22 22222222
00
12 ..2
44
Fb PFm
bb
=×⎡⎤⎡⎤ −+−+ ⎢⎥⎢⎥ ⎣⎦⎣⎦
ωω
ωωωωωω
2
0
1 cosT
t dtT
ω∫ = 0
1 1 (cos2 1)2
T
t dtT
ω +∫ = 1 .2
TT =
12
~ÓÇ
0
1sin2T
tdtT∫ω
= 0
1 cos22
Tt
T⎡ ⎤−⎢ ⎥⎣ ⎦
ωω = 1 0
T× = 0 ܲyÓ˚îñ T =
2πω
107
=
()220
2 222204
bF
mb ⎡⎤ −+ ⎢⎥ ⎣⎦
ωωωω
...4.27
xyÓyÓ˚ ~•z 4.27 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ
P
~Ó˚ üyö ω Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°–
P
~Ó˚ •ˆÏÓ˚
ˆÎˆÏ•ï%˛ ˆÜ˛Ó°üye Óà≈§Çáƒy í˛z˛õ!fliï˛ Ó˚ˆÏÎ˚ˆÏåÈñ ï˛y•z •ˆÏÓ˚Ó˚ §Ó≈!ö¡¨ üyö ˛õyÄÎ˚y ÎyˆÏÓñ Îáö ω = ω0– ~•z
§üÎ˚ ≤ÃyÆ
P
~Ó˚ üyö §ˆÏÓ≈yFã˛ ~ÓÇ ˆ§•z üyöˆÏܲ
mP
!öˆÏÎ˚ §)!ã˛ï˛ ܲÓ˚ˆÏ° ˆòáy ÎyÎ˚ ˆÎñ
220022
0
.4 m
bFP
bmω
ω=
SÎáö ω = ω0V
= 2
0
4Fbm
...4.28
°«˛ƒ ܲÓ˚&öñ
mP
~Ó˚ üyö xÌ≈yÍñ ~ܲܲ
§üˆÏÎ˚ ã˛y°Ü˛ ˆÌˆÏܲ ã˛y!°ï˛ ÓƒÓfliy!ê˛ˆÏï˛
§ˆÏÓ≈yFã˛ •hflÏyhsˇ!Ó˚ï˛ ¢!_´ xÓü®ö (b) Óy û˛Ó˚
(m) Ó,!k˛ ˆ˛õˆÏ° ܲˆÏü ÎyÎ˚– ï˛ˆÏÓ ≤ÃÎ%_´ ӈϰÓ˚
!ÓhflÏyÓ˚ F0 ~Ó˚ Ó,!k˛ˆÏï˛
mP
~Ó˚ üyö Ó,!k˛
˛õyÎ˚– 4.6 !ã˛ˆÏe àí˛¸ •hflÏyhsˇ!Ó˚ï˛ «˛üï˛y
P
–
~Ó˚ §ˆÏD ܲ#û˛yˆÏÓ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ñ ï˛y
ˆ°á!ã˛ˆÏeÓ˚ §y•yˆÏ΃ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˆòá%ö
ω = ω0 •ˆÏ° à!ï˛ˆÏÓà xö%öyò ˛õyÄÎ˚y ÎyÎ˚
~ÓÇ ˙ ~ܲ•z ¢ï≈˛ ˛õ)Ó˚î •ˆÏ° ¢!_´ •hflÏyhsˇˆÏÓ˚Ó˚
àí˛¸ •yÓ˚Ä §ˆÏÓ≈yFã˛ •Î˚– ~ê˛y §Ω˛Ó •Î˚ñ ܲyÓ˚î
˙ ¢ï≈˛ (ω = ω0) ˛õ)Ó˚î •ˆÏ° ܲ¡õö¢#°
ï˛ˆÏsfÓ˚ à!ï˛ˆÏÓà ~ÓÇ ≤ÃÎ%_´ ӈϰÓ˚ üˆÏôƒ ò¢y ˛õyÌ≈ܲƒ ¢)öƒ •Î˚– ˙ §üÎ˚ í˛zû˛ˆÏÎ˚Ó˚•z ï˛ˆÏsfÓ˚ §Ó˚ˆÏîÓ˚ ˆÌˆÏܲ
2π
ò¢y ˆÜ˛yˆÏî ~!àˆÏÎ˚ ÌyˆÏܲ– ~•z ܲyÓ˚ˆÏî !ÓhflÏyÓ˚ xö%öyò xˆÏ˛õ«˛y à!ï˛ˆÏÓà xö%öyò ˆÓ!¢ =Ó˚&c˛õ)î≈ •Î˚–
108
4.7 xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ~ÓÇxö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ~ÓÇxö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ~ÓÇxö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ~ÓÇxö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ~ÓÇ Q =îyB˛=îyB˛=îyB˛=îyB˛=îyB˛ (Sharpness of resonance & Q factor)
4.6 ~ÓÇ 4.7 xö%ˆÏFåȈÏòÓ˚ xyˆÏ°yã˛öy ˆÌˆÏܲ xyüÓ˚y ˆòˆÏá!åÈ ˆÎñ ω = ω0 •ˆÏ° ˆÎüö à!ï˛ˆÏÓà xö%öyò
âˆÏê˛ñ ˆï˛üö•z
P
~Ó˚ üyöÄ §ˆÏÓ≈yFã˛ •Î˚– !ܲv ˆÜ˛Ó°
xö%öyò# ˆÜ˛Ô!îܲ ܲ¡õyB˛ Èω0
ÙÈ•z ï˛ˆÏsfÓ˚ ~ܲüye ˜Ó!¢‹T
öÎ˚ñ xyˆÏÓ˚ܲ!ê˛ !Ó£ÏÎ˚Ä xï˛ƒhsˇ =Ó˚&c˛õ)î≈– xˆÏöܲ §üˆÏÎ˚
ˆòáy ÎyÎ˚ñ ˆÜ˛yöÄ ï˛ˆÏsf ã˛y°Ü˛ ӈϰÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛
ω Î!ò ω0 ˆÌˆÏܲ §yüyöƒ !Óã%˛ƒï˛ •Î˚ñ ï˛ˆÏÓ
P
~Ó˚ üyö
mP
ˆÌˆÏܲ á%Ó o&ï˛ Ü˛ˆÏü ÎyÎ˚– xyÓyÓ˚ ˆÜ˛yöÄ ˆÜ˛yöÄ
ˆ«˛ˆÏe ~•z •…y§ ï˛ï˛ê˛y o&ï˛ âˆÏê˛ öy– 4.7 !ã˛ˆÏe !Ó£ÏÎ˚!ê˛
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~•z ˆ°á!ã˛e!ê˛ Ü˛ï˛Ü˛ê˛y 4.6 !ã˛ˆÏeÓ˚
xö%Ó˚*˛õ– ï˛ˆÏÓ ~áyˆÏö ω ~Ó˚ §ˆÏD
m
PP
~•z
xö%˛õyï˛!ê˛Ó˚ ˛õ!Ó˚Óï˛≈ö ï%˛ˆÏ° ôÓ˚y •ˆÏÎ˚ˆÏåÈ– ˆòáy ÎyˆÏFåÈ ˆÎñ ò%!ê˛ ˆ«˛ˆÏe xö%öyò# ܲ¡õyB˛ §üyö (ω0) • Ï°Äñ
~ܲ!ê˛ ˆ«˛ˆÏeÓ˚ (A) ï%˛°öyÎ˚ x˛õÓ˚!ê˛ˆÏï˛ (B) ˙ xö%˛õyˆÏï˛Ó˚ üyö ω ~Ó˚ •…y§Ó,!k˛Ó˚ §ˆÏD xˆÏöܲ o&ï˛ 0.5
Óy §ˆÏÓ≈yFã˛ üyˆÏöÓ˚ xˆÏô≈ˆÏܲ ˆöˆÏü ~ˆÏ§ˆÏåÈ– ú˛°ï˛ñ B ˆ°á!ã˛e!ê˛ ï%˛°öyÎ˚ ï˛#«¯˛ xÌ≈yÍ xö%öyò# ܲ¡õyB˛ ˆÌˆÏܲ
§yüyöƒ !Óã%˛ƒ!ï˛ ~ˆÏ«˛ˆÏe ¢!_´ •hflÏyhsˇˆÏÓ˚Ó˚ •yÓ˚ˆÏܲ xˆÏöܲ ˆÓ!¢ ≤â!üï˛ Ü˛ˆÏÓ˚– ˛õ!Ó˚û˛y£ÏyÎ˚ ~ˆÏܲ Ó°y •Î˚ñ
xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y (Sharpness of resonance)–
ˆÜ˛yöÄ Ü˛¡õö¢#° ï˛ˆÏsf ã˛y°Ü˛ ˆÌˆÏܲ ã˛y!°ˆÏï˛Ó˚ !òˆÏܲ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yÓ˚ ω ~Ó˚ •…y§ Ó,!k˛Ó˚ §ˆÏD
Ü˛ï˛ o&ï˛ ï˛yÓ˚ §ˆÏÓ≈yFã˛ üyö ˆÌˆÏܲ xˆÏô≈ܲ üyˆÏö ˆöˆÏü xy§ˆÏåÈñ ï˛yÓ˚ §y•yˆÏ΃ xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛yÓ˚ ~ܲê˛y
˛õ!Ó˚üyîàï˛ !•§yÓ Ü˛Ó˚y ÎyÎ˚– ˆÎüö 4.7 !ã˛ˆÏe ˆòá%ö Îáöñ ω = ω0 ï˛áö
an
mx
PP
~Ó˚ üyö A Ä B ò%•z
ˆ«˛ˆÏe•z 1.0ñ ܲyÓ˚î ï˛áö xyüÓ˚y È P ÙÈÓ˚ §ˆÏÓ≈yFã˛ üyö
mP
≤Ãï˛ƒ«˛ ܲÓ˚Ó–
~ÓyÓ˚ ˆòá%öñ B ˆ°á!ã˛ˆÏe ω ~Ó˚ üyö ܲˆÏü ω1 Óy Ó,!k˛ ˆ˛õˆÏÎ˚ ω2 ~Ó˚ §üyö •ˆÏ°
m
PP
~Ó˚ üyö òÑyí˛¸yÎ˚
0.5 xÌ≈yÍñ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ àí˛¸ •yÓ˚
P
~áö §ˆÏÓ≈yFã˛ üyˆÏöÓ˚ xˆÏô≈ܲ– xˆÏ˛õ«˛yÜ,˛ï˛ fli(° A ˆ°ˆÏáÓ˚ ˆ«˛ˆÏe
~ܲ•z âê˛öy âˆÏê˛ Îáöñ ω = ω'1 Óy ω'2– ˆ°á!ã˛e!ê˛ °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓñ ï˛#«¯˛ï˛yÓ˚ ˆ°á B ~Ó˚ ˆ«˛ˆÏe
ω1 Ä ω2 ˛õyÌ≈ܲƒñ xÌ≈yÍ (ω2 – ω1) ~Ó˚ üyö (ω'2 – ω'1) ~Ó˚ ˆã˛ˆÏÎ˚ ܲü–
ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ ˆÎ ˛õyÕ‘yÓ˚ üˆÏôƒ •hflÏyhsˇ!Ó˚ï˛ «˛üï˛y §ˆÏÓ≈yFã˛ üyˆÏöÓ˚ xˆÏô≈ܲ Óy ï˛ò%k≈˛ ÌyˆÏܲñ xÌ≈yÍ
ω'1 ω'2
109
(ω'2 – ω'1) Óy (ω2 – ω1) ˆÜ˛ Ó°y •Î˚ xô≈«˛üï˛yÓ˚ ÓƒyˆÏu˛Ó˚ ˛õ)î≈ ≤çyÓ˚ (half power band width)– ~!ê˛ˆÏܲ
xô≈«˛üï˛yÎ˚ ÓƒyˆÏ[˛Ó˚ ˛õ)î≈ ≤çyÓ˚ (Full width at half power) xÌÓy ܲӰ ÓƒyˆÏ[˛Ó˚ ≤çyÓ˚Ä (band width)
Ó°y •Î˚–
xyˆÏàÓ˚ ~ܲˆÏܲ 3.5.3 xLjϢ xy˛õ!ö ~ܲ!ê˛ ï˛ˆÏsfÓ˚ Q =îyˆÏB˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö– ˛≤ÈÏîy!òï˛
ܲ¡õˆÏöÓ˚ ˆ«˛ˆÏe xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y ˛õ!Ó˚üyîàï˛ û˛yˆÏÓ !öˆÏò≈¢ ܲÓ˚yÓ˚ çöƒ xyüÓ˚y Q =îyˆÏB˛Ó˚ xyÓ˚Ä ~ܲ!ê˛
§ÇK˛y ÓƒÓ•yÓ˚ ܲ!Ó˚–
~!ê˛ •° ≠ Q =
xòò%yî# ÑÔ!íÑ Ñ¡ôyAÑ˛
xï≈«õìyÎ˚ ÓƒyÏuÓ ≤âyÓ˚
=
0
21
ωωω−
...4.29
m
PP
~Ó˚ üyö ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ Îï˛ ˆÓ!¢ ˛õyÕ‘yÓ˚ üˆÏôƒ 0.5 Óy ï˛ò)k≈˛ •Î˚ñ xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛yÄ ï˛ï˛
ܲü •Î˚– §%ï˛Ó˚yÇñ (ω2 – ω1) ˆÓ!¢ •ˆÏ°ñ ï˛#«¯˛ï˛y ܲü •ˆÏÓ– xyÓyÓ˚ 4.29 §)e xö%ÎyÎ˚# Q ~Ó˚ üyöÄ Ü˛ü
•ˆÏÓ– xï˛~Óñ QÈÙÈ~Ó˚ í˛zFã˛ï˛Ó˚ üyö ï˛#«¯˛ï˛Ó˚ xö%öyò §)!ã˛ï˛ ܲˆÏÓ˚– 4.29 §)ˆÏe Q =îyˆÏB˛Ó˚ ˆÎ Ó˚y!¢üy°y ˆòÄÎ˚y
•ˆÏÎ˚ˆÏåÈ xyüÓ˚y ~ÓyÓ˚ ï˛yÓ˚ ~ܲ!ê˛ !Óܲ“ ˛Ó˚y!¢üy°y !öî≈ˆÏÎ˚Ó˚ ˆã˛‹Ty ܲÓ˚Ó– 4.27 ~ÓÇ 4.28 §ü#ܲÓ˚î ò%•z!ê˛
ˆÌˆÏܲ xyüÓ˚y ˛õy•zñ
m
PP
=
()22
2 22220
4
4
b
b
ωωωω ⎡⎤ −+ ⎢⎥ ⎣⎦Îáö ÓÑy!òˆÏܲÓ˚ xö%˛õyï˛!ê˛ 0.5 ~Ó˚ §üyöñ ï˛áö ˆòáy ÎyÎ˚ñ
0.5 =
()22
2 22220
4
4
b
b
ωωωω −+
Óyñ (ω02 – ω 2)2 = 4b2ω 2
∴ ω02 – ω 2 = ± 2bω ...4.30
í˛yö!òˆÏܲÓ˚ ôöydܲ Ä îydܲ !ã˛ˆÏ•´Ó˚ çöƒ xyüÓ˚y 4.30 ˆÌˆÏܲ ò%!ê˛ !mâyï˛ §ü#ܲÓ˚î (quadratic
equation) ˛õy•z ≠
ω2 – 2bω – ω02 = 0 ...4.31
~ÓÇ ω2 + 2bω – ω0
2 = 0 ...4.32
4.31 §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y ˛õy•zñ ω = 2 20b b ω± + – ~Ó˚ üˆÏôƒ ˛õòyÌ≈!ÓòƒyÎ˚ ï˛yͲõÎ≈•#ö îydܲ
üyö!ê˛ Óyò !òˆÏ° xy˛õ!ö ˛õyˆÏÓöñ
ω2 = 2 20 b bω + +
110
ˆÎ!ê˛ xÓ¢ƒ•z ω0 xˆÏ˛õ«˛y Ó,•_Ó˚–
xyÓyÓ˚ 4.3 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ ω =
220 bbω −±+
ñ ÎyÓ˚ üˆÏôƒ îydܲ üyö!ê˛ •° ≠
ω1 = 2 20 b bω + − – ~!ê˛ ω
0 xˆÏ˛õ«˛y «%˛oï˛Ó˚– xy˛õ!ö Ó%é˛ˆÏï˛•z ˛õyÓ˚ˆÏåÈö ˆÎñ ~áyˆÏö xyüÓ˚y ω ~Ó˚
ω0 xˆÏ˛õ«˛y Ó,•_Ó˚ Ä «%˛oï˛Ó˚ üyö ò%!ê˛ˆÏܲ•z ÎÌye´ˆÏü ω
2 Ä ω
1 !•§yˆÏÓ ¢öy_´ ܲÓ˚°yü– ~•z ò%•z ˆÜ˛Ô!îܲ
ܲ¡õyˆÏB˛Ó˚ ÓƒÓôyöñ
ω2 – ω1 = 2b ...4.33
~áö 4.33 §ü#ܲÓ˚ˆÏîÓ˚ §y•yˆÏ΃ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
Q =
xòò%yî# ÑÔ!íÑ Ñ¡ôyAÑ˛
xï≈«õìyÎ˚ ÓƒyÏuÓ ≤âyÓ˚
=
0
21
ωωω−
=
0
2bω
...4.34
4.34 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ
ω0 Ó,!k˛ ˆ˛õˆÏ° Q Ó,!k˛ ˛õyÎ˚ !ܲv !ö!ò≈‹T
fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛ xÓü®ö =îyB˛ b Ó,!k˛
ˆ˛õ Ï° Q ~Ó˚ üyö ܲˆÏü ÎyÎ˚– xÌ≈yÍñ
xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y •…y§ ˛õyÎ˚– 4.8 !ã˛ˆÏe ~•z
!Ó£ÏÎ˚!ê˛ ~ܲ!ê˛ ï˛ˆÏsfÓ˚ ω ~ÓÇ av
P
ˆ°á!ã˛ˆÏeÓ˚ üyôƒˆÏü ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– °«˛ƒ
ܲÓ˚&öñ ≤Ã!ï˛ ˆ«˛ˆÏe•z xö%öyò# ˆÜ˛Ô!îܲ
ܲ¡õyB˛ ω0 !ܲv b ~Ó˚ üyö Ó,!k˛Ó˚ §ˆÏD
§ˆÏD ˆÜ˛Ó°
mP
ÈÙÈ~Ó˚ üyö•z ܲˆÏü!öñ ˆ°á!ã˛e=!° x!ôܲï˛Ó˚ fli(° •ˆÏÎ˚ˆÏåÈñ xÌ≈yÍ xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y •…y§
ˆ˛õˆÏÎ˚ˆÏåÈ ~ÓÇ Q ~Ó˚ üyö ܲˆÏü ˆàˆÏåÈ–
Q =îyˆÏB˛Ó˚ xyÓ˚Ä ~ܲ!ê˛ §ÇK˛y xy˛õöyÓ˚ ܲyˆÏç °yàˆÏï˛ ˛õyˆÏÓ˚– xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈñ ~ܲ!ê˛ Ü˛¡õö¢#°
ï˛sfñ ≤Ã!ï˛!ê˛ ˛õÎ≈yˆÏÎ˚ (cycle) xÓü®ö ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ ܲyÎ≈ ܲˆÏÓ˚ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ï˛sf!ê˛ ¢!_´ •yÓ˚yÎ˚– xöƒ!òˆÏܲñ
ã˛y°Ü˛ Ó° myÓ˚y Ü,˛ï˛ ܲyˆÏÎ≈Ó˚ ú˛ˆÏ° ܲ¡õö¢#° ï˛ˆÏsf ¢!_´ §!MÈ˛ï˛ •Î˚– ˆÜ˛yöÄ ~ܲ!ê˛ ï˛sf ï˛yÓ˚ §!MÈ˛ï˛ ¢!_´Ó˚
Îï˛ «%˛o û˛@¿yÇ¢ ~ܲ!ê˛ ˛õÎ≈yˆÏÎ˚ xÓü®ö ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ ÓƒÎ˚ ܲˆÏÓ˚ñ ï˛yÓ˚ Q =îyB˛ ï˛ï˛•z x!ôܲ •Î˚– ~•z
ò,!‹Tû˛!D ˆÌˆÏܲ Q ~Ó˚ ˆÎ §ÇK˛y ˛õyÄÎ˚y ˆ§!ê˛ •° ≠
Q = 2π
Ñ¡ôòü#ú ì˛sÏ˛f ¢!Mì ü!=˛
~Ñ!› ôí)≈ ôÎy≈ÏÎ Óƒ!Îì ü!=–
...4.35
111
xyüÓ˚y ˛õˆÏÓ˚ ˆòáÓ ˆÎñ ˜Óò%ƒ!ï˛Ü˛ Óï≈˛ö#ˆÏï˛ ≤ÃÓy•üyeyÓ˚ xö%öyˆÏòÓ˚ !Ó£ÏÎ˚!ê˛ û˛y° ܲˆÏÓ˚ Ó%é˛ÓyÓ˚ çöƒ Q
=îyB˛ ~ܲ!ê˛ x!ï˛ ≤ÈÏÎ˚yçö#Î˚ ôyÓ˚îy–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -5 : 10Nm–1 !fl±Ç ô &Óܲ !Ó!¢‹T ~ܲ!ê˛ xö%û)˛!üܲ !fl±ÇÈÙÈ~Ó˚ ~ܲ!òܲ xyÓk˛ Ä xöƒ!òˆÏܲ
!fl±ÇÈÙÈ~Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ã˛°ö¢#° 100g û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ Ó› xyê˛Ü˛yˆÏöy xyˆÏåÈ– û˛Ó˚!ê˛Ó˚ IJõÓ˚ ~ܲܲ ˆÓà !˛õå%È
0.1N xÓü®ö Ó° !e´Î˚y ܲˆÏÓ˚– Ó›!ê˛Ó˚ IJõÓ˚ ï˛yÓ˚ §Ó˚ˆÏîÓ˚ !òܲ ÓÓ˚yÓÓ˚ cos ωt !öí˛zê˛ö ã˛y°Ü˛ Ó° ≤ÈÏÎ˚yà
ܲÓ˚y •°– Ó›!ê˛Ó˚ fliyÎ˚# xÓfliyÓ˚ ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ !öî≈Î˚ ܲÓ˚&öñ Îáö (i) ω = 1 rad s–1 Ä (ii) ω = 50
rad s–1– ï˛sf!ê˛Ó˚ Q =îyB˛ !öî≈Î˚ ܲÓ˚&ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -6 : ≤Ãüyî ܲÓ˚&ö ˆÎñ ˆÜ˛yöÄ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe ≤Ã!ï˛ ã˛ˆÏe´ ˆÎ àí˛¸ ¢!_´ ã˛y!°ï˛ ï˛sf
°yû˛ ܲˆÏÓ˚ ï˛y •°ñ
< E > =
() 2220
14
mB ωω+
ˆÎáyˆÏö !ã˛•´=!° ≤Ãã˛!°ï˛ xÌ≈Ó•– ˆòáyö ˆÎ ~ˆÏ«˛ˆÏe
Q = 2 2
0
4bω ω
ω+
4.8 ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° §• ◊î# §üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ (L), ôyÓ˚ܲôyÓ˚ܲôyÓ˚ܲôyÓ˚ܲôyÓ˚ܲ (C) ~ÓÇ~ÓÇ~ÓÇ~ÓÇ~ÓÇ
ˆÓ˚yôˆÓ˚yôˆÓ˚yôˆÓ˚yôˆÓ˚yô (R) §¡∫!°ï˛ Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–§¡∫!°ï˛ Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–§¡∫!°ï˛ Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–§¡∫!°ï˛ Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–§¡∫!°ï˛ Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!òï˛ Ü˛¡õö Ä xö%öyò–
xyüÓ˚y ~•z ~ܲˆÏܲ ~ ˛õÎ≈hsˇ Îy!sfܲ ï˛ˆÏsf ≤ÈÏîy!òï˛
ܲ¡õö Ä xö%öyò !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– !ܲv ~•z
âê˛öy=!° ˆÜ˛Ó° Îy!sfܲ ï˛ˆÏsf•z §#üyÓk˛ öÎ˚– ˆ◊î#
§üÓyˆÏÎ˚ Î%_´ xyˆÏÓ¢ (L)ñ ôyÓ˚ܲ (C) ~ÓÇ ˆÓ˚yô (R)
§¡∫!°ï˛ Óï≈˛ö#ñ xÌ≈yÍ L–CR ˆ◊î# §üÓyÎ˚ Óï≈˛ö#ˆÏï˛ (L–
CR series circuit) Î!ò ~ܲ!ê˛ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó°
(alternating emf) ≤ÈÏÎ˚yà ܲˆÏÓ˚ ï˛yÓ˚ ܲ¡õyB˛ ˛õ!Ó˚Óï≈˛ö
ܲÓ˚y ÎyÎ˚ñ ï˛y•ˆÏ° ˆ§áyˆÏöÄ ≤ÈÏîy!òï˛ ˜Óò%ƒ!ï˛Ü˛ ܲ¡õö Ä
xö%öyò ˛õÎ≈ˆÏÓ«˛î ܲÓ˚y §Ω˛Ó– 4.9 !ã˛ˆÏe ~•z LCR Óï≈˛ö#!ê˛
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
ï,˛ï˛#Î˚ ~ܲˆÏܲ xy˛õ!ö ~ܲ•z ôÓ˚ˆÏöÓ˚ Óï≈˛ö#ˆÏï˛ xy!•ï˛ ôyÓ˚ˆÏܲÓ˚ xyôyö §üˆÏÎ˚Ó˚ §ˆÏD ܲ#û˛yˆÏÓ •…y§ ˛õyÎ˚
112
ï˛y ˆòˆÏáˆÏåÈö– ~áyˆÏö ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ ӈϰÓ˚ í˛z˛õ!fli!ï˛ˆÏï˛ xyüÓ˚y !ܲå%Èê˛y !û˛ß¨ xÓfliy °«˛ƒ ܲÓ˚Ó–
≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° ≤ÃÜ,˛ï˛˛õˆÏ«˛ ã˛y°Ü˛ ӈϰÓ˚ û)˛!üܲy ˛õy°ö ܲˆÏÓ˚–
xyüÓ˚y çy!ö ˆÎñ ~ܲ!ê˛ !ö!ò≈‹T §üÎ˚ t ˆï˛ Î!ò Óï≈˛ö#Ó˚ ≤ÃÓy•üyey I •Î˚ ï˛y•ˆÏ°ñ L, R ~ÓÇ CÈÙÈˆï˛ !Óû˛Ó
˛õï˛ˆÏöÓ˚ §ü!‹TˆÏܲ !ö¡¨!°!áï˛ §ü#ܲÓ˚ˆÏîÓ˚ §y•yˆÏ΃ Î%_´ ܲÓ˚y ÎyÎ˚–
qdlL RIdt C
+ + = E0 cos ωt ...4.36
~•z §ü#ܲÓ˚ˆÏî ÓÑyÈÙÈ!òˆÏܲÓ˚ !ï˛ö!ê˛ ˛õò ÎÌye´ˆÏü L, R ~ÓÇ CÈÙÈ~Ó˚ ò%•z ≤ÃyˆÏhsˇÓ˚ üˆÏôƒ !Óû˛Ó˛õï˛ö §)!ã˛ï˛
ܲÓ˚ˆÏåÈ– ~áyˆÏö E0 •ˆÏFåÈ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ ӈϰÓ˚ §ˆÏÓ≈yFã˛ üyö Óy !ÓhflÏyÓ˚ñ ω ï˛yÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛–
'q' §)!ã˛ï˛ ܲÓ˚ˆÏåÈ 't' §üˆÏÎ˚ ôyÓ˚ C ÈÙÈ~Ó˚ ˆÜ˛yöÄ ~ܲ ˛õyˆÏï˛ xyôyˆÏöÓ˚ ˛õ!Ó˚üyî–
~áö I =
dqdt
!°áˆÏ° 4.36 §ü#ܲÓ˚î ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yÎ˚ñ
2
21 dqdq LRq
dtC dt++
= E0 cos ωt ...4.37
§ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚˛õ«˛ˆÏܲ L !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ˛õy•zñ
2
21d q dqR q
L dt LCdt+ + = 0 cos
Et
Lω ...4.38a
~áöñ RL
= 2b ~ÓÇ 1LC
= ω0
2 Ó!§ˆÏÎ˚ ˆ°áy ÎyÎ˚ñ22
0 22 dqdq bqdt dt
ω ++
= 0 cosE
tL
ω ...4.38b
~•z §ü#ܲÓ˚î!ê˛ !öÿ˛Î˚•z xy˛õöyÓ˚ á%Ó ˛õ!Ó˚!ã˛ï˛ üˆÏö •ˆÏFåÈ– ~!ê˛ ~•z ~ܲˆÏܲÓ˚ 4.3 §ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ–
˙ §ü#ܲÓ˚ˆÏî ˆÎüö xÓü®ö §• ≤ÈÏîy!òï˛ Îy!sfܲ ˆòy°ˆÏöÓ˚ ~ܲ!ê˛ à!ï˛¢#° xÓfliy Ó!î≈ï˛ •ˆÏÎ˚!åÈ°ñ !ë˛Ü˛
~ܲ•zû˛yˆÏÓ 4.38b §ü#ܲÓ˚î!ê˛Ä LCR Óï≈˛ö#ˆÏï˛ ˜Óò%ƒ!ï˛Ü˛ xyôyˆÏöÓ˚ ≤ÈÏîy!òï˛ ˆòy°ö ˆÓyé˛yˆÏFåÈ– °«˛ƒ ܲÓ˚&öñ
q ~áyˆÏö 4.3 §ü#ܲÓ˚ˆÏîÓ˚ x ~Ó˚ çyÎ˚ày !öˆÏÎ˚ˆÏåÈ ~ÓÇ E0 cos ωt~áyˆÏö ã˛y°Ü˛ Ó° 'F0 cos ωt' ÈÙÈ~Ó˚
û)˛!üܲy ˛õy°ö ܲÓ˚ˆÏåÈ– ï˛y•z 4.15 §ü#ܲÓ˚î xö%§Ó˚î ܲˆÏÓ˚ fliyÎ˚# xÓfliyÓ˚ §üyôyö !•§yˆÏÓ ~áyˆÏö ˆ°áy ÎyÎ˚ñ
q(t) =
()
0
122 2222
0
cos()
4
Et
L
b
⎛⎞− ⎜⎟ ⎝⎠
⎡⎤ −+ ⎢⎥ ⎣⎦
ωθ
ωωω
...4.39
°«˛ƒ ܲÓ˚&öñ 4.38b ~ÓÇ 4.39 §ü#ܲÓ˚î ò%!ê˛ ˆÌˆÏܲ §•ˆÏç•z ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ~•z Óï≈˛ö#ˆÏï˛ L Îy!sfܲ
ï˛ˆÏsfÓ˚ û˛Ó˚ Óy m ~Ó˚ §üï%˛° û)˛!üܲy ˛õy°ö ܲÓ˚ˆÏåÈ ~ÓÇ R ˛õy°ö ܲÓ˚ˆÏåÈ γ Óy ~ܲܲ à!ï˛ˆÏÓˆÏà xÓü®ö
113
ӈϰÓ˚ û)˛!üܲy– ï˛sf!ê˛Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ 01LC
ω ⎛ ⎞=⎜ ⎟⎝ ⎠ ˛õyÄÎ˚y ÎyˆÏFåÈñ L ~ÓÇ C ˆÌˆÏܲ ~ÓÇ ~áyˆÏö
1C
˛õy°ö ܲÓ˚ˆÏåÈ Îy!sfܲ ï˛ˆÏsfÓ˚ 'K'ÈÙÈ~Ó˚ xÌ≈yÍñ ≤Ãï˛ƒyöÎ˚ö =îyˆÏB˛Ó˚ û)˛!üܲy–
(4.39) §ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚ ˛õ«˛ˆÏܲ t Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö (differentiate) ܲˆÏÓ˚ ≤ÃÓy•üyey I ˛õyÄÎ˚y ÎyÎ˚–
I =
dqdt
=
()0
122 2222
0
(/)sin()
4
ELt
b
− −⎡⎤ −+ ⎢⎥ ⎣⎦
ωωθ
ωωω
= 01
2 22 22
2
cos( )
1
E t
L RLC L
−
⎡ ⎤⎛ ⎞− +⎢ ⎥⎝ ⎠⎣ ⎦
ω ψ
ωωω
, ˆÎáyˆÏö ψ = 2− πθ
= 01
2 22
cos( )
1
E t
L RC
ω ψ
ω ω
−
⎡ ⎤⎛ ⎞− +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
= I0 cos (ωt – ψ) ...4.40
~áyˆÏöñ I0 =
01
222 1
E
LRC
ωω⎡⎤ ⎛⎞ −+ ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ ...4.41a
~ÓÇñ tan ψ = – cot θ =
220
2bωω
ω− −
= 1LC
R
ω ω−...4.41b
ψ ˆÜ˛yî!ê˛ ≤ÃÎ%_´ ï˛!í˛¸Fã˛y°Ü˛ Ó° (E = E0 cos ωt) ~ÓÇ ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ (I) üˆÏôƒ ò¢y ˛õyÌ≈ܲƒ– ~•z
ò¢y ˛õyÌ≈ܲƒ ˆÜ˛Ó° L, C ~ÓÇ R ÈÙÈ•z öÎ˚ñ ω ~Ó˚ IJõÓ˚Ä !öû≈˛Ó˚¢#°–
4.40 §ü#ܲÓ˚ˆÏî I0 ≤ÃÓyˆÏ•Ó˚ !ÓhflÏyÓ˚ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– ~•z !ÓhflÏyÓ˚ Èω ÙÈ~Ó˚ §ˆÏD ˛õ!Ó˚Óï≈˛ö¢#°– ωÈÙÈ~Ó˚
˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ú˛ˆÏ° I0 ~Ó˚ üyö Îáö §ˆÏÓ≈yFã˛ •Î˚ñ ï˛áö Óï≈˛ö#ˆÏï˛ xö%öyò ˛õyÄÎ˚y ÎyÎ˚–
I0 ~Ó˚ Ó˚y!¢üy°y!ê˛ °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ï˛yÓ˚ •Ó˚
114
221L R
Cω ω
⎛ ⎞− +⎜ ⎟⎝ ⎠
ÎyÓ˚ üˆÏôƒ
21Lc
ω ω⎛ ⎞−⎜ ⎟⎝ ⎠
~ܲ!ê˛ Óà≈§Çáƒy– ~Ó˚ §Ó≈!ö¡¨ üyö ¢)öƒ– x˛õÓ˚ xÇ¢ R, ω ÈÙÈ~Ó˚ IJõÓ˚ !öû≈˛Ó˚
ܲˆÏÓ˚ öy– ï˛y•z •ˆÏÓ˚Ó˚ §Ó≈!ö¡¨ Óy I0 ~Ó˚ §ˆÏÓ≈yFã˛ üyö ˛õyÄÎ˚y ÎyÎ˚ Îáöñ
1 LC
ωω−
= 0 Óyñ ω2 = 1LC
...4.42
!ܲv xyüÓ˚y çy!ö ˆÎñ
1LC
= ω02 – xï˛~Ó Ó°y ÎyÎ˚ ˆÎñ Îáö xyˆÏÓ˚y!˛õï˛ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ ӈϰÓ˚
ܲ¡õyB˛ ω1 Óï≈˛ö#Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ω0 ~Ó˚ §üyö •ˆÏÓñ ï˛áö•z xö%öyò ˛õyÄÎ˚y ÎyˆÏÓ– xö%öyˆÏòÓ˚ §üÎ˚
I0ÈÙÈÓ˚ §ˆÏÓ≈yFã˛ üyö •Î˚
I0m =
0 ER
...4.43
xÌ≈yÍñ xö%öyˆÏòÓ˚ §üÎ˚ Óï≈˛ö# ˆÜ˛Ó°üye ˆÓ˚yôÎ%_´ Óï≈˛ö#Ó˚ üï˛ xyã˛Ó˚î ܲˆÏÓ˚–
~ˆÏ«˛ˆÏe ò¢y ˛õyÌ≈ܲƒ ψ =
1
1
tanL
CR
ωω −−
= 0– ˆû˛ˆÏÓ ˆòá%öñ ~ê˛y !ܲv ≤Ãï˛ƒy!¢ï˛•z !åÈ°– ܲyÓ˚î !Ó÷k˛
ˆÓ˚yôÎ%_´ Óï≈˛ö#ˆÏï˛ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° Ä
≤ÃÓy•üyeyÓ˚ ò¢yÓ˚ ˛õyÌ≈ܲƒ ¢)öƒ •Î˚– xyÓ˚Ä
°«˛ƒ ܲÓ˚&öñ I0 Ó˚ §ˆÏÓ≈yFã˛ üyö R ~Ó˚ IJõÓ˚
!öû˛≈Ó˚ ܲˆÏÓ˚– R ܲˆÏü ˆàˆÏ° I0m
~Ó˚ üyöÄ
ÓyˆÏí˛¸– ܲyÓ˚î xyˆÏà•z ˆòˆÏáˆÏåÈöñ R ~•z
Óï≈˛ö#ˆÏï˛ xÓü®ˆÏöÓ˚ û)˛!üܲy õy°ö ܲˆÏÓ˚– ï˛ˆÏÓ
ˆáÎ˚y° Ó˚yáy òÓ˚ܲyÓ˚ ˆÎñ R ÈÙÈ~Ó˚ üyö ¢)öƒ •ˆÏ°
Óy Óï≈˛ö#ˆÏï˛ xy°yòy ܲˆÏÓ˚ ˆÜ˛yöÄ ˆÓ˚yô Î%_´
öy ܲÓ˚ˆÏ°Ä I0m
~Ó˚ üyö x§#ˆÏüÓ˚ !òˆÏܲ ã˛ˆÏ°
ÎyˆÏÓ öy– ܲyÓ˚î ÓyhflÏÓ Óï≈˛ö#ˆÏï˛ xyˆÏÓ¢ L ~Ó˚
§ˆÏD !ܲå%È ˆÓ˚yô §Ó≈òy•z ˆÌˆÏܲ ÎyÎ˚ ~ÓÇ ˆ§•z
ˆÓ˚yô•z I0m
~Ó˚ üyö !öÎ˚sfî ܲˆÏÓ˚– 4.10 !ã˛ˆÏe
!Ó!û˛ß¨ R ~Ó˚ çöƒ Èω ÙÈÓ˚ §ˆÏD I0 ÈÙÈÓ˚
˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˆ°á!ã˛e!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
I 0
115
xyüÓ˚y xyüyˆÏòÓ˚ ˛õ!Ó˚!ã˛ï˛ §ÇK˛y xö%ÎyÎ˚# ~•z Óï≈˛ö#Ó˚ ˆ«˛ˆÏeÄ Q =îyB˛ ï˛Ìy xö%öyˆÏòÓ˚ ï˛#«¯˛ï˛y !öî≈Î˚
ܲÓ˚ˆÏï˛ ˛õy!Ó˚– 4.34 §)e xö%§Ó˚î ܲˆÏÓ˚ ~•z Óï≈˛ö#Ó˚ Q =îyB˛!ê˛ ˆ°áy ÎyÎ˚ ≠
Q = 0
2bω =
0
RL
ω
=
0LR
ω
...4.44
xï˛~Ó ˆòáy ÎyˆÏFåÈ ˆÎñ Óï≈˛ö#Ó˚ Q =îyB˛ xö%öyò# ܲ¡õyB˛ ω0ñ xy ÏÓ¢ L ~ÓÇ ˆÓ˚yô RÈÙÈ~Ó˚ üyˆÏöÓ˚
IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– R •…yˆÏ§Ó˚ §ˆÏD §ˆÏD ˆÜ˛Ó° I0 öÎ˚ñ Q ~Ó˚ üyöÄ Ó,!k˛ ˛õyÎ˚– 4.42 §ü#ܲÓ˚î xö%ÎyÎ˚#
ω0 =
1LC
!°áˆÏ° Q =îyˆÏB˛Ó˚ xyˆÏÓ˚ܲ!ê˛ !Óܲ“ Ó˚y!¢üy°y ˛õyÄÎ˚y ÎyÎ˚ ≠
Q =
0LR
ω
=
1LLC
R
= 1 LR C
...4.45
§%ï˛Ó˚yÇñ Óï≈˛ö#Ó˚ Q =îyˆÏB˛Ó˚ üyö xyˆÏÓ¢ L, ôyÓ˚ܲc C Ä ˆÓ˚yô R, ~•z !ï˛ö!ê˛ Ä˛õÓ˚•z !öû≈˛Ó˚¢#°–
ˆ◊î# §üÓyˆÏÎ˚ Î%_´ LCR Óï≈˛ö#ˆÏï˛ xyüÓ˚y ˆÎ xö%öyˆÏòÓ˚ âê˛öy °«˛ƒ ܲÓ˚°yüñ ï˛yÓ˚ çöƒ ~•z Óï≈˛ö#ˆÏܲ
ˆ◊î# §Iy xö%öyò# Óï≈˛ö# (Series resonant circuit) Ó°y •Î˚– ~•z Óï≈˛ö#Ó˚ Q =îyˆÏB˛Ó˚ xyˆÏÓ˚ܲ!ê˛ ï˛yͲõÎ≈ƒ
~•z ≤çˆÏD í˛zˆÏÕ‘áˆÏÎyàƒ– Q =îyˆÏB˛Ó˚ üyö x!ôܲ •ÄÎ˚yÓ˚ ú˛ˆÏ°ñ LCR Óï≈˛ö#ˆÏï˛ Ü˛¡õyˆÏB˛Ó˚ ~ܲ!ê˛ x˛õ!Ó˚§Ó˚
ÓƒyˆÏu˛ xö%öyò âˆÏê˛– ~Ó˚ ú˛ˆÏ° Ó‡ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ !ü◊ ï˛Ó˚D ˆÌˆÏܲ ~•z Óï≈˛ö# ~ܲ!ê˛ !ӈϢ£Ï ܲ¡õyB˛
!öÓ≈yã˛ö ܲÓ˚ˆÏï˛ §«˛ü •Î˚– ï˛y•z ˆÓï˛yÓ˚ @ˇÃy•Ü˛ Óy ˆÓ˚!í˛Ä ΈÏsf ~Ó˚ !ӈϢ£Ï û)˛!üܲy Ó˚ˆÏÎ˚ˆÏåÈ– ˆÓ˚!í˛ÄÓ˚ xƒyˆÏrê˛öyÎ˚
à,•#ï˛ ï˛Ó˚ˆÏD öyöy ܲ¡õyˆÏB˛Ó˚ ï˛Ó˚D í˛z˛õ!fliï˛ ÌyܲˆÏ°Äñ LCR Óï≈˛ö#Ó˚ §y•yˆÏ΃ !ӈϢ£Ï ܲ¡õyB˛ ˆÓˆÏåÈ ˆöÄÎ˚y
ÎyÎ˚– Ó›ï˛ C ~ÓÇ L ˆÜ˛ ˛õ!Ó˚Ó!ï≈˛ï˛ ܲˆÏÓ˚ xyüÓ˚y Óï≈˛ö#Ó˚ xö%öyò# ܲ¡õyB˛ !öÎ˚sfî ܲ!Ó˚ ~ÓÇ ï˛yÓ˚ myÓ˚y
!Ó!û˛ß¨ ˆÓï˛yÓ˚ ˆÜ˛ˆÏwÓ˚ §¡±ã˛yÓ˚ ÷öˆÏï˛ ˛õy!Ó˚– Q ~Ó˚ üyö Îï˛ ˆÓ!¢ •Î˚ñ ~ܲ!ê˛ !ӈϢ£Ï ˆÓï˛yÓ˚ ˆÜ˛ˆÏwÓ˚
§¡±ã˛yÓ˚ ï˛ï˛ §%flõ‹T û˛yˆÏÓ ôÓ˚y §Ω˛Ó–
xy§%ö ~ܲˆÏܲÓ˚ xyˆÏ°yã˛öy ˆ¢£Ï ܲÓ˚yÓ˚ xyˆÏà ~•z xö%¢#°ö#!ê˛ ˆã˛‹Ty ܲˆÏÓ˚ ˆòá%ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 7 : ~ܲ!ê˛ ˆ◊î#§Iy ÈÙÈ xö%öyò# Óï≈˛ö#ˆÏï˛ L = 0.5mH, R = 30 Ω– ~•z Óï≈˛ö#ˆÏï˛ 1.2v
!ÓhflÏyÓ˚ Ä 105Hz ܲ¡õyB˛ !Ó!¢‹T ~ܲ!ê˛ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° Î%_´ ܲÓ˚y •°– C ~Ó˚ üyö Ü˛ï˛ •ˆÏ°
xö%öyò ˛õyÄÎ˚y ÎyˆÏÓ⁄ ï˛áö ≤ÃÓy•üyeyÓ˚ §ˆÏÓ≈yFã˛ üyö Ü˛ï˛ •ˆÏÓ⁄ ˙ Óï≈˛ö#Ó˚ QÈÙÈ~Ó˚ üyö !öî≈Î˚ ܲÓ˚&ö–
4.9 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
1. ܲ¡õö¢#° ï˛ˆÏsfÓ˚ ܲ¡õö xÓü®ˆÏöÓ˚ ú˛ˆÏ° ô#ˆÏÓ˚ ô#ˆÏÓ˚ °Î˚≤ÃyÆ •Î˚– Óy•zˆÏÓ˚ ˆÌˆÏܲ ˛≤ÈÏÎ˚yà ܲÓ˚y ˛õÎ≈yÓ,_
ӈϰÓ˚ §y•yˆÏ΃ xÓü®ˆÏöÓ˚ í˛z˛õ!fli!ï˛ˆÏï˛Ä ܲ¡õö ÓçyÎ˚ Ó˚yáy §Ω˛Ó •Î˚–
2. ã˛y°Ü˛ ӈϰÓ˚ ≤Ãû˛yˆÏÓ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ §,!‹T •Î˚– ~•z ܲ¡õˆÏöÓ˚ §üÎ˚ ò)Ó˚c §ü#ܲÓ˚î !öˆÏã˛Ó˚ xÓܲ°
§ü#ܲÓ˚ˆÏîÓ˚ §üyôyˆÏöÓ˚ üyôƒˆÏü ˛õyÄÎ˚y ÎyÎ˚ ≠
22
0 22 dxdx bxdt dt
ω ++
= f0 cos ωt
116
~áyˆÏöñ û˛Ó˚ 2b = ( )mmγ = ω
02 =
km
, f0 =
0 Fm
~ÓÇ ω = ã˛y°Ü˛ ӈϰÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛–
3. ≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚ §üÎ˚ ò)Ó˚c §ü#ܲÓ˚ˆÏîÓ˚ ò%!ê˛ xÇ¢– ~Ó˚ ≤ÃÌü!ê˛ (μ) fl∫“fliyÎ˚# §üyôyö ~Ó˚ x“«˛î
˛õˆÏÓ˚ !Ó°%Æ •ˆÏÎ˚ ÎyÎ˚– !mï˛#Î˚!ê˛ (x2) fliyÎ˚# xÓfliyÓ˚ §üyôyöñ Îy ã˛y°Ü˛ Ó° Îï˛«˛î í˛z˛õ!fliï˛ ÌyˆÏܲñ ï˛ï˛«˛î•z
°«˛ƒ ܲÓ˚y ÎyÎ˚– §ü#ܲÓ˚î!ê˛Ó˚ §¡õ)î≈ §üyôyöñ
x = x1 + x
2 = A
0e–bt cos (ω't + Q) _ B cos (ωt – θ)
ˆÎáyˆÏöñ B =
()0
122 2222
04
f
mb ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
4. xÓü®ˆÏöÓ˚ çöƒ ã˛y°Ü˛ Ó° ~ÓÇ §Ó˚ˆÏîÓ˚ üˆÏôƒ ~ܲ!ê˛ ò¢y ˛õyÌ≈ܲƒ ˆòáy ÎyÎ˚– ~•z ò¢y ˛õyÌ≈ܲƒ 'b',
ω ~ÓÇ ω0 ÈÙÈÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°–
5. ã˛y°Ü˛ ӈϰÓ˚ ܲ¡õyB˛ ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ï˛yÓ˚ !ӈϢ£Ï üyˆÏöÓ˚ çöƒ xö%öyò ˛õyÄÎ˚y ÎyÎ˚– ˆÎüö ωγ =
2202b ω−
•ˆÏ°ñ fliyÎ˚# xÓfliyÎ˚ §ˆÏÓ≈yFã˛ !ÓhflÏyÓ˚ °«˛ƒ ܲÓ˚y ÎyÎ˚ ~ÓÇ !ÓhflÏyÓ˚ xö%öyò ˛õyÄÎ˚y ÎyÎ˚– ω =
ω0 •ˆÏ° à!ï˛ˆÏÓà xö%öyò ˛õyÄÎ˚y ÎyÎ˚– ~•z §üÎ˚ ܲ¡õö¢#° ӛܲîyÓ˚ à!ï˛ˆÏÓà §ˆÏÓ≈yFã˛– !ÓhflÏyÓ˚ Ä ˆÓˆÏàÓ˚
§ˆÏÓ≈yFã˛ üyö=!° xÓü®ö =îyB˛ b myÓ˚y !öÎ˚!sfï˛ •Î˚–
6. ã˛y°Ü˛ Ó° ã˛y!°ï˛ ï˛sfˆÏܲ ¢!_´ §Ó˚ÓÓ˚y• ܲˆÏÓ˚ ܲ¡õö ÓçyÎ˚ Ó˚yˆÏá– ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ àí˛¸ •yÓ˚ §ˆÏÓ≈yFã˛
•ÄÎ˚yÓ˚ ˛çöƒ ≤ÈÏÎ˚yçö#Î˚ ¢ï≈˛ ω = ω0–
7. ï˛#«¯˛ï˛y xö%öyˆÏòÓ˚ ~ܲ!ê˛ =Ó˚&c˛õ)î≈ ˜Ó!¢‹Tƒ– Q =îyB˛ !öî≈ˆÏÎ˚Ó˚ §y•yˆÏ΃ ~•z ï˛#«¯˛ï˛yÓ˚ ˛õ!Ó˚üy˛õ ܲÓ˚y
ÎyÎ˚– Î!ò ω2 Ä ω1 ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ àí˛¸ •hflÏyhsˇ!Ó˚ï˛ «˛üï˛y §ˆÏÓ≈yFã˛ üyˆÏöÓ˚ xˆÏô≈ܲ •Î˚ ï˛ˆÏÓñ
Q =îyB˛ = 0
2 1
ωω ω− =
0
2bω
8. ï˛!í˛¸Í Óï≈˛ö#ˆÏï˛ ≤ÈÏîy!ï˛ Ü˛¡õö °«˛ƒ ܲÓ˚y ÎyÎ˚– ~ܲ!ê˛ LCR ˆ◊î#§Iy ÈÙÈ Óï≈˛ö#ˆÏï˛ ˛õ!Ó˚Óï≈˛ö¢#°
ܲ¡õyB˛ !Ó!¢‹T ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛ Ó° ≤ÈÏÎ˚yà ܲˆÏÓ˚ ≤ÃÓy•üyeyÓ˚ xö%öyò ˛õyÄÎ˚y §Ω˛Ó– ~•z Óï≈˛ö#Ó˚ xö%öyò#
ܲ¡õyB˛ ω0 •ˆÏ°ñ ω0 =
1LC
,
§ˆÏÓ≈yFã˛ ≤ÃÓy•ñ I0m
=
0 ER
~ÓÇ
ω Óï≈˛ö#Ó˚ Q =îyB˛ =
0LR
ω
=
1LRC
117
4.10 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ≤Ãüyî ܲÓ˚&ö ˆÎñ xÓü!®ï˛ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe ܲ¡õö¢#° ï˛ˆÏsfÓ˚ ˆüyê˛ ¢!_´ ô &Óܲ öÎ˚– ~•z
§ˆÏD ˆòáyö ˆÎñ ï˛sf!ê˛Ó˚
˛õ)î≈ ˛õÎ≈yˆÏÎ˚
Üv !fiÌ!ì ü!=˛˛
Üv Ü!ì ü!=˛
=
20
2
ωω
2. ˆòáyö ˆÎñ ˆÜ˛yöÄ °â% xÓü®ö §• ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe ï˛ˆÏsfÓ˚ !ÓhflÏyÓ˚ Ä ò¢yñ Q =
0
2bω
ÓƒÓ•yÓ˚ ܲˆÏÓ˚ !ö¡¨!°!áï˛ û˛yˆÏÓ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚ñ
B(ω) =
0
0122
02
0
1
B
Q
ωω
ωωωω
⎡⎤ ⎛⎞ −+ ⎢⎥ ⎜⎟ ⎝⎠ ⎢⎥ ⎣⎦
~ÓÇ tan θ =
0
0
1Q
ωωωω
⎛⎞ − ⎜⎟ ⎝⎠
– ~áyˆÏö B0 =
20
Fmω
3. ~ܲ!ê˛ ˆ◊î# §üÓyˆÏÎ˚Ó˚ LCR Óï≈˛ö#ˆÏï˛ C = 50μF, R = 10Ω, ˙ Óï≈˛ö#ˆÏï˛ Î%_´ ≤Ãï˛ƒyÓï≈˛# ï˛!í˛¸Fã˛y°Ü˛
ӈϰÓ˚ !ÓhflÏyÓ˚ 20v ~ÓÇ Ü˛¡õyB˛ 60 Hz– LÈÙÈ~Ó˚ ˆÜ˛yö üyˆÏöÓ˚ çöƒ Óï≈˛ö#ˆÏï˛ ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ üyö §ˆÏÓ≈yFã˛
•ˆÏÓ⁄ ˙ §ˆÏÓ≈yFã˛ ï˛!í˛¸Í ≤ÃÓyˆÏ•Ó˚ üyö ܲï˛⁄
4.11 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1 : ˆòÄÎ˚y xyˆÏåÈ m = 0.2kg, k = 200Nm–1, F0 = 12N, ω = 50 rad s–1
xÓü®ö =îyB˛ γ = 2Nsm–1
ω0
2 =
km
=
200.2
= 1000 rad2 s–2
f0 =
0 Fm
=
120.2
= 60 N kg–1
118
2b =
mγ
=
20.2
= 10s–1 ∴ b = 5s–1
∴ fliyÎ˚# xÓfliyÓ˚ !ÓhflÏyÓ˚ B =
()0
122 2222
04
f
b ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
=
1222
60
(10002500)4.5.2500 ⎡⎤ −+ ⎣⎦
=
260
15.8110 ×
= .038 m = 3.8 cm
§Ó˚î Ä ≤ÃÎ%_´ ӈϰÓ˚ üˆÏôƒ ò¢y ˆÜ˛yî θ •ˆÏ°
tan θ = 2 20
2bωω ω−
= 2.5.501000 2500− = – 0.33
∴ θ = 180º – 18.4º = 161.6º xÌ≈yÍñ §Ó˚î ӈϰÓ˚ ï%˛°öyÎ˚ 161.6º ò¢yˆÏܲyˆÏî ~!àˆÏÎ˚ ÌyܲˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -2 : ôÓ˚y Îyܲ !fl±ÇÈÙÈ~Ó˚ ≤çyÓ˚î Îáö ¢)öƒñ ï˛áö û˛Ó˚!ê˛ z = 0 xÓfliyˆÏö xyˆÏåÈ– z ò)Ó˚c
˙ !Ó®% ˆÌˆÏܲ !öˆÏã˛Ó˚ !òˆÏܲ üy˛õy •ˆÏ°ñ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ xÓܲ° §ü#ܲÓ˚î
2
2dzdz mkz
dt dtγ ++
= mg
~áyˆÏöñ m = 0.1kg, γ = 5Nsm–1, k = 150 Nm–1
xÓܲ° §ü#ܲÓ˚î!ê˛Ó˚ ≤Ã!ï˛!ê˛ ˛õò m !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ 4.5 ~Ó˚ üï˛ ˆ°áy ÎyÎ˚ñ
22
0 22 dzdz bzdt dt
ω ++
= g
Óyñ
22
02' '2 'd z dzb z
dtdtω+ + = 0 ˆÎáyˆÏöñ z' = mgz
k−
2b = mγ =
50.1
= 50s–1, b = 25s–1, ω02 =
km
=
1500.1
= 1500s–2
∴ xÓü!®ï˛ ˆòy°ˆÏöÓ˚ ܲ¡õyB˛
'2ω
π
=
220
12
b − ω π
=
()1
22 11500254.72
Hz π−≈
xyˆÏÓ˚y!˛õï˛ Ó° F = 3 cos 20t N
119
∴ F0 = 3N, f0 = 0Fm
=
3.1
= 30ms–2, ω = 20 s–1
fliyÎ˚# xÓfliyÎ˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ (4.14 §ü#ܲÓ˚î ˆòá%öV
B =
()0
122 2222
04
f
b ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
=
122
30
(1500400)4.625.400 ⎡⎤ −+ ⎣⎦
= .02m Óyñ 2cm
ò¢yÓ˚ ˛õyÌ≈ܲƒ θ (4.13b §ü#ܲÓ˚î xö%ÎyÎ˚#V =
122
0
2 tanbωωω
−
−
= 1 2.25.20tan1500 400
−
− = tan–1 0.91 = 42º
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -3 : ~áyˆÏö k = 200 Nm–1, m = 200g = 0.2 kg
ω02 =
km = 1000s–2
xÓü®ö Ó° – yv = –v ∴ γ = 1 Nsm–1
2b =
mγ
=
10.2
= 5s–1 ∴ b =
52
= 2.5
F0 = 5N, ω = 30s–1
!ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ ܲ¡õyB˛ ωr •ˆÏ°ñ (4.18) §ü#ܲÓ˚î xö%ÎyÎ˚#ñ
ωr =
2202b ω−
= 2510002
− = 44.6 rad s–1
§ˆÏÓ≈yFã˛ à!ï˛¢!_´ (KE)max
(4.23) §ü#ܲÓ˚î xö%ÎyÎ˚# ˛õyÄÎ˚y ÎyÎ˚ñ
(KE)max = 2
028
Fb m
= 2.5J.
120
fliyÎ˚# xÓfliyÓ˚ ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ B,
B =
()0
122 2222
04
F
mb ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
=
25
(0.2)1.810 ××
= 13.8 ×10–2 m
= 13.8 cm
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 4 : ~áyˆÏö m = 70g = 0.07kg, k = 14Nm–1, γ = 0.7 Nsm–1
∴ b = 2mγ =
0.72.07 ×
= 5s–1 ω0
2 =
km
=
2 14.07
s−
= 200s–2
!ÓhflÏyÓ˚ xö%öyˆÏòÓ˚ çöƒ ≤ÃÎ%_´ ӈϰÓ˚ ≤ÈÏÎ˚yçö#Î˚ ܲ¡õyB˛
ωr =
2202b ω−
= 2200 2.5− = 12.2s–1
à!ï˛ˆÏÓà xö%öyˆÏòÓ˚ çöƒ ≤ÃÎ%_´ ӈϰÓ˚ ≤ÈÏÎ˚yçö#Î˚ ܲ¡õyB˛ ω = ω0
∴ ω = ω0 = 1200 s− ∴ ω = 14.1 s–1
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ~•z í˛zòy•Ó˚ˆÏî ωr Ä ω0 ÈÙÈ~Ó˚ üˆÏôƒ ˆÓ¢ !ܲå%Èê˛y ˛õyÌ≈ܲƒ xyˆÏåÈ– ~Ó˚
ܲyÓ˚îñ ~áyˆÏö ω0 Ä b Ó˚y!¢ ò%•z!ê˛ ï%˛°ö#Î˚ üyˆÏöÓ˚ xÌ≈yÍñ ~áyˆÏö xÓü®ö ˆüyˆÏê˛•z öàîƒ öÎ˚–
≤ÃÎ%_´ ӈϰÓ˚ !ÓhflÏyÓ˚ F0 = 8N
4.20 §ü#ܲÓ˚î ˆÌˆÏܲ §ˆÏÓ≈yFã˛ à!ï˛¢!_´ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚
Vm
=
2200
2220 8()
Fmbb −
ωω
= 2
2 28 200
8 .07 5 (200 5 )×
× × −
= 5.2 J
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -5 : m = 100g = 0.1kg, k = 10 Nm–1, γ = 0.1 Nsm–1, F0 = 1N
2b = mγ =
0.10.1
= 1 s–1 ω0
2 =
km
=
100.1
= 100 s–2
f0 =
0 Fm
=
10.1
= 10 ms–2 ~ÓÇ 4b2 = 1 s–2
121
(i) Îáö ω = 1 rad s–1, fliyÎ˚# xÓfliyÎ˚ ˛õy•zñ
!ÓhflÏyÓ˚ B =
()0
122 2222
04
f
b ωωω ⎡⎤ −+ ⎢⎥ ⎣⎦
=
122
10
(1001)1.1 ⎡⎤ −+ ⎣⎦
=
122
10
991metre
⎡⎤ + ⎣⎦
= 1099.01 = 0.1 m = 10 cm
(ii) Îáö ω = 50 rad s–1
B =
12222
10
(10050)50 ⎡⎤ −+ ⎣⎦
102400
= 0.004 m = 4 mm
í˛zû˛Î˚ ˆ«˛ˆÏe•z Q =îyB˛ = 0
2bω =
101
= 10
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 6 : ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe ˆÜ˛yöÄ §üÎ˚ ï˛ˆÏsfÓ˚ !fli!ï˛¢!_´ ≠
V =
2 12
kx
=
2220
1cos()2
mBt ωωθ−
ôÓ˚y Îyܲñ ˛õÎ≈yÎ˚ܲy°
2πω
= T0
˛õ)î≈ ˛õÎ≈ˆÏÎ˚ àí˛¸ !fli!ï˛¢!_´ ôÓ˚y Îyܲ
0222
000
1cos()2
T
VmBtdtT
ωωθ =− ∫
= 0
2 2 20
0 0
1 1. [cos ( ) 1]2 2
T
m B t dtT
ω ω θ− +∫
= 2 2 0
00
1 . .2 2
Tm B
Tω ⎛ ⎞
⎜ ⎟⎝ ⎠ =
220
14
mB ω
xyÓyÓ˚ à!ï˛¢!_´ T = 212
mx =
222 1sin()2
mBt ωωθ−
122
~ܲ•zû˛yˆÏÓ ˛õ)î≈ ˛õÎ≈yˆÏÎ˚ àí˛¸ à!ï˛¢!_´
T
•ˆÏ°
T
=
222
0
1sin()2
T
mBtdtT
− ∫ωωθ
= 2 214
m Bω
≤Ã!ï˛ ã˛ˆÏe´ ˆüyê˛ àí˛¸ ¢!_´ E •ˆÏ°
E
=
V
+
T
=
220
14
mB ω
+ 2 214
m Bω
= 2 2 20
1 ( )4
m Bω ω+
≤Ã!ï˛ ˛õÎ≈yˆÏÎ˚ xÓü®ˆÏöÓ˚ !ÓÓ˚&ˆÏk˛ ܲyÎ≈ ܲÓ˚yÓ˚ ú˛ˆÏ° ¢!_´Ó˚ •…y§ •Î˚– ~ܲܲ à!ï˛ˆÏÓˆÏà xÓü®ö Ó° γ•ˆÏ°ñ «˛üï˛y •…yˆÏ§Ó˚ üyö = SÓ° × ò)Ó˚cV ÷ §üÎ˚ = Ó° γ v × ˆÓà v = γ v2
§%ï˛Ó˚yÇ xÓü®ö ӈϰÓ˚ !ÓÓ˚&ˆÏk˛ Óƒ!Î˚ï˛ «˛üï˛y
2vγ = bmω2B2
ˆÜ˛ööy γ = 2bm, v2 = B2ω2 sin2 (ωt – θ) ~ÓÇ 222 1
2vBω =
~áö xyüÓ˚y ˆòˆÏá!åÈ ˆÎñ Q =îyB˛ˆÏܲ ˆ°áy ÎyÎ˚ (4.35 ˆÌˆÏܲV
Q = 2π = ×ì˛ˆsÏ ˛f ¢!M˛ì˛ ü!=˛˛˛
˛ôí) ≈ ˛ôÎ≈yˆÎÏ Óƒ!Îì˛ ü!=˛ S ˆîyúòÑ˛yú Óƒ!Îì˛ «˛õì˛yV˛
=
() 2220
22
14 22.
mB
bmB
+ ωωππω ω
=
220
4b+ ωω
ω
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -7 : ~áyˆÏö ܲ¡õyB˛ f = 105 Hz, ω0 = 2fπs–1, E
0 = 1.2V
ˆÎˆÏ•ï%˛ñ ω0 =
1LC
∴ C =
20
1L ω
~ˆÏ«˛ˆÏe L = 0.5 mH, ω0 = 2π × 105 s–1, R = 30Ω
= 10 31
40 10 (0.5) 10F−× × ≅ 5.1 × 10–9F = 5.1nF
123
(I0)
max = 0E
R =
1.230
A
= .04 A = 40 mA
Q =
0LRω
=
53 2100.51030
− ××× π
= 10.5
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe fliyÎ˚# xÓfliyÎ˚ ܲ¡õö¢#° û˛ˆÏÓ˚Ó˚ !fli!ï˛¢!_´ 21
2V kx=
= ( )2 2 20
1 cos2
m B t −ω ω θ
à!ï˛¢!_´
() 2222 11sin22
TmxmBt ==− ωωθ
∴ ˆüyê˛ ¢!_´
()() 222220
1cossin2
EVTmBtt ⎡⎤ =+=−+− ⎣⎦ ωωθωωθ
ˆòáy ÎyˆÏFåÈñ ˆüyê˛ ¢!_´ E §üÎ˚ ‘t’ ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#° ~ÓÇ ô Óܲ öÎ˚– ~áö ~ܲ!ê˛ ˛õ)î≈ ã˛ˆÏe´ !fli!ï˛¢!_´
Ä à!ï˛¢!_´Ó˚ àí˛¸ üyö !öî≈Î˚ ܲÓ˚y Îyܲ–
˛õ)î≈ ã˛ˆÏe´ àí˛¸ !fli!ï˛¢!_´ = ()() 222
0
11sin2
T
avdtPEmBt
T=− ∫ωωθ
=
22 14
mB ω
˛õ)î≈ã˛ˆÏe´Ó˚ àí˛¸ à!ï˛¢!_´ = = −zKET
m B tav dt.
sin ( )1 12
2 2 2
0
1
ω ω θ
∴ ~Ñ˛!›˛ ˛ô)í ≈ â˛ˆÑÏ ˛ Üv˛ !fiÌ!ì˛ü!=˛˛
˛~Ñ˛!›˛ ˛ôí) ≈ â˛ˆÏÑ˛ Üv˛ Ü!ì˛ü!=˛˛ =
22200
222
1414
mB
mB=
ωωω ω
2. °â% xÓü®ö §• ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏe xyüÓ˚y çy!ö ˆÎ fliyÎ˚# xÓfliyÓ˚ !ÓhflÏyÓ˚
( )( )
0
01 1
2 22 222 2 2 200
200 0
44
FF mB
bm b
= =⎡ ⎤ ⎡ ⎤⎛ ⎞− +⎢ ⎥ − +⎣ ⎦ ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
ωω ωω ω ω ωω ω ω ω
124
~áöñ Q = 0
2bω ~ÓÇ
020
F Bmω
=
ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚ñ
( ) 00
2 10 2
0
B B
Q
=
⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
ωω ωω ωω ω
xyÓyÓ˚ñ tanθ ωω ω
ω
ωωωω
ωω
=−
=−
FHG
IKJ
2 2
02 2
00
0
b b
ˆÎˆÏ•ï%˛ñ Q = 0
2bω ∴
ω=0
21 bQ
∴
0
0
1
tanQ =⎛⎞ − ⎜⎟ ⎝⎠
θωωωω3. xyüÓ˚y çy!öñ
() 0cos IIt =− ωψ002
21
EI
RLC
=⎛⎞ +− ⎝⎠ ωω
~Ó˚ §ˆÏÓù≈yFã˛ üyö ˛õyÄÎ˚y ÎyÎ˚ Îáöñ
1LC
ω ω= Óyñ 2 1
LCω =
~áy Ïö 1 150 , 2 60 120C F s s− −= = =μ ω π π
∴
() 226
110.141205010
LHc−
===×× ωπ
0I − Ó˚ §ˆÏÓù≈yFã˛ üyö
0Im
•ˆÏ°ñ
00
20210 m
EIA
R===
125
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 5 Î%!@¬ï˛ ˆòy°öÎ%!@¬ï˛ ˆòy°öÎ%!@¬ï˛ ˆòy°öÎ%!@¬ï˛ ˆòy°öÎ%!@¬ï˛ ˆòy°ö (Coupled oscillation)
àë˛öàë˛öàë˛öàë˛öàë˛ö
5.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
5.2 •y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö
5.2.1 xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y
5.2.2 Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#
5.3 Î%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢ö–
5.4 ò%!ê˛ Î%!@¬ï˛ §Ó˚° ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî Ä ˛õÎ≈yˆÏ°yã˛öy
5.5 fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛
5.6 §yÓ˚yÇ¢
5.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
5.8 í˛z_Ó˚üy°y
5.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
~•z ˛õÎ≈yˆÏÎ˚ xyüÓ˚y ~ ˛õÎ≈hsˇ ~ܲܲ ˆòy°ˆÏܲÓ˚ ˆòy°ö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ~Ó˚ üˆÏôƒ ˛õ)Ó≈Óï≈˛# ~ܲˆÏܲ
≤ÈÏîy!òï˛ ˆòy°ˆÏöÓ˚ xyˆÏ°yã˛öyÎ˚ xyüyÓ˚ ˆòˆÏá!åÈñ ã˛y°Ü˛ Ó° ܲ#û˛yˆÏÓ Óy•zˆÏÓ˚ ˆÌˆÏܲ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ üyôƒˆÏü
ã˛y!°ï˛ ï˛ˆÏsf ܲ¡õö ÓçyÎ˚ Ó˚yˆÏá– ~ˆÏ«˛ˆÏe ¢!_´Ó˚ •hflÏyhsˇÓ˚ ˆÜ˛Ó°üye ~ܲ!òˆÏܲ xÌ≈yÍ ã˛y°Ü˛ ˆÌˆÏܲ ã˛y!°ˆÏï˛Ó˚
!òˆÏܲ âˆÏê˛ ÌyˆÏܲÈÙÙÙÈã˛y°ˆÏܲÓ˚ !òˆÏܲ ¢!_´ ˆú˛Ó˚ï˛ xyˆÏ§ öy ~ÓÇ ~Ó˚ ú˛ˆÏ° ã˛y°ˆÏܲÓ˚ xÓfliyÓ˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö
âˆÏê˛ öy– ï˛áö•z xyüÓ˚y í˛zˆÏÕ‘á ܲˆÏÓ˚!åÈ°yü ˆÎñ Î!ò ¢!_´Ó˚ •hflÏyhsˇÓ˚ í˛zû˛Î˚ü%á# •ÄÎ˚y §Ω˛Ó˛õÓ˚ •Î˚ñ ï˛y•ˆÏ°
ã˛y°Ü˛ Ä ã˛y!°ï˛ˆÏܲ xy°yòyû˛yˆÏÓ !ã˛!•´ï˛ ܲÓ˚y ÎyˆÏÓ öy– ï˛áö ˆÎ ܲ¡õö ˆòáy ÎyˆÏÓñ ï˛yÓ˚ !Ó!û˛ß¨ ˛õÎ≈yˆÏÎ˚
¢!_´ •hflÏyhsˇˆÏÓ˚Ó˚ x!û˛ü%á ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÓ ~ÓÇ ~ܲ!ê˛ ï˛sfˆÏܲ ~ܲÓyÓ˚ ã˛y!°ˆÏï˛Ó˚ û)˛!üܲyÎ˚ ˆòáˆÏ° !ܲå%È ˛õˆÏÓ˚•z
ˆ§!ê˛ ã˛y°ˆÏܲÓ˚ xy§ö @ˇÃ•î ܲÓ˚ˆÏÓ– ~•z ˛õ!Ó˚Óï≈˛ö ã˛°ˆÏï˛•z ÌyܲˆÏÓ Îï˛«˛î öy ܲ¡õö §¡õ)î≈ Ó¶˛ •ˆÏFåÈ– ~•z
ôÓ˚ˆÏöÓ˚ ܲ¡õö xyüyˆÏòÓ˚ ˛õ)Ó≈ ˛õ!Ó˚!ã˛ï˛ ܲ¡õö ˆÌˆÏܲ !û˛ß¨ ~ÓÇ ˛õòyÌ≈!ÓòƒyÓ˚ ˛õ!Ó˚û˛y£ÏyÎ˚ ~ˆÏܲ Ó°y •Î˚ Î%!@¬ï˛
ˆòy°ö (coupled oscillation)–
~•z ~ܲˆÏܲ xyüÓ˚y ≤Ãôyöï˛ ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ ܲ¡õö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó– ï˛ˆÏÓ ~•z xyˆÏ°yã˛öyÎ˚
xy˛õ!ö ˆÎ ày!î!ï˛Ü˛ ˛õk˛!ï˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÓöñ ˆ§!ê˛ ˆÎ ˆÜ˛yöÄ §ÇáƒÜ˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe ≤ÈÏÎ˚yà
ܲÓ˚y ÎyÎ˚–
≤ÃÜ,˛!ï˛ˆÏï˛ Î%!@¬ï˛ ˆòy°ˆÏöÓ˚ xˆÏöܲ í˛zòy•Ó˚î ˆòáˆÏï˛ ˛õyÄÎ˚y ÎyÎ˚– çˆÏ°Ó˚ xî%ˆÏï˛ x!:ˆÏçö ˛õÓ˚üyî% ò%!ê˛
•y•zˆÏí»˛yˆÏçö ˛õÓ˚üyî%Ó˚ ≤Ã!ï˛!ê˛Ó˚ §ˆÏD !ü!°ï˛ •ˆÏÎ˚ ~ܲ ~ܲ!ê˛ ˆòy°Ü˛ Ó˚ã˛öy ܲˆÏÓ˚– x!:ˆÏçö ˛õÓ˚üyî%Ó˚ üyôƒˆÏü
~•z ò%!ê˛ ˆòy°Ü˛ Î%!@¬ï˛ ÌyˆÏܲ– ò%!ê˛ §Ó˚° ˆòy°Ü˛ Î!ò ~ܲ•z ê˛yö ܲˆÏÓ˚ Ó˚yáy Ó˚I% ˆÌˆÏܲ ˆé˛y°yˆÏöy ÌyˆÏܲñ
126
ï˛y•ˆÏ° ˙ Ó˚I%Ó˚ fliyöã%˛ƒ!ï˛Ó˚ üyôƒˆÏü ˆòy°Ü˛ ò%!ê˛ Î%!@¬ï˛ •Î˚ S!ã˛e 5.1)– !fli!ï˛fliy˛õܲ üyôƒˆÏü §!ߨ!•ï˛
ӛܲîy=!° ˛õÓ˚flõÓ˚ Î%!@¬ï˛ ÌyˆÏܲ ӈϰ•z ~•z çyï˛#Î˚ üyôƒˆÏü !fli!ï˛fliy˛õܲ ï˛Ó˚ˆÏDÓ˚ í˛zͲõ!_ •Î˚– ~ ˆÌˆÏܲ
ˆÓyé˛y ÎyÎ˚ ˆÎñ Î%!@¬ï˛ ˆòy°ö ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ ˆòáˆÏ° xï˛ƒhsˇ =Ó˚&c˛õ)î≈–
í˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒ
Î%!@¬ï˛ ˆòy°ö !Ó£ÏÎ˚ܲ ~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° xy˛õ!ö ˆÎ ܲyç=!° ܲÓ˚yÓ˚ ò«˛ï˛y xç≈ö ܲÓ˚ˆÏÓö ˆ§=!°
•° ÈÙÙÙÈ
Î%@¿ˆÏöÓ˚ ú˛ˆÏ° ~ܲy!ôܲ !Ó!FåÈߨ ˆòy°ˆÏܲÓ˚ ˜Ó!¢ˆÏ‹Tƒ ܲ# ôÓ˚ˆÏöÓ˚ ˛õ!Ó˚Óï≈˛ö âˆÏê˛ ï˛yÓ˚ ˛õÎ≈yˆÏ°yã˛öy ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
xö%˜Ïòâ≈ƒû˛yˆÏÓ Ü˛!¡õï˛ Î%!@¬ï˛ ˆòy°Ü˛ ï˛ˆÏsfÓ˚ àï˛#Î˚ §ü#ܲÓ˚ˆÏîÓ˚ (equation of motion) ≤Ã!ï˛¤˛y
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
Î%!@¬ï˛ ˆòy°ˆÏöÓ˚ §üO§ !öˆÏò≈¢yB˛ (normal coordinates of coupled oscillation) Ä fl∫û˛yÓ
ܲ¡õˆÏöÓ˚ ôÓ˚ö (normal modes of vibration) ~ÓÇ Ü˛¡õyB˛ àîöy Ä ˆ§=!° !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î
˛õk˛!ï˛Ó˚ ˛õÎ≈yˆÏ°yã˛öy–
ï%˛°öyü)°Ü˛ xyˆÏ°yã˛öyÓ˚ üyôƒˆÏü !ܲå%È ˛õ!Ó˚!ã˛ï˛ ÓyhflÏÓ ï˛ˆÏsfÓ˚ fl∫û˛yÓ Ü˛¡õˆÏöÓ˚ ôÓ˚ö àîöy–
5.2 •y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö–•y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö–•y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö–•y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö–•y°Ü˛y !fl±Ç myÓ˚y Î%_´ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ Î%!@¬ï˛ ܲ¡õö–
Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ !Óhfl,Ïï˛ xyˆÏ°yã˛öyÓ˚ ~•z xö%ˆÏFåȈÏò xyüÓ˚y ~ܲ!ê˛ !ӈϢ£Ï !fl±ÇÈÙÈû˛Ó˚ ï˛sfˆÏܲ ˆÓˆÏåÈ !ö!FåÈ–
5.2 !ã˛ˆÏe ~•z ï˛sf!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö ò%!ê˛ §üyö û˛Ó˚ m, k' !fl±Ç ô Óܲ !Ó!¢‹T ò%•z!ê˛ !fl±Ç ~Ó˚ ≤ÃyˆÏhsˇÓ˚
§ˆÏD Î%_´– ~•z !fl±Ç ò%!ê˛Ó˚ xöƒ ≤Ãyhsˇ=!° ò,ì˛¸ xÓ°¡∫ˆÏöÓ˚ §ˆÏD xyÓk˛– ò%!ê˛ û˛Ó˚ m ˆÜ˛ Î%_´ ܲˆÏÓ˚ˆÏåÈ ï,˛ï˛#Î˚
127
~ܲ!ê˛ !fl±Ç ÎyÓ˚ !fl±Ç ô Óܲ k– ~•z !fl±Ç ò%!ê˛ û˛ˆÏÓ˚Ó˚ üˆÏôƒ Î%@¬ˆÏöÓ˚ (coupling) ܲyç ܲÓ˚ˆÏåÈ– k ~Ó˚ üyˆÏöÓ˚
IJõÓ˚ Î%@¬ˆÏöÓ˚ üyey !öû≈˛Ó˚
ܲÓ˚ˆÏÓ– §yüƒ xÓfliyˆÏö §Ó
ܲ!ê˛ !fl±Ç !öç !öç
fl∫yû˛y!Óܲ ˜òâ≈ƒ ÓçyÎ˚ Ó˚yˆÏá
~ÓÇ û˛Ó˚=!° !fliÓ˚ xÓfliyÎ˚
ÌyˆÏܲ–
xy˛õ!ö ˆû˛ˆÏÓ ˆòáyˆÏï˛
˛õy ÏÓ˚öñ ~ôÓ˚ ÏöÓ˚ ~ܲ!ê˛ ï˛ Ïsf
ܲ# ôÓ˚ˆÏöÓ˚ ܲ¡õö âê˛ˆÏï˛
˛õyˆÏÓ˚– û˛Ó˚ ò%!ê˛Ó˚ ˆÎ ˆÜ˛yöÄ
~ܲ!ê˛ñ xÌÓy ò%!ê˛ˆ Ïܲ•z
§yüƒyÓfliy ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚
ˆåȈÏí˛¸ !òˆÏ°ñ ˆ§=!° ܲ!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ ~ÓÇ !fl±Ç=!°Ä û˛Ó˚=!°Ó˚ §Ó˚î xö%ÎyÎ˚# §B%˛!ã˛ï˛ Ä ≤çy!Ó˚ï˛
•ˆÏï˛ ÌyܲˆÏÓ– û˛Ó˚ ò%!ê˛Ó˚ ≤ÃyÌ!üܲ !Óã˛ƒ%!ï˛ Î!ò !fl±Ç=!°Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ xÌ≈yÍ í˛yö!òˆÏܲ Óy ÓÑy!òˆÏܲ •Î˚ñ ï˛ˆÏÓ
ˆ§=!°Ó˚ ܲ¡õöÄ !fl±Ç ~Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ •ˆÏÓ– ~•z ôÓ˚ˆÏöÓ˚ ܲ¡õöˆÏܲ xyüÓ˚y xö%˜Ïòâ≈ƒ ܲ¡õö Ó!°– û˛Ó˚
ò%!ê˛Ó˚ ~ܲ!ê˛ Óy í˛zû˛Î˚ˆÏܲ Î!ò !fl±Ç ~Ó˚ ˜òˆÏâ≈ƒÓ˚ °¡∫ ÓÓ˚yÓÓ˚ !Óã%˛ƒï˛ ܲÓ˚yÓ˚ ˛õÓ˚ ˆåȈÏí˛¸ ˆòÄÎ˚y •Î˚ñ ï˛ˆÏÓ û˛Ó˚=!°Ä
˙ °¡∫ ÓÓ˚yÓÓ˚ñ xÌ≈yÍ xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ •ˆÏÓ– ï˛ˆÏÓ ~áyˆÏö xyüÓ˚y ˆÜ˛Ó°üye xö%˜Ïòâ≈ƒ ܲ¡õˆÏö•z
xyˆÏ°yã˛öy §#üyÓk˛ Ó˚yáÓ–
xÓ¢ƒ ~ê˛y xyüyˆÏòÓ˚ üˆÏö Ó˚yáˆÏï˛•z •ˆÏÓ ˆÎñ û˛Ó˚=!°Ó˚ ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ñ ˛õyÓ˚flõ!Ó˚ܲ ò¢yÓ˚ ≤ÈÏû˛ò ≤Ãû˛,!ï˛
˜Ó!¢‹Tƒ ܲ¡õˆÏöÓ˚ ≤ÃyÌ!üܲ ¢ï≈˛=!°Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– §%ï˛Ó˚yÇñ xyüÓ˚y ܲ¡õö¢#° Ó›ï˛ˆÏsfÓ˚ à!ï˛ §ü#ܲÓ˚î
≤Ã!ï˛¤˛y ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏ°Äñ ï˛yÓ˚ §üyôyˆÏöÓ˚ çˆÏöƒ xyüyˆÏòÓ˚ ≤ÃyÌ!üܲ ¢ï≈˛=!° çyöy ≤ÈÏÎ˚yçö •ˆÏÓ
5.2 xÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yxÓܲ° §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛y
5.2 !ã˛ˆÏe ˆòáyˆÏöy !fl±Ç ÈÙÈû˛Ó˚ ï˛ˆÏsfÓ˚ §ˆÏD xy˛õ!ö xyˆÏà•z ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö– ~áyˆÏö xyüÓ˚y O !Ó®% Ïܲ
ü)°!Ó®% ~ÓÇ !fl±Ç ~Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ‘x’ x«˛ ôˆÏÓ˚ ˆöÓ– ôÓ˚y Îyܲñ XA ~ÓÇ X
B ÎÌye´ˆÏü ò%!ê˛ û˛Ó˚ A
~ÓÇ B ~Ó˚ û˛Ó˚ˆÏܲˆÏwÓ˚ fliyöyB˛ !öˆÏò≈¢ ܲÓ˚ˆÏåÈ– ~áö A Ä B û˛Ó˚ ò%!ê˛ˆÏܲ í˛yö!òˆÏܲ ÎÌye´ˆÏü x1 Ä x2 ò)Ó˚c
ˆê˛ˆÏö §!Ó˚ˆÏÎ˚ ôˆÏÓ˚ Ó˚yáy •°– ~Ó˚ ú˛ˆÏ° í˛yö!òˆÏܲÓ˚ !fl±Ç!ê˛ §B%˛!ã˛ï˛ •ˆÏÓ– Óyü!òˆÏܲÓ˚ !fl±Ç Ä üyˆÏé˛Ó˚ !fl±Çñ
ÎyˆÏܲ ˛õ!Ó˚û˛y£ÏyÎ˚ Ó°y •Î˚ Î%@¬ö !fl±Ç (coupling spring), ~Ó˚ ú˛ˆÏ° ≤çy!Ó˚ï˛ •ˆÏÓ– ~áö A Ä B ˆÜ˛
ˆåȈÏí˛¸ !òˆÏ° §ü@ˇÃ ÓƒÓfliy!ê˛Ó˚ xö%˜Ïòâ≈ƒ ܲ¡õö âê˛ˆÏï˛ ÌyܲˆÏÓ–
Î!ò t §üˆÏÎ˚ ü)°!Ó®%Ó˚ §yˆÏ˛õˆÏ«˛ A ~ÓÇ B û˛ˆÏÓ˚Ó˚ xÓfliyö ÎÌye´ˆÏü XA ~ÓÇ X
B •Î˚ñ §yüƒyÓfliy ˆÌˆÏܲ
XB
XB
XA
XA
128
ï˛yˆÏòÓ˚ ò)Ó˚c •ˆÏÓ ÎÌye´ˆÏü x1=x
A -X
A ~ÓÇ x
2 =x
B -X
B– xyüÓ˚y xyˆÏà•z í˛zˆÏÕ‘á ܲˆÏÓ˚!åÈ ˆÎñ ò%’≤ÃyˆÏhsˇÓ˚
ò%!ê˛ !fl±Ç ~Ó˚ !fl±Ç ô Óܲ k′ ~ÓÇ üyˆÏé˛Ó˚ Î%@¿ö !fl±Ç ~Ó˚ ˆ«˛ˆÏe ˙ üyö k– §%ï˛Ó˚yÇñ A Ä B ~Ó˚ IJõÓ˚
!e´Î˚y¢#° Ó°à%!° Ü˛ï˛ •ˆÏÓ ï˛y !öˆÏã˛Ó˚ §yÓ˚!îˆÏï˛ ˆòáyˆÏöy •° :
ӈϰÓ˚˛ õ!Ó˚ã˛Î˚ӈϰÓ˚˛ õ!Ó˚ã˛Î˚ӈϰÓ˚˛ õ!Ó˚ã˛Î˚ӈϰÓ˚˛ õ!Ó˚ã˛Î˚ӈϰÓ˚˛ õ!Ó˚ã˛Î˚ A ~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó° B ~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°~Ó˚ IJõÓ˚ !e´Î˚y¢#° Ó°
(i) ≤Ãï˛ƒyöÎ˚ܲ Ó° –k′(xA –X
A)=–k′x
1–k′(x
B–X
B) =–k′x
2
(ii) Î%@¿ö Ó° ( ) ( )B A B AK x x X X⎡ ⎤− − −⎣ ⎦
()() ABAB kxxXX ⎡⎤ −−− ⎣⎦
(coupling force) = k(x2–x
1) = –k(x
2 –x
1)
xyüÓ˚y ôˆÏÓ˚ !öˆÏÎ˚!åÈñ í˛zû˛Î˚ û˛Ó˚•z §üyö xÌ≈yÍñ mA = m
B = m ~ÓÇ í˛zû˛ˆÏÎ˚•z â£Ï≈î•#ö ï˛ˆÏ° ã˛°yˆÏú˛Ó˚y
ܲÓ˚ˆÏåÈ– xï˛~Óñ !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚ §)e xö%ÎyÎ˚# ˆòáy ÎyÎ˚ ˆÎñ û˛Ó˚ A -~Ó˚ ˆ«˛ˆÏe
()2
121 2′ =−+− A dxmkxkxx
dt
........ 5.1
ˆÎˆÏ•ï%˛ XA ~ܲ!ê˛ !fliÓ˚ xÓfliyö ˆÓyé˛yˆÏFåÈñ §üˆÏÎ˚Ó˚ §ˆÏD ï˛y ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÓ öy– xÌ≈yÍñ
2
2 0Ad Xdt
=
∴ ()22 2
11 222
AA
dxdx dxXdtdtdt
=+=
§ü#ܲÓ˚î 5.1 ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ òÑyí˛¸yÎ˚
( )2
11 2 12 ′= − + −d x
m k x k x xdt
í˛zû˛Î˚˛õ«˛ˆÏܲ m !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ~ÓÇ ˛õò=!°ˆÏܲ §y!çˆÏÎ˚ !öˆÏÎ˚ ˛õy•z
( )2
2 210 1 2 12 0c
d xx x x
dtω ω+ − − = ........ 5.2
~áy Ïö
ω02=′ k
m
~ÓÇ ω ckm
2 =
!ë˛Ü˛ ~ܲ•zû˛yˆÏÓ B û˛Ó˚!ê˛Ó˚ çöƒ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚
( )2
2 1 22 'Bd xm k x k x x
dt= − + − ........ 5.3a
129
m !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ñ
2
2 0Bd Xdt
= Ó!§ˆÏÎ˚ Ä §y!çˆÏÎ˚ !öˆÏÎ˚ ˛õy•z
()2
22 2202210 c
dxxxx
dt++−= ωω
........ 5.3b
~áyˆÏöÄ 20ω Ä
2c ω
~ܲ•z xÌ≈ Ó•ö ܲÓ˚ˆÏåÈ–
5.2 ~ÓÇ 5.3b §ü#ܲÓ˚î ò%!ê˛ û˛yˆÏ°y ܲˆÏÓ˚ °«˛ƒ ܲÓ˚&ö– ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ò%!ê˛Ó˚ ˆÜ˛yöê˛y•z §Ó˚° ˆòy°à!ï˛Ó˚
xÓܲ° §ü#ܲÓ˚î öÎ˚– xï˛~Óñ A Ä B û˛Ó˚ ò%!ê˛Ó˚ à!ï˛ˆÏܲ §Ó˚° ˆòy°à!ï˛ Ó°y ÎyÎ˚ öy– ~•z xÓfliyÓ˚ çöƒ
òyÎ˚# ò%•z §ü#ܲÓ˚ˆÏîÓ˚•z ï,˛ï˛#Î˚ ˛õòñ ˆÎ=!° xyüÓ˚y ˆ˛õˆÏÎ˚!åÈ Î%@¿ˆÏöÓ˚ çöƒ– ï˛yåÈyí˛¸y x1 ~ÓÇ x2 í˛zû˛ˆÏÎ˚•z ò%!ê˛
§ü#ܲÓ˚ˆÏî í˛z˛õ!fliï˛ ÌyܲyÎ˚ §ü#ܲÓ˚î ò%!ê˛Ó˚ §üyôyˆÏö !ܲå%Èê˛y x§%!Óôy ˆòáy !òˆÏFåÈ– ï˛y•ˆÏ° ~•z xÓfliyÎ˚ ܲ#û˛yˆÏÓ
xyüÓ˚y §ü#ܲÓ˚î ò%!ê˛Ó˚ §üyôyö ܲÓ˚Ó⁄ ˆÎ !Ó£ÏÎ˚!ê˛ üˆÏö ˆÓ˚ˆÏá xyüyˆÏòÓ˚ ã˛°ˆÏï˛ •ˆÏÓ ï˛y •°ñ xyüÓ˚y x1 Ä
x2 -Ó˚ ÓòˆÏ° ˆÜ˛yˆÏöy §%!Óôyçöܲ Ó˚y!¢ !öˆÏÎ˚ ܲyç ܲÓ˚ˆÏï˛ ˛õy!Ó˚ !ܲöy– ~çöƒ ≤Ã̈Ïü xyüÓ˚y 5.2 ~ÓÇ 5.3b
ˆÜ˛ ˆÎyà ܲˆÏÓ˚ ~ÓÇ ï˛yÓ˚˛õÓ˚ ≤ÃÌü!ê˛ ˆÌˆÏܲ !mï˛#Î˚!ê˛ˆÏܲ !ÓˆÏÎ˚yà ܲˆÏÓ˚ ˛õy•z
()()2
22120120 dxxxx
dt+++= ω
........ 5.4a
~ÓÇ ( ) ( )( )2
2 22 1 2 0 1 22 0c
d x x x xdt
− + + − =ω ω ........ 5.4b
~•z §ü#ܲÓ˚î ò%!ê˛ ˆÓyô •Î˚ xy˛õöyÓ˚ ˛õ!Ó˚!ã˛ï˛ üˆÏö •ˆÏFåÈ– ˆòá%ö ~ÓyÓ˚ xyüÓ˚y xyˆÏÓ˚ܲ!ê˛ í˛z˛õyˆÏÎ˚ ~ˆÏòÓ˚
xyÓ˚Ä ˛õ!Ó˚!ã˛ï˛ ܲˆÏÓ˚ ï%˛°ˆÏï˛ ˛õy!Ó˚– xyüÓ˚y ˛õy!Ó˚ñ
1 2 1x x+ = ξ ........ 5.5a
x1 – x2 =
2 ξ
........ 5.5b
~•z ò%!ê˛ Ó˚!¢
1 ξ
~ÓÇ
2 ξ
ˆÜ˛ Î!ò (5.4a) ~ÓÇ (5.4b) §ü#ܲÓ˚ˆÏî ÎÌye´ˆÏü (x1+x2) ~ÓÇ (x1 –x2)Ó˚
çyÎ˚àyÎ˚ Ó§yˆÏöy ÎyÎ˚ñ ï˛y•ˆÏ° xyüÓ˚y ˛õy•zñ
22 1
2110ddt
+= ξωξ
........ 5.6
~ÓÇñ
222
2 2 2 0ddt
+ =ξ ω ξ ........ 5.7
~áyˆÏöñ2 21 0
km′= =ω ω ........ 5.8
130
~ÓÇñ2 2 22 0
22 ck k
m+′= + =ω ω ω ........ 5.9
°«˛ƒ ܲÓ˚&öñ 1x Ä x2 -~Ó˚ ÓòˆÏ°
1 ξ
~ÓÇ
2 ξ
Ó˚y!¢ ò%!ê˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ (5,6) ˆyÓÇ (5,7) §ü#ܲÓ˚î ò%!ê˛
àë˛ö ܲÓ˚yÓ˚ ú˛ˆÏ° §ü@ˇÃ ï˛ˆÏsfÓ˚ à!ï˛!ê˛ xyüyˆÏòÓ˚ §%˛õ!Ó˚!ã˛ï˛ §Ó˚° ˆòy°à!ï˛Ó˚ ò%!ê˛ §ü#ܲÓ˚ˆÏîÓ˚ üyôƒˆÏü ≤Ãܲy¢
ܲÓ˚y ÎyˆÏFåÈ– ~áyˆÏö §üˆÏÎ˚Ó˚ (t) §ˆÏD ˛õ!Ó˚Óï≈˛ö¢#°
1 ξ
Ä
2 ξ
Ó˚y!¢ ò%!ê˛ ˜Ïï˛!Ó˚ •ˆÏÎ˚ˆÏåÈ x1 ~ÓÇ x
2 ~Ó˚ §üß∫ˆÏÎ˚–
5.6 Ä 5.7 §ü#ܲÓ˚î ˆÎ §Ó˚° ˆòy°à!ï˛=!° §)!ã˛ï˛ ܲÓ˚ˆÏåÈñ ï˛yˆÏòÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ÎÌye´ˆÏü
1 ω
Ä
2 ω
–
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ~•z ò%!ê˛ ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ üˆÏôƒ
21 ωω>
– ˛õÓ˚Óï≈˛# xö%ˆÏFåȈÏò xyüÓ˚y
5.6 Ä 5.7 §ü#ܲÓ˚î ò%!ê˛Ó˚ §üyôyö ܲÓ˚Ó ~ÓÇ
1 ξ
,
2 ξ
~ÓÇ ò%!ê˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛
1 ω
Ä
2 ω
ÈÙÈ~Ó˚ ï˛yͲõÎ≈
!ӈϟ’£Ïî ܲÓ˚Ó–
5.2.2 Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#
xyˆÏàÓ˚ xö%ˆÏFåȈÏò xyüÓ˚y x1 Ä x
2 ~Ó˚ §ÇˆÏÎyˆÏà à!ë˛ï˛ ò%!ê˛ ã˛°Ó˚y!¢
1 ξ
Ä
2 ξ
ˆ˛õˆÏÎ˚!åÈ– ~•z
1 ξ
~ÓÇ
2 ξ
Ó˚y!¢ò%!ê˛ˆÏܲ Ó°y •Î˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ (normal coordinates)– x1 ~ÓÇ x
2 -˛õ!Ó˚ÓˆÏï≈˛ ~•z ò%!ê˛ ã˛°
Ó˚y!¢ ÓƒÓ•yÓ˚ ܲÓ˚yÎ˚ 5.6 ~ÓÇ 5.7 §ü#ܲÓ˚î ò%!ê˛ ày!î!ï˛Ü˛û˛yˆÏÓ §Ó˚°ï˛Ó˚ •ˆÏÎ˚ ˆàˆÏåÈ–
Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ ˆ«˛ˆÏe fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ó°ˆÏï˛ ~üö ܲˆÏÎ˚ܲ!ê˛ !öÓ≈y!ã˛ï˛ ã˛°Ó˚y!¢ˆÏܲ ˆÓyé˛yÎ˚ñ ˆÎ=!°Ó˚
§y•yˆÏ΃ ܲ¡õö¢#° ÓƒÓfliy!ê˛Ó˚ à!ï˛ §ü#ܲÓ˚î=!°ˆÏܲ §ü§ÇáƒÜ˛ ˜Ó˚!áܲ xÓܲ° §ü#ܲÓ˚î !•§yˆÏÓ ˆ°áy
ÎyÎ˚– ˆÜ˛Ó° ï˛y•z öÎ˚ñ ~•z §ü#ܲÓ˚î=!°ˆÏï˛ ≤Ã!ï˛!ê˛ ˛õˆÏòÓ˚ §•à !•§yˆÏÓ ô &Óܲ Ó˚y!¢ ˛õyÄÎ˚y ÎyÎ˚ ~ÓÇ
ã˛°Ó˚y!¢=!°Ä ˆÜ˛Ó°üye §üˆÏÎ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#° •Î˚– ~Ó˚ ú˛ˆÏ° ï˛ˆÏsfÓ˚ §yü!@ˇÃܲ à!ï˛!ê˛ ày!î!ï˛Ü˛û˛yˆÏÓ
!ӈϟ’£Ïî ܲÓ˚y §•ç •ˆÏÎ˚ ÎyÎ˚– 5.6 Ä 5.7 §ü#ܲÓ˚ˆÏî ˆÎ ò%!ê˛ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛
1 ξ
Ä
2 ξ
Ó˚ˆÏÎ˚ˆÏåÈñ ˆ§=!°
xyÓ˚ ~ܲ!ê˛ Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°# (normal mode of vibration) ˆÓyé˛yÎ˚– ~•z
!öˆÏò≈¢yB˛=!° ÎÌye´ˆÏü
1 ω
Ä
2 ω
˛ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ ܲ!¡õï˛ •Î˚–
1
2ωπ
~ÓÇ
2
2ωπ
Ó˚y!¢ ò%!ê˛ˆÏܲ xyüÓ˚y
fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ ܲ¡õyB˛fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ ܲ¡õyB˛fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ ܲ¡õyB˛fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ ܲ¡õyB˛fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ ܲ¡õyB˛ (normal mode frequency) Ó!°–
Î%!@¬ï˛ ï˛ˆÏsfÓ˚ ܲ¡õˆÏö ~ܲ§ˆÏD ~ܲ Óy ~ܲy!ôܲ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°# Óï≈˛üyö ÌyܲˆÏï˛ ˛õyˆÏÓ˚– Î!ò
~ܲ!ê˛üye ܲ¡õö˜Ï¢°# í˛z˛õ!fliï˛ ÌyˆÏܲñ ï˛ˆÏÓ Î%!@¬ï˛ ≤Ã!ï˛!ê˛ ï˛ˆÏsfÓ˚ ܲ¡õö ~ܲ•z ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛ˆÏï˛
•Î˚– ï˛ˆÏÓ ~üö âê˛öy !ӈϢ£Ï xÓfliyˆÏï˛•z âê˛ˆÏï˛ ˛õyˆÏÓ˚– §yôyÓ˚îû˛yˆÏÓ Î%!@¬ï˛ ï˛sf!ê˛ ~ܲy!ôܲ ܲ¡õö˜Ï¢°#ˆÏï˛
ܲ!¡õï˛ •Î˚ñ ú˛ˆÏ° ≤Ã!ï˛ ï˛ˆÏsf ~ܲy!ôܲ ܲ¡õyˆÏB˛Ó˚ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ö âˆÏê˛– ~ÓyÓ˚˛ xyüÓ˚y ≤ÃÌü
~ܲˆÏܲ ˆÎû˛yˆÏÓ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ˆÌˆÏܲ ï˛yÓ˚ §üÎ˚ ÈÙÙÙÈò)Ó˚c §ü#ܲÓ˚î !öî≈Î˚ ܲˆÏÓ˚!åÈñ ˆ§û˛yˆÏÓ
5.6 Ä 5.7 xÓܲ° §ü#ܲÓ˚î=!°Ó˚ §üyôyö !°áˆÏï˛ ˛õy!Ó˚– ˙ ò%!ê˛ §üyôyö ÎÌye´ˆÏü :
()() 1111 cos tAt =+ ξωφ
........ 5.10
131
()() 2222 cos tAt ξωφ =+
........ 5.11
~áy Ïö A1 ~ÓÇ A
2 ò%!ê˛ xyˆÏ®y°ˆÏöÓ˚ !ÓhflÏyÓ˚ ~ÓÇ 1φ Ä
2 φ
ò%!ê˛ ˆ«˛ˆÏe ≤ÃyÌ!üܲ ò¢y §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
5.10 ~ÓÇ 5.11 §ü#ܲÓ˚î myÓ˚y ≤Ãܲy!¢ï˛
() 1t ξ
~ÓÇ ( )2 tξ ~Ó˚ üyö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y ≤ÃyÌ!üܲ
!öˆÏò≈¢yB˛ ( )1x t ~ÓÇ
() 2 xt
§•ˆÏç•z ˆ˛õˆÏï˛ ˛õy!Ó˚ :
()()() 11212
xttt ⎡⎤ =+ ⎣⎦ ξξ
= ( ) ( )1 1 1 2 2 21 cos cos2
A t A t⎡ ⎤+ + +⎣ ⎦ω φ ω φ ........ 5.12
~ÓÇñ
()()() 21212
xttt ⎡⎤ =− ⎣⎦ ξξ
= ( ) ( )1 1 1 2 2 21 cos cos2
A t A t⎡ ⎤+ + +⎣ ⎦ω φ ω φ ........ 5.12
5.12 ~ÓÇ 5.13 ˆï˛ ≤Ãܲy!¢ï˛ §¡õÜ≈˛ ò%!ê˛ ˆÌˆÏܲ x1(t) ~ÓÇ x2(t) !öî≈Î˚ ܲÓ˚ˆÏï˛ ˆàˆÏ° xyüyˆÏòÓ˚ ã˛yÓ˚!ê˛
ô Óܲ Ó˚y!¢ A1, A
2,
φ
~ÓÇ
2 φ
~Ó˚ üyö çyöy òÓ˚ܲyÓ˚– ~çöƒ xyüÓ˚y û˛Ó˚=!°Ó˚ §Ó˚î Ä ˆÓˆÏàÓ˚ ≤ÃyÌ!üܲ
üyö=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ Ìy!ܲ– ~=!°ˆÏܲ ≤ÃyÌ!üܲ ¢ï≈˛ (initial condition) Ó°y •Î˚– ã˛yÓ˚!ê˛ xK˛yï˛ ô Óܲ
Ó˚y!¢Ó˚ üyö !öî≈ˆÏÎ˚Ó˚ çöƒ ã˛yÓ˚!ê˛ ≤ÃyÌ!üܲ ¢ˆÏï≈˛Ó˚ ≤ÈÏÎ˚yçö •Î˚– ~ܲ!ê˛ í˛zòy•Ó˚î ˆÌˆÏܲ !Ó£ÏÎ˚!ê˛ flõ‹T •ˆÏÓ–
ôÓ˚y Îyܲ t = 0 §ü ÏÎ˚
1212 ,,0
dxdxxaxa
dtdt==−==
– xÌ≈yÍñ ≤ÃyÌ!üܲ xÓfliyÎ˚ A Ä B û˛Ó˚ ò%!ê˛
§yüƒyÓfliy ˆÌˆÏܲ ÎÌye´ˆÏü í˛yö!òˆÏܲ Ä ÓÑy!òˆÏܲ
a
ò)Ó˚ˆÏc §ˆÏÓ˚ !fliÓ˚ xÓfliyÎ˚ Ó˚ˆÏÎ˚ˆÏåÈ S!ã˛e 5.3)– 5.12 Ä
5.13 §ü#ܲÓ˚ˆÏî ~•z ¢ï≈˛=!° Ó!§ˆÏÎ˚ xyüÓ˚y ˛õy•z :
() 11122 20coscos2 xAAa =+= φ
........ 5.14a
( )2 1 1 2 22 0 cos cos 2x A A a= − = −φ φ ........ 5.14b
( )11 1 1 2 2 2
0
1 sin sin 02
t
dxA A
dt=
⎛ ⎞ = − + =⎜ ⎟⎝ ⎠ ω φ ω φ ........ 5.14c
( )21 1 1 2 2 2
0
1 sin sin 02
t
dxA A
dt=
⎛ ⎞ = − + =⎜ ⎟⎝ ⎠ ω φ ω φ ........ 5.14d
132
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §üyôyö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎí˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §üyôyö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎí˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §üyôyö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎí˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §üyôyö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎí˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §üyôyö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎ˚ :
1 1 1 1 1cos 0, sin 0,A A= =φ ω φ
22222 cos2,sin0 AaA == φωφ
xÌ≈yÍñ
10, A=
22, Aa =
120 φφ ==
~•z üyö=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆ°áy ÎyÎ˚
() 12 cos xtat ω =
........ 5.15a
( )2 2cosx t a tω= − ........ 5.15b
§•ˆÏç•z ˆÓyé˛y ÎyÎ˚ñ ~ˆÏ«˛ˆÏe ( ) ( )1 2 20, 2 cost t a tξ ξ ω= = –
~áyˆÏö !fl±Ç û˛Ó˚ ï˛ˆÏsfÓ˚ ܲ¡õˆÏö ~ܲ!ê˛üye fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ í˛z˛õ!fliï˛ Ó˚ˆÏÎ˚ˆÏåÈ– ~ÓÇ ˆ§!ê˛ •°
( )2 tξ – ~Ó˚ ú˛ˆÏ° ï˛ˆÏsfÓ˚ ( )1x t Ä
() 2 xt
fliyöyB˛=!° ~ܲ•z ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ •ˆÏFåÈ– ~•z ܲ¡õyB˛!ê˛
•°
2
2ω
π
, Îy
() 2t ξ
~Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛–
~ÓyÓ˚ ~•z ôÓ˚ˆÏöÓ˚ ~ܲ!ê˛ xö%¢#°ö# xy˛õ!ö !öˆÏç•z ܲÓ˚ˆÏï˛ ˆã˛‹Ty ܲÓ˚&ö–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ1 : !öˆÏã˛ ˆòÄÎ˚y ≤ÃyÌ!üܲ ¢ï≈˛=!° !fl±Ç û˛Ó˚ ï˛ˆÏsfÓ˚ 5.12 Ä 5.13 à!ï˛ §ü#ܲÓ˚ˆÏîÓ˚ ô &Óܲ=!°
!öî≈Î˚ ܲÓ˚&ö :
(i) ( )1 0 ,x A= (ii) () 20, xA = (iii) 1
0
0t
dxdt=
⎛⎞= ⎜⎟ ⎝⎠
~ÓÇ (iv) 2
0
0t
dxdt =
⎛ ⎞ =⎜ ⎟⎝ ⎠
~•z xö%¢#°ö#ˆÏï˛ ˆÎ ≤ÃyÌ!üܲ ¢ï≈˛ xÌ≈yÍ t= 0 §üˆÏÎ˚ §Ó˚î Ä à!ï˛ˆÏÓˆÏàÓ˚ ˆÎ üyö ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈñ ï˛y
~ܲ!ê˛ !ӈϢ£Ï í˛z˛õyˆÏÎ˚ §,‹T Î%!@¬ï˛ ܲ¡õö §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– 5.3 !ã˛ˆÏe ˛õ!Ó˚!fli!ï˛!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö í˛zû˛Î˚
û˛Ó˚ˆÏܲ•z ï˛yÓ˚ §yüƒyÓfliyˆÏöÓ˚ §yˆÏ˛õˆÏ«˛ í˛yö !òˆÏܲ §ü˛õ!Ó˚üyî (=A) !Óã%˛ƒï˛ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ò%ôyˆÏÓ˚Ó˚ !fl±Ç ò%!ê˛Ó˚
ô Óܲ ~ܲ ~ÓÇ §ˆÏB˛yã˛ö Óy ≤çyÓ˚ˆÏîÓ˚ üyö ~ܲ •ÄÎ˚yÎ˚ ï˛yÓ˚y Î%!@¬ï˛ û˛Ó˚ ò%!ê˛Ó˚ IJõÓ˚ ~ܲ•z !òˆÏܲ §üyö
Ó° ≤ÈÏÎ˚yà ܲˆÏÓ˚– ~Ó˚ ú˛ˆÏ° Î%_´Ü˛yÓ˚# üyˆÏé˛Ó˚ !fl±Ç!ê˛Ó˚ ˆÜ˛yˆÏöy §ˆÏB˛yã˛ö Óy ≤çyÓ˚î •Î˚ öy ~ÓÇ ˆÎˆÏ•ï%˛
!fl±Ç=!°ˆÏܲ •y°Ü˛y xÌ≈yÍ û˛Ó˚•#ö ôˆÏÓ˚ ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈñ ï˛y•z û˛yÓy ÎyÎ˚ ˆÎñ ~•zû˛yˆÏÓ Ü˛¡õö §¡õy!òï˛
•ÄÎ˚yÓ˚ §üÎ˚ ò%!ê˛ û˛ˆÏÓ˚Ó˚ üyˆÏé˛Ó˚ !fl±Ç!ê˛Ó˚ ˆÎö x!hflÏc•z ˆö•z– ú˛ˆÏ° Î%@¿ˆÏöÓ˚ ˆÜ˛yˆÏöy ≤Ãû˛yÓ•z ~•z ܲ¡õˆÏö
ˆö•z– xy˛õ!ö Ó%é˛ˆÏï˛•z ˛õyÓ˚ˆÏåÈöñ ~!ê˛ !fl±ÇÈÙÈû˛Ó˚ ï˛sf!ê˛Ó˚ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#– ~ˆÏ«˛ˆÏe fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚
ˆÎ ܲ¡õyB˛ ˛õyÄÎ˚y ÎyÎ˚ñ ï˛y ≤Ã!ï˛!ê˛ û˛ˆÏÓ˚Ó˚ !öçfl∫ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ §üyö–
133
5.3 Î%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢öÎ%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢öÎ%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢öÎ%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢öÎ%!@¬ï˛ ܲ¡õˆÏö !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢ö
xy˛õ!ö •Î˚ï˛ °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ 5.2 xö%ˆÏFåȈÏòÓ˚ xyˆÏ°yã˛öyÎ˚ Ä xö%¢#°ö#ˆÏï˛ xyüÓ˚y !fl±ÇÈÙÈû˛Ó˚ ï˛ˆÏsfÓ˚
à!ïÓ˚ ˆÎ ò%!ê˛ í˛zòy•Ó˚î ˆ˛õˆÏÎ˚!åÈñ ï˛yˆÏï˛ !ӈϢ£Ï ò%’ôÓ˚ˆÏöÓ˚ ≤ÃyÌ!üܲ ¢ï≈˛ xyˆÏÓ˚y˛õ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ò%!ê˛ ≤ÃyÌ!üܲ
¢ˆÏï≈˛ û˛Ó˚ ò%!ê˛Ó˚ xÎ%!@¬ï˛ ˆòy°à!ï˛Ó˚ ~ܲ ~ܲ!ê˛ xÓfliy ܲ“öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ï˛ˆÏÓ ~ܲ!ê˛ˆÏï˛ñ xÌ≈yÍ Îáö
( ) ( )1 20 0x x= , ï˛áö ò¢y ˛õyÌ≈ܲ¢)öƒ xyÓ˚ xöƒ!ê˛ˆÏï˛ñ Îáö ( ) ( )1 20 0x x= − , ï˛áö ò¢y ˛õyÌ≈ܲƒ π ôÓ˚y
•ˆÏÎ˚ˆÏåÈ– ~•z !ӈϢ£Ï ≤ÃyÌ!üܲ ¢ï≈˛ !öÓ≈yã˛ˆÏöÓ˚ ú˛ˆÏ° xyüÓ˚y !fl±Ç ÈÙÈû˛Ó˚ ï˛sf!ê˛ˆÏܲ ~ܲ ~ܲ!ê˛ !Ó÷m
ܲ¡õö˜Ï¢°#ˆÏï˛ Ü˛!¡õï˛ •ˆÏï˛ ˆòˆÏá!åÈ– ~ˆÏï˛ xÓ¢ƒ xyüyˆÏòÓ˚ xyˆÏ°yã˛öy !ܲå%Èê˛y §Ó˚° •ˆÏÎ˚ˆÏåÈ–
!ܲv xy˛õ!ö !öÿ˛Î˚•z ≤ß¿ ï%˛°ˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ !fl±Ç ÈÙÈû˛Ó˚ ï˛sf!ê˛ !ܲ ~ܲ•z §ˆÏD ò%•z ܲ¡õö˜Ï¢°#ˆÏï˛ Ü˛!¡õï˛
•ˆÏï˛ ˛õyˆÏÓ˚ öy⁄ à!îˆÏï˛Ó˚ û˛y£ÏyÎ˚ Ó°y ÎyÎ˚,
1 ξ
~ÓÇ
2 ξ
í˛zû˛Î˚•z !ܲ xÈÙÈ¢)öƒ (non zero) •ˆÏï˛ ˛õyˆÏÓ˚ öy⁄
~ܲ ܲÌyÎ˚ ~Ó˚ í˛z_Ó˚ •Ñƒyñ !öÿ˛Î˚•z ï˛y •ˆÏï˛ ˛õyˆÏÓ˚– ï˛ˆÏÓ xy˛õ!ö •Î˚ï˛ ~ܲ!ê˛ §¡õ)î≈ í˛zòy•Ó˚ˆÏîÓ˚ §y•yˆÏ΃
~•z xÓfliy!ê˛ Ó%é˛ˆÏï˛ ã˛y•zˆÏåÈö– ï˛y•z ~áyˆÏö xyüÓ˚y ~ܲ!ê˛ ˛õ,Ìܲ ≤ÃyÌ!üܲ ¢ï≈˛ ôˆÏÓ˚ !öˆÏÎ˚ 5.12 Ä 5.13
§ü#ܲÓ˚ˆÏîÓ˚ ô &Óܲ=!° !öî≈Î˚ ܲÓ˚Ó–
~•z ≤ÃyÌ!üܲ ¢ï≈˛à%!° •°ñ t = 0 §üˆÏÎ˚
()()112
0
02,00,0t
dxxax
dt=
⎛⎞ === ⎜⎟ ⎝⎠
~ÓÇ =
⎛ ⎞ =⎜ ⎟⎝ ⎠2
0
0t
dxdt
xÌ≈yÍñ ≤ÃÌü û˛Ó˚!ê˛ í˛yö !òˆÏܲ 2a ò)Ó˚ˆÏc ~ÓÇ !mï˛#Î˚ û˛Ó˚!ê˛ §yüƒyÓfliyˆÏö !fliÓ˚ xÓfliyÎ˚ xyˆÏåÈ S!ã˛e 5.4)
~áö xyüÓ˚y 5.14 (a –d) §ü#ܲÓ˚î=!°Ó˚ xö%Ó˚*˛õ ã˛yÓ˚!ê˛ §ü#ܲÓ˚î !°áˆÏï˛ ˛õy!Ó˚ :
( ) ( )1 1 1 2 210 cos cos 22
x A A a= + =φ φ
( ) ( )2 1 1 2 210 cos cos 02
x A A= − =φ φ
( )11 1 1 2 2 2
0
1 sin sin 02
t
dxA A
dt=
⎛ ⎞ = − + =⎜ ⎟⎝ ⎠ ω φ ω φ
( )21 1 1 2 2 2
0
1 sin sin 02
t
dxA A
dt=
⎛ ⎞ = − + =⎜ ⎟⎝ ⎠ ω φ ω φ
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î=!° §y•yˆÏ΃ ~áö xy˛õ!ö !öˆÏç•z ˆòáyˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ
( ) ( )1 1 2cos cosx t a t t= +ω ω ........ 5.16a
134
()() 212 coscos xtatt =− ωω
........ 5.16b
!eˆÏܲyî!ü!ï˛Ó˚ §)e ≤ÈÏÎ˚yà ܲˆÏÓ˚ x1 Ä x2 -~Ó˚ Ó˚y!˘¢üy°yˆÏܲ ˆ°áy ÎyÎ˚
()211222coscos 22 xtatt
−+=
ωωωω
........ 5.17a
Ä
()2122sin2 xtat
− =ωω
12 sin2+ ωω
t ........ 5.17b
xy˛õöyÓ˚ •Î˚ï˛ üˆÏö xyˆÏåÈ ˆÎñ
21
' km
ω=
~ÓÇ
22
'2 kkm
ω+ =
– ~áyˆÏö 2 1ω ω> xÌ≈yÍñ
21 − ωω
Ó˚y!¢!ê˛
ôöydܲ– xyüÓ˚y
21
2+ ωω
, xÌ≈yÍ ò%•z ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ àí˛¸ˆÏܲ
ω
~ÓÇ
21 − ωω
xÌ≈yÍñ ò%•z ˆÜ˛Ô!îܲ
ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒˆÏܲ
ωΔ
!•§yˆÏÓ !°áÓ– û˛Ó˚ ò%•z!ê˛Ó˚ üˆÏôƒ Î%@¬ö Î!ò xy°ày •Î˚ñ ï˛ˆÏÓ
' kk <<
~ÓÇ
ωΔ
Ó˚y!¢Ó˚ üyö
ω
~Ó˚ ï%˛°öyÎ˚ ˆåÈyê˛ •ˆÏÓ– ~•z xÓfliyÎ˚
2ωΔ
t ~Ó˚ §y•zö Óy ˆÜ˛y§y•zö
t ω
~Ó˚ §y•zö
Óy ˆÜ˛y§y•zˆÏöÓ˚ ï%˛°öyÎ˚ ô#Ó˚ à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛ •ˆÏÓ– xyüÓ˚y ôˆÏÓ˚ !öˆÏï˛ ˛õyÓ˚Ó ˆÎñ
() 1 xt
Ä
() 2 xt
~Ó˚
!ÓhflÏyÓ˚ •ˆÏÓ ÎÌye´ˆÏü
2cos2 m Aat ωΔ =
~ÓÇ 2 sin2mB a tωΔ= – xyüÓ˚y ~ÓyÓ˚ ( )1x t Ä
() 2 xt
ˆÜ˛
!°áˆÏï˛ ˛õy!Ó˚ : () 12cos.coscos2Δ ==m xtattAt ωωω
........ 5.18a
Ä
() 22sin.sinsin2Δ ==m xtattBt ωωω
........ 5.18b
5.18a Ä b §)ˆÏe xyüÓ˚y
1 x
Ä
2 x
~Ó˚ ˆÎ Ó˚y!¢üy°y ò%!ê˛ ˆ˛õ°yüñ ï˛yÓ˚ ˆÌˆÏܲ !fl±Ç ÈÙÈû˛Ó˚ ï˛ˆÏsfÓ˚
xyˆÏ®y°ˆÏöÓ˚ ã˛!Ó˚e!ê˛ ˆÓyé˛y ˆÎˆÏï˛ ˛õyˆÏÓ˚– 5.5 a Ä b !ã˛ˆÏe §üˆÏÎ˚Ó˚ §ˆÏD
1 x
Ä x2 ~Ó˚ ˛õ!Ó˚Óï≈˛ö ˆ°á!ã˛ˆÏeÓ˚
§y•yˆÏ΃ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
0 t=
§üˆÏÎ˚
2,0, mm AaB ==
§%ï˛Ó˚yÇ
12 2,0 xax ==
§%ï˛Ó˚yÇñ ~•z §üˆÏÎ˚
1 x
Ó˚y!¢!ê˛ 2a !ÓhflÏyˆÏÓ˚ Ä
ω
ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ ܲ!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ– x2 Ó˚y!¢Ó˚
!ÓhflÏyÓ˚ ~•z ¢)öƒ– §%ï˛Ó˚yÇñ A û˛Ó˚!ê˛ 2a !ÓhflÏyˆÏÓ˚ Ä
ω
ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ ܲ!¡õï˛ •ˆÏ°Äñ B û˛Ó˚!ê˛ ~•z ü%•)ˆÏï≈˛
!fliÓ˚ ÌyܲˆÏÓ–
!ܲv §üˆÏÎ˚Ó˚ §ˆÏD
m A
!ÓhflÏyÓ˚!ê˛ Ü˛üˆÏï˛ ÌyܲˆÏÓ ~ÓÇ Îáö
22t ωπ Δ=
xÌ≈yÍñ
, tπω =Δ
ï˛áö
0 m A=
135
•ˆÏÓ– x˛õÓ˚!òˆÏܲ, Bm ~Ó˚ üyö e´ü¢ Ó,!k˛ ˆ˛õˆÏÎ˚ t = πωΔ
§üˆÏÎ˚ 2a •ˆÏÓ– §%ï˛Ó˚yÇñ ~•z ü)‡ˆÏï≈˛ A û˛Ó˚!ê˛
!fliÓ˚ xÓfliyÎ˚ xy§ˆÏÓ ~ÓÇ B û˛Ó˚!ê˛ §ˆÏÓù≈yFã˛ 2a !ÓhflÏyˆÏÓ˚
ω
ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ ܲ!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ–
xy˛õ!ö !öÿ˛Î˚ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ~Ó˚˛õÓ˚ Îáö
ωπ Δ=2
t
•ˆÏÓñ xÌ≈yÍ πω
=Δ
t §üˆÏÎ˚ xyÓyÓ˚
= −2mA a ~ÓÇ
=0 m B
•ˆÏÓ– xӈϢˆÏ£Ïñ πω
=Δ4
t §üˆÏÎ˚ñ = 2mA a ~ÓÇ
=0 m B
•ˆÏÓ xÌ≈yÍñ ï˛sf!ê˛
≤ÃyÌ!üܲ xÓfliyÎ˚ !ú˛ˆÏÓ˚ xy§ˆÏÓ– mA Ä
m B
!ÓhflÏyˆÏÓ˚Ó˚ ~•z •…y§ÈÙÈÓ,!k˛ˆÏܲ xyüÓ˚y ü!í˛í˛zˆÏ°¢ö (modulation)
Ó!°– xyüÓ˚y ˆòá°yüñ ü!í˛í˛zˆÏ°¢ˆÏöÓ˚ ˛õÎ≈yÎ˚ܲy°
πω
=Δ4
m T
Óy ü!í˛í˛zˆÏ°¢ˆÏöÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛
π ωω Δ= =22m
mT – x1 Ä x2 ~Ó˚ ü!íí˛zˆÏ°¢ö ܲÓ˚y !ÓhflÏyÓ˚=!°ˆÏܲ ˆ°áy ÎyÎ˚ñ
ω= 2 cosm mA a t
Ä
ω =2sin mm Bat
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲÓ˚ˆÏåÈö ˆÎñ !ÓhflÏyÓ˚ ò%!ê˛Ó˚ ü!í˛í˛zˆÏ°¢ˆÏö
π2
ò¢y ˛õyÌ≈ܲƒ xyˆÏåÈñ ˆÜ˛ööy
π ωω⎛⎞ =+ ⎜⎟ ⎝⎠cossin
2 mm tt~ÓyÓ˚ A Ä B û˛ˆÏÓ˚Ó˚ ˆòy°à!ï˛Ó˚ xyÓ˚ ~ܲ!ê˛ ˜Ó!¢ˆÏ‹TƒÓ˚ !òˆÏܲ ò,!‹T ˆú˛Ó˚yˆÏöy Îyܲ– 5.16 a Ä b §ü#ܲÓ˚î
ˆÌˆÏܲ ~!ê˛ flõ‹T ˆÎñ û˛Ó˚ ò%!ê˛Ó˚ à!ï˛ ò%!ê˛ ˛õ,Ìܲ ܲ¡õyˆÏB˛Ó˚ ~ܲˆÏÓ˚á#Î˚ §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ°
í˛zͲõߨ •ˆÏÎ˚ˆÏåÈ– ~•z ˛õÎ≈yˆÏÎ˚Ó˚ !mï˛#Î˚ ~ܲˆÏܲ xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ~ çyï˛#Î˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ° !Ó ˆÏê˛Ó˚ í˛zqÓ
•Î˚ xyüÓ˚y A Ä B û˛ˆÏÓ˚Ó˚ ˆòy°ˆÏöÓ˚ !ÓhflÏyˆÏÓ˚ ˆÎ Äë˛yöyüy °«˛ƒ ܲÓ˚°yüñ ï˛y ~•z !Óê˛ åÈyí˛¸y xyÓ˚ !ܲå%È öÎ˚–
5.4 ò%!ê˛ Î%!@¬ï˛ §Ó˚° òy°ˆÏܲÓ˚ «˛ˆÏe fl∫yò%!ê˛ Î%!@¬ï˛ §Ó˚° òy°ˆÏܲÓ˚ «˛ˆÏe fl∫yò%!ê˛ Î%!@¬ï˛ §Ó˚° òy°ˆÏܲÓ˚ «˛ˆÏe fl∫yò%!ê˛ Î%!@¬ï˛ §Ó˚° òy°ˆÏܲÓ˚ «˛ˆÏe fl∫yò%!ê˛ Î%!@¬ï˛ §Ó˚° òy°ˆÏܲÓ˚ «˛ˆÏe fl∫yûûûûû y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî õÎ≈yˆÏ°yã˛öyy!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî õÎ≈yˆÏ°yã˛öyy!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî õÎ≈yˆÏ°yã˛öyy!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî õÎ≈yˆÏ°yã˛öyy!Óܲ ܲ¡õö˜Ï¢°#Ó˚ !ӈϟ’£Ïî õÎ≈yˆÏ°yã˛öy
~ÓyÓ˚ xyüÓ˚y xyüyˆÏòÓ˚ á%Ó ˛õ!Ó˚!ã˛ï˛ ~ܲ!ê˛ ˆ«˛ˆÏe Î%@¬ˆÏöÓ˚ û)˛!üܲy ˛õÎ≈yˆÏ°yã˛öy ܲÓ˚Ó– §Ó˚° ˆòy°Ü˛ ~ÓÇ
ï˛yÓ˚ ˆòy°ö §Çe´yhsˇ !Ó£ÏÎ˚=!° !öˆÏÎ˚ öï%˛ö ܲˆÏÓ˚ !öÿ˛Î˚•z xyÓ˚ !ܲå%È Ó°yÓ˚ ˆö•z– ôˆÏÓ˚ !ööñ ò%!ê˛ §üyö ˜ÏòˆÏâ≈ƒÓ˚
§Ó˚° ˆòy°Ü˛ˆÏܲ ˛õy¢y˛õy!¢ é%˛!°ˆÏÎ˚ !˛õ[˛ (bob) ò%!ê˛ˆÏܲ 5.4 !ã˛ˆÏe ˆÎüö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ ˆ§•zû˛yˆÏÓ ~ܲ!ê˛
•y°Ü˛y !fl±Ç ~Ó˚ §y•y΃ Î%!@¬ï˛ ܲÓ˚y •°– ôˆÏÓ˚ !ööñ !˛õ[˛ ò%!ê˛Ó˚ üˆÏôƒ ò)Ó˚c ~üö ˆÎñ Îáö ï˛yÓ˚y §yüƒ xÓfliyÎ˚
xyˆÏåÈ ï˛áö ï˛yˆÏòÓ˚ §ÇˆÏÎyàܲyÓ˚# !fl±Ç!ê˛ ï˛yÓ˚ fl∫yû˛y!Óܲ ˜òˆÏâ≈ƒ Ó˚ˆÏÎ˚ˆÏåÈ xÌ≈yÍñ §B%˛!ã˛ï˛ Óy ≤çy!Ó˚ï˛ •Î˚!ö–
~ÓyÓ˚ !˛õ[˛ ò%!ê˛ˆÏܲ ܲyàˆÏçÓ˚ ï˛ˆÏ° ˆòy°yˆÏöyÓ˚ ˆã˛‹Ty ܲÓ˚y •ˆÏ° xyüÓ˚y ܲ# ˆòáˆÏï˛ ˛õyÓñ ˆ§!ê˛•z xyüyˆÏòÓ˚
xyˆÏ°yã˛ƒ !Ó£ÏÎ˚–
136
5.5 (a) !ã˛ˆÏe ˆÎû˛yˆÏÓ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ˆ§•zû˛yˆÏÓ ˆòy°Ü˛ò%!ê˛ˆÏܲ ï˛yˆÏòÓ˚
§yüƒÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ !fliÓ˚ xÓfliy ˆÌˆÏܲ ˆåȈÏí˛¸ !òˆÏ° ï˛yˆÏòÓ˚ ˆòy°à!ï˛
˘÷Ó˚& •ˆÏÓ– !ã˛e 5.5 (b) ~ÓÇ (c) ˆï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ ˆÎñ ~•z ˆòy°ö ÷Ó˚&Ó˚
ܲyçê˛y !ܲå%Èê˛y !û˛ß¨û˛yˆÏÓÄ Ü˛Ó˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– 5.5 (a) ˆï˛ ˆòáy ÎyˆÏFåÈ
ˆÎñ ˆòy°Ü˛ ò%!ê˛ §ü ò¢yÎ˚ (in same phase) ï˛yˆÏòÓ˚ ˆòy°ö ÷Ó˚& ܲˆÏÓ˚ˆÏåÈñ xyÓ˚
(b) ~ÓÇ (c) -ˆï˛ ï˛yˆÏòÓ˚ ˆòy°ö ÷Ó˚& •ˆÏFåÈ !Ó˛õÓ˚#ï˛ ò¢y (opposite phase)
ˆÌˆÏܲ–
~•z ˛õÎy≈ˆÏÎ˚Ó˚ ≤ÃÌü ~ܲˆÏܲ xyüÓ˚y ˆòˆÏá!åÈ ˆÎ §Ó˚° ˆòy°ˆÏܲÓ˚ ˆòy°à!ï˛Ó˚ çöƒ ˆÎ xÓܲ° §ü#ܲÓ˚î
ˆ°áy ÎyÎ˚ ï˛y •° : =2
2
d x mgm x
dt l ˆÎáyˆÏöñ m = !˛õˆÏ[˛Ó˚ û˛Ó˚ñ g = x!û˛Ü˛£Ï≈ç cÓ˚î , l = §Ó˚° ˆòy°ˆÏܲÓ˚
ܲyÎ≈ܲÓ˚# ˜òâ≈ƒ– §ü#ܲÓ˚ˆÏîÓ˚ í˛yö!òˆÏܲÓ˚ Ó˚y!¢!ê˛ !˛õˆÏ[˛Ó˚ í˛z˛õÓ˚ ≤ÃÎ%_´ ≤Ãï˛ƒyöÎ˚ܲ Ó°– !˛õ[˛ ò%•z!ê˛ Î!ò §yüƒyÓfliyö
ˆÌˆÏܲ x1 Ä x2 ˛õ!Ó˚üyî !Óã%˛ƒï˛ •Î˚ñ ï˛ˆÏÓ !fl±Ç ~Ó˚ ≤çyÓ˚î •ˆÏÓ x2 -x1– ~áö !fl±Ç ô Óܲ Î!ò k •Î˚ñ
ï˛ˆÏÓ !fl±Ç ~Ó˚ ≤çyÓ˚ˆÏîÓ˚ çöƒ Óyü Ä í˛yö!òˆÏܲÓ˚ A Ä B !˛õˆÏ[˛Ó˚ í˛z˛õÓ˚ ÎÌye´ˆÏü
() −21 kxx
~ÓÇ
– k(x2 – x
1) Ó° ܲyç ܲÓ˚ˆÏÓ– ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ §ˆÏD ~•z ò%!ê˛ Ó° ˆÎyà ܲˆÏÓ˚ A Ä B !˛õˆÏ[˛Ó˚ à!ï˛
§ü#ܲÓ˚î ˆ°áy ÎyÎ˚ !˛õ[˛
() ⎛⎞ =−+− ⎜⎟ ⎝⎠
21
121 2 :dxmg
Amxkxxdtl
....... 5.19a
!ã˛e 5.4
!ã˛e 5.5
137
~ÓÇ, ( )⎛ ⎞= − − −⎜ ⎟⎝ ⎠
22
2 2 12
d x mgm x k x x
dt l....... 5.19b
°«˛ƒ ܲÓ˚&öñ ~•z ò%!ê˛ §ü#ܲÓ˚ˆÏîÓ˚ ˆÜ˛yˆÏöy!ê˛•z !ܲv §Ó˚° ˆòy°à!ï˛ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ öy– §ü#ܲÓ˚î ò%!ê˛Ó˚
í˛zû˛Î˚˛õ«˛ˆÏܲ ‘m’ !òˆÏÎ˚ û˛yà ܲˆÏÓ˚ ~ÓÇ ˛õò=!°ˆÏܲ §y!çˆÏÎ˚ !öˆÏÎ˚ ˛õyÄÎ˚y ÎyÎ˚ :
( )ω ω+ + − =2
2 210 1 1 22 0x
d xx x x
dt....... 5.20a
( )2
2 220 2 1 22 0+ − − =c
d xx x x
dtω ω ....... 5.20b
~áyˆÏöñ ω =20
g
l ~ÓÇ.
ω=2c
k
m
5.20a ~ÓÇ 5.20b §ü#ܲÓ˚î ò%!ê˛ ÎÌye´ˆÏü 5.2b ~ÓÇ 5.3b -~Ó˚ xö%Ó˚*˛õ– xï˛~Óñ ~•z ˛õÎ≈yÎ˚ ˆÌˆÏܲ
xyüÓ˚y 5.2 xö%ˆÏFåȈÏòÓ˚ !ӈϟ’£Ïî ˛õk˛!ï˛ xö%§Ó˚î ܲÓ˚ˆÏï˛ ˛õy!Ó˚– ~•z ܲyçê˛y ~ܲê˛y xö%¢#°ö#Ó˚ üyôƒˆÏü ܲˆÏÓ˚
ˆòáˆÏ° ˆÜ˛üö •Î˚⁄
xö%¢#°öy xö%¢#°öy xö%¢#°öy xö%¢#°öy xö%¢#°öy -2 : 5:20a ~ÓÇ 5.20b §ü#ܲÓ˚î ò%!ê˛ ˆÌˆÏܲ ÷Ó˚& ܲˆÏÓ˚ ò%!ê˛ Î%!@¬ï˛ §Ó˚° ˆòy°ˆÏܲÓ˚ fl∫yû˛y!Óܲ
!öˆÏò≈¢yB˛ Ä fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚&ö–
í˛z˛õˆÏÓ˚Ó˚ xö%¢#°ö#!ê˛Ó˚ §üyôyö ܲÓ˚yÓ˚ §ˆÏD §ˆÏD xy˛õ!ö ~•z Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ ˆ«˛ˆÏe ò%!ê˛ xö%Ó˚*˛õ ˆòy°Ü˛
!öˆÏÎ˚ à!ë˛ï˛ ~•z ï˛sf!ê˛Ó˚ à!ï˛¢!_´ K ~ÓÇ !fli!ï˛¢!_´ V-~Ó˚ àîöy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– ˆÎˆÏ•ï%˛ û˛Ó˚ò%!ê˛Ó˚ üˆÏôƒ
ÓƒÓ•*ï˛ Î%@¬ö !fl±Ç!ê˛ û˛Ó˚•#öñ ï˛y•z à!ï˛¢!_´ ˆÜ˛Ó°üye !˛õ[˛ ò%!ê˛Ó˚ ˆ«˛ˆÏe•z ˛õyÄÎ˚y ÎyˆÏÓ– xyÓ˚ !fli!ï˛¢!_´
ˆÜ˛Ó° !˛õ[˛ ò%!ê˛Ó˚ çöƒ öÎ˚ñ Î%@¬ö !fl±ÇˆÏÎ˚Ó˚ çöƒÄ ˛õyÄÎ˚y ÎyˆÏÓ– ï˛y•z ˆ°áy ÎyÎ˚ ˆÎñ
à!ï˛¢!_´
2212
12
⎡⎤ =+ ⎣⎦ kmxx
....... 5.21
~áyˆÏöñ
11=dxx
dt
~ÓÇ
22=dxx
dt
!fli!ï˛¢!_´Ó˚ ˆ«˛ˆÏe xyüÓ˚y çy!ö ˆÎñ ˆÜ˛yˆÏöy §Ó˚° ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ û˛Ó˚ m ~ÓÇ §yüƒyÓfliyö ˆÌˆÏܲ
ï˛yÓ˚ !Óã%˛ƒ!ï˛ x •ˆÏ°ñ !˛õ[˛!ê˛Ó˚ !fli!ï˛¢!_´Ó˚ üyö
220
12
mx ω
ˆÎáyˆÏö, 20 = g
lω – xyÓyÓ˚ xyüÓ˚y çy!öñ !fl±Ç
138
~Ó˚ !fli!ï˛¢!_´Ó˚ üyö ( ×12
k §ˆÏB˛yã˛ö Óy ≤çyÓ˚ˆÏîÓ˚ Óà≈V– ~ˆÏ«˛ˆÏe ˆüyê˛ !fli!ï˛¢!_´ •ˆÏFåÈñ !˛õ[˛ ò%!ê˛Ó˚
!fli!ï˛¢!_´ ~ÓÇ !fl±Ç ~Ó˚ !fli!ï˛¢!_´Ó˚ §ü!‹TÓ˚˛ §üyö– xï˛~Ó ˆòáy ÎyÎ˚ñ
!fli!ï˛¢!_´
() ⎛⎞⎡⎤ =++− ⎜⎟⎣⎦ ⎝⎠2 22
1212
1122
mgVxxkxx
l
....... 5.22
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲÓ˚ˆÏåÈö ˆÎñ 5.21 Ä 5.22 §ü#ܲÓ˚ˆÏî xyüÓ˚y à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ ˆÎ Ó˚y!¢üy°y
ˆ˛õ°yüñ ï˛y fl∫yû˛y!Óܲ !öˆÏò≈¢yˆÏB˛Ó˚ §y•yˆÏ΃ ˆ°áy •Î˚!ö– ˛õˆÏÓ˚Ó˚ xö%¢#°ö#ˆÏï˛ xy˛õ!ö !öˆÏç•z ~•z ܲyç!ê˛
ܲˆÏÓ˚ ˆòá%ö–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ 3 ≠ ≠ ≠ ≠ ≠ 5.21 Ä 5.22 §)e ò%!ê˛ˆÏï˛ à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ ˆÎ Ó˚y!¢üy°y ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈñ
ˆ§=!°ˆÏܲ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ ξ = +1 1 2x x ~ÓÇ
ξ=− 212 xx
~Ó˚ §y•yˆÏ΃ ≤Ãܲy¢ ܲÓ˚&ö– ˆòáyö ˆÎñ
ξ1
Ä
ξ2
-˛~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛
1.2m = ηξ
Ä
22 .2m = ηξ
ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° à!ï˛¢!_´ Ä !fli!ï˛¢!_´ˆÏܲ xyÓ˚Ä §Ó˚°û˛yˆÏÓ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚–
~•z xö%ˆÏFåÈò!ê˛ ˆ¢£Ï ܲÓ˚yÓ˚ xyˆÏà fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#Ó˚ xyÓ˚ ~ܲ!ê˛ ôˆÏü≈Ó˚ ܲÌy í˛zˆÏÕ‘á ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
xö%¢#°ö# 2 ~Ó˚ í˛z_Ó˚ ܲÓ˚yÓ˚ §üÎ˚ xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ Î%!@¬ï˛ §Ó˚° ˆòy°ˆÏܲÓ˚ 5.20a Ä b §ü#ܲÓ˚î ò%!ê˛ˆÏܲ
fl∫yû˛y!Óܲ !öˆÏò≈¢yˆÏB˛Ó˚ !•§yˆÏÓ ˆ°áy ÎyÎ˚
22 101 20
d
dt+=
ξωξ
....... 5.23a
Äñ ( )2
2 220 22 0c
d
dt+ + =
ξ ω ω ξ ....... 5.23b
ˆÎáyˆÏö, ξ ξ= + = −1 1 2 2 1 2,x x x x – í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î 5.23a Ä b ˆÌˆÏܲ flõ‹T•z ˆÓyé˛y ÎyÎ˚ ˆÎñ
ξ1
Ä
ξ2
ÎÌye´ˆÏü
ω0
~ÓÇ
() ωωω +=1
22202 2c
ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •Î˚– ~=!°Ó˚
§üyôyö •ˆÏÓ :
( )1 2 1 1 0 1cosx x A t+ = = +ξ ω α
~ÓÇñ
() 122222 cos xxAt −==+ ξωα
~áö Î!ò ˆÜ˛Ó°üye
ξ1
ܲ¡õö˜Ï¢°# í˛z˛õ!fliï˛ ÌyˆÏܲñ xÌ≈yÍ
ξ=20
•Î˚ñ ï˛ˆÏÓ
=12 xx
•ˆÏÓ ~ÓÇ
1 x
Ä
2 x
~Ó˚ üyö ˛õyÄÎ˚y ÎyˆÏÓ
139
() ωα ==+ 12101
1cos
2xxAt
~Ó˚ xÌ≈ x1 Ä x2 §ü ò¢yÎ˚ ܲ!¡õï˛ •ˆÏÓ ~ÓÇ í˛zû˛ˆÏÎ˚Ó˚•z ˆÜ˛Ô!îܲ ܲ¡õyB˛ •ˆÏÓ ω0 –
xyÓyÓ˚ñ Î!ò ˆÜ˛Ó°üye
ξ2
ܲ¡õö˜Ï¢°#!ê˛ ÌyˆÏܲ ~ÓÇ,
ξ=10
•Î˚ñ ï˛ˆÏÓ
=− 12 xx
•ˆÏÓ ~ÓÇ
1 x
Ä
x2 ~Ó˚ üyö •ˆÏÓ
() ωα =+ 1222
1cos
2xAt
( ) ( )ω α ω α π= − + = + +2 2 2 2 2 2 2
1 1cos cos
2 2x A t A t
xÌ≈yÍñ 1x Ä
2 x
!Ó˛õÓ˚#ï˛ ò¢yÎ˚ Óy
π
ò¢y ˛õyÌ≈ˆÏܲƒ ܲ!¡õï˛ •ˆÏÓ ~ÓÇ í˛zû˛ˆÏÎ˚Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛
ω2
•ˆÏÓ– í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öy ˆÌˆÏܲ xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ~ܲ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛
ï˛ˆÏsfÓ˚ ~ܲ ~ܲ!ê˛ xLjϢÓ˚ ܲ¡õö ~ܲ•z ܲ¡õyˆÏB˛ •ˆÏ°Ä ï˛yˆÏòÓ˚ üˆÏôƒ ~ܲ ~ܲ ò¢y ˛õyÌ≈ܲƒ ÌyܲˆÏï˛ ˛õyˆÏÓ˚–
5.5 fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛
xyüÓ˚y ~áö ˛õÎ≈hsˇ Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ xyˆÏ°yã˛öyÎ˚ ôˆÏÓ˚ !öˆÏÎ˚!åÈ ˆÎñ Î%!@¬ï˛ û˛Ó˚ ò%!ê˛ ˛õÓ˚flõÓ˚ §üyö–
≤Ã!ï˛§yˆÏüƒÓ˚ ú˛ˆÏ° Î!òÄ §ü§ƒy!ê˛ xˆÏöܲê˛y §Ó˚° •ˆÏÎ˚ˆÏåÈñ ï˛Ó% fl∫yû˛y!Óܲû˛yˆÏÓ•z ~•z ≤ß¿ ï%˛°ˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ
ò%!ê˛ û˛Ó˚ §üyö öy •ˆÏ° xyüÓ˚y ܲ#û˛yˆÏÓ §ü§ƒy!ê˛Ó˚ §üyôyö ܲÓ˚Ó⁄ Ó›ï˛ ò%!ê˛ û˛Ó˚ §üyö ôˆÏÓ˚ !öˆÏ° Ó°y ã˛ˆÏ°
öy ˆÎñ xyüÓ˚y ˆÜ˛yˆÏöy §yôyÓ˚î ˛õk˛!ï˛ ÓƒÓ•yÓ˚ ܲÓ˚!åÈ ÓÓ˚Ç ï˛y ~ܲ!ê˛ !ӈϢ£Ï ˆ«˛ˆÏe•z (special case) ≤ÈÏÎyçƒ
•ˆÏFåÈ– ï˛y•z Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ~ÓÇ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚
xö%˛õyï˛ !öî≈ˆÏÎ˚Ó˚ §yôyÓ˚î ˛õk˛!ï˛Ó˚ ≤Ãò¢≈ö ܲÓ˚ˆÏï˛ xyüÓ˚y ôˆÏÓ˚ ˆöÓ ˆÎñ xyˆÏàÓ˚ xö%ˆÏFåȈÏò xyˆÏ°y!ã˛ï˛ Î%!@¬ï˛
§Ó˚° ˆòy°Ü˛ ò%!ê˛Ó˚ ˆ«˛ˆÏe l ~ܲ •ˆÏ°Ä !˛õ[˛ ò%!ê˛Ó˚ û˛Ó˚ x§üyö– ôÓ˚y Îyܲñ A Ä B !˛õˆÏ[˛Ó˚ û˛Ó˚ ÎÌye´ˆÏü
a m
Ä
b m
–
fl∫yû˛y!Óܲ ܲ¡õyB˛ Ä û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyï˛ !öî≈ˆÏÎ˚Ó˚ çöƒ ~ÓyÓ˚ xyüÓ˚y ܲˆÏÎ˚ܲ!ê˛ !ö!ò≈‹T ôyˆÏ˛õ x@ˇÃ§Ó˚
•Ó– ~=!° xyüÓ˚y §Çáƒy !òˆÏÎ˚ !ã˛!•´ï˛ ܲˆÏÓ˚ !°á!åÈñ ÎyˆÏï˛ xy˛õ!ö §•ˆÏç ôy˛õ=!° xö%§Ó˚î ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö–
1. ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y–ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y–ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y–ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y–ò%!ê˛ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ !˛õˆÏ[˛Ó˚ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y–
!˛õˆÏ[˛Ó˚ û˛Ó˚ ma Ä mb ~ÓÇ !fl±Ç ô Óܲ = k ôˆÏÓ˚ !öˆÏÎ˚
A !˛õˆÏ[˛Ó˚ çöƒ :
() 1121 =−+− aa
gmxmxkxx
l
B !˛õˆÏ[˛Ó˚ çöƒ : ( )2 2 2 1
8= − − −b bm x m x k x xl
140
~áöñ ω ω= =2 20 , h
a
g k
l m ~ÓÇ ω= 2b
h
k
m !°ˆÏá ˛õyÄÎ˚y ÎyÎ˚–
( )2 21 0 1 2 1= − + −ax x x xω ω ....... 5.24a
~ÓÇ ( )x x x xb2 02
22
2 1− − −ω ω ....... 5.24b
~•z ò%!ê˛•z •° xyüyˆÏòÓ˚ í˛z!j‹T ò%!ê˛ xÓܲ° §ü#ܲÓ˚î–
2. ܲyöÄ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ §Ó=!° !öˆÏò≈¢yB˛•z ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ܲ!¡õï˛ •Î˚– ~•z ï˛ˆÏ̃Ó˚ˆÜ˛yöÄ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ §Ó=!° !öˆÏò≈¢yB˛•z ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ܲ!¡õï˛ •Î˚– ~•z ï˛ˆÏ̃Ó˚ˆÜ˛yöÄ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ §Ó=!° !öˆÏò≈¢yB˛•z ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ܲ!¡õï˛ •Î˚– ~•z ï˛ˆÏ̃Ó˚ˆÜ˛yöÄ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ §Ó=!° !öˆÏò≈¢yB˛•z ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ܲ!¡õï˛ •Î˚– ~•z ï˛ˆÏ̃Ó˚ˆÜ˛yöÄ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛ §Ó=!° !öˆÏò≈¢yB˛•z ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ܲ!¡õï˛ •Î˚– ~•z ï˛ˆÏ̃Ó˚
≤ÈÏÎ˚yà ܲˆÏÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öî≈Î˚–≤ÈÏÎ˚yà ܲˆÏÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öî≈Î˚–≤ÈÏÎ˚yà ܲˆÏÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öî≈Î˚–≤ÈÏÎ˚yà ܲˆÏÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öî≈Î˚–≤ÈÏÎ˚yà ܲˆÏÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öî≈Î˚–
1x Ä
2 x
!öˆÏò≈¢yB˛ ò%!ê˛ Î!ò ~ܲ•z ˆÜ˛Ô!îܲ ܲ¡õyB˛ ‘w’ ˆï˛ ܲ!¡õï˛ •Î˚ñ ï˛ˆÏÓ ~=!°Ó˚ Ó˚*˛õ •ˆÏÓ
() ωφ =+ 111 cos xAt
Ä ( )ω φ= +2 2 2cosx A t
≤Ã!ï˛!ê˛ˆÏܲ ò%ÓyÓ˚ xÓܲ°ö ܲÓ˚ˆÏ° xy˛õ!ö ˛õyˆÏÓö :
21 1= −x xω ~ÓÇ
22 2= −x xω
5.24a Ä 5.24b §ü#ܲÓ˚ˆÏî ~•z üyö ò%!ê˛ Ó§yˆÏ° ˛õyÄÎ˚y ÎyÎ˚ :
( )ω ω ω− = − + −2 2 21 1 2 1a ax x x x
() ωωω −=−−− 22220221 b xxxx
~ ò%!ê˛ §ü#ܲÓ˚îˆÏܲ §Ó˚° ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚ :
() 222212 aaa xx ωωωω +−=
~ÓÇ, ( )2 2 2 20 2 1b bx xω ω ω ω+ − =
2ω ~Ó˚ üyö !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ ~•z ò%•z §ü#ܲÓ˚î ˆÌˆÏܲ
1
2
xx
~Ó˚ üyö !öî≈Î˚ ܲˆÏÓ˚ ò%!ê˛ˆÏܲ §üyö ӈϰ ˆ°áy
ˆÎˆÏï˛ ˛õyˆÏÓ˚– xÌ≈yÍ,
22220 1
22222
ab
aab
xx
ωωωωωωωω
+− ==+−
....... 5.25
ÓL=îö ܲˆÏÓ˚ §y!çˆÏÎ˚ !°áˆÏ° xy˛õ!ö 2ω ~Ó˚ ~ܲ!ê˛ !mâyï˛ §ü#ܲÓ˚î ˛õyˆÏÓö;
()() 422222222000 20 abab ωωωωωωωωω −+++++=
~Ó˚ ˆÌˆÏܲ
2 ω
~Ó˚ ò%!ê˛ üyö ˛õyÄÎ˚y ÎyÎ˚– ~=!° •° :
141
()() ωωωωωωωωωωω⎡⎤
=++±++−++ ⎢⎥⎢⎥ ⎣⎦
122 22222222222
000012242ababab
= 2ω xÌÓyñ
2220ab ωωω ++ 2 ω
~Ó˚ ò%!ê˛ üyˆÏöÓ˚ ôöydܲ Óà≈ü)°=!° !öˆÏ°
ω
- ~Ó˚ ò%!ê˛ üyö •°
10 ωω=
....... 5.26a
~ÓÇ
()1
222220 =++ab ωωωω
....... 5.26b
1ω ~ÓÇ
2 ω
ÈÙÈ•z •° Î%!@¬ï˛ ˆòy°Ü˛ ï˛ˆÏsfÓ˚ ò%•z fl∫yû˛y!Óܲ ܲ¡õyB˛–
3. Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyˆÏï˛Ó˚ Ó˚y!¢üy°yÎ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛=!°Ó˚ üyˆÏöÓ˚Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyˆÏï˛Ó˚ Ó˚y!¢üy°yÎ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛=!°Ó˚ üyˆÏöÓ˚Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyˆÏï˛Ó˚ Ó˚y!¢üy°yÎ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛=!°Ó˚ üyˆÏöÓ˚Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyˆÏï˛Ó˚ Ó˚y!¢üy°yÎ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛=!°Ó˚ üyˆÏöÓ˚Î%!@¬ï˛ û˛Ó˚=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyˆÏï˛Ó˚ Ó˚y!¢üy°yÎ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛=!°Ó˚ üyˆÏöÓ˚
≤Ã!ï˛fliy˛õö–≤Ã!ï˛fliy˛õö–≤Ã!ï˛fliy˛õö–≤Ã!ï˛fliy˛õö–≤Ã!ï˛fliy˛õö–
xyüÓ˚y •z!ï˛üˆÏôƒ•z 5.25 §ü#ܲÓ˚ˆÏî Î%!@¬ï˛ ˆòy°Ü˛ ò%!ê˛Ó˚ !˛õ[˛ A Ä B ~Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyï˛
1
2
xx
~Ó˚
ò%!ê˛ Ó˚y!¢üy°y ˆ˛õˆÏÎ˚!åÈ– ~=!°Ó˚ üyö
2 ω
~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– §%ï˛Ó˚yÇñ ~ܲ ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°#ˆÏï˛
~•z xö%˛õyï˛ ~ܲ ~ܲ Ó˚ܲü •ˆÏÓ–
Îáöñ
220 ωω=
S§ü#ܲÓ˚î 5.26a)
21
22220
1 a
aa
xx
ωωωω
==+−
, xÌ≈yÍ ~•z ܲ¡õö˜Ï¢°#ˆÏï˛ x1= x2
§%ï˛Ó˚yÇñ !˛õ[˛ ò%!ê˛Ó˚ ~ܲ•z !òˆÏܲ §ü˛õ!Ó˚üyî §Ó˚î •ˆÏÓ ~ÓÇ ï˛yˆÏòÓ˚ ܲ¡õˆÏöÓ˚ ò¢y §üyö •ˆÏÓ–
xyÓyÓ˚ Îáö, 2 2 2 20 a bω ω ω ω= + +
()22
12 22222
200
aab
a b aab
m xxm
ωωω ωωωωω
===−+−++
....... 5.27
xÌ≈yÍñ ~•z ܲ¡õö˜Ï¢°#ˆÏï˛ !˛õ[ ˛ò%!ê˛Ó˚ §Ó˚î ÷ô% !Ó˛õÓ˚#ï˛ü%á# öÎ˚ñ ï˛yˆÏòÓ˚ üyöÄ û˛Ó˚=!°Ó˚ ÓƒhflÏyö%˛õyï˛#–
xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈñ §Ó˚î ò%!ê˛ !Ó˛õÓ˚#ï˛ü%á# •ÄÎ˚yÓ˚ xÌ≈ñ ï˛yˆÏòÓ˚ ˆòy°à!ï˛ˆÏï˛ ò¢yÓ˚ ˛õyÌ≈ܲƒ π –
°«˛ƒ ܲÓ˚&ö,
1
2
xx
xö%˛õyˆÏï˛Ó˚ x˛õÓ˚ Ó˚y!¢üy°y!ê˛ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ°Ä ~ܲ•z ú˛° ˛õyÄÎ˚y ÎyˆÏÓ–
142
~˛ ˛õÎ≈hsˇ ˆÎ xyˆÏ°yã˛öy ܲÓ˚y •°ñ ï˛y ˆÌˆÏܲ Î%!@¬ï˛ ˆòy°ˆÏܲÓ˚ ˆ«˛ˆÏe §yôyÓ˚î ˛õk˛!ï˛Ó˚ ≤ÈÏÎ˚yà xy˛õ!ö
!öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈö– ~•z ˛õk˛!ï˛ˆÏï˛ ˆÎ ú˛° ˛õyÄÎ˚y ˆà°ñ ˆ§=!° xyˆÏà ˆòy°ˆÏܲÓ˚ !˛õ[˛ ò%!ê˛Ó˚ û˛Ó˚
§üyö ôˆÏÓ˚ !öˆÏÎ˚ ˆÎ ú˛° ˛õyÄÎ˚y !àˆÏÎ˚!åÈ°ñ ï˛yÓ˚ §ˆÏD §D!ï˛˛õ)î≈ !ܲöy ˆ§!ê˛ xy˛õ!ö ˆòˆÏá !öˆÏï˛ ˛õyˆÏÓ˚ö–
§yôyÓ˚î ˛õk˛!ï˛ˆÏï˛ !˛õ[˛ ò%!ê˛Ó˚ !û˛ß¨ û˛ˆÏÓ˚Ó˚ çöƒ
2 ω
~Ó˚ ˆÎ üyö xyüÓ˚y ˆ˛õˆÏÎ˚!åÈ xÌ≈yÍñ
()1
222220ab ωωωω =++
, ˆ§!ê˛ 2 2,a b a b
km m m
mω ω= = = = !°áˆÏ° òÑyí˛¸yÎ˚
12
20
2km
ω⎛⎞ + ⎜⎟ ⎝⎠
, Îy 5.9
§ü#ܲÓ˚ˆÏîÓ˚
2 ω
~Ó˚ §üyö–
xyÓyÓ˚ 5.27 §ü#ܲÓ˚ˆÏî
ab mm=
!°áˆÏ°
1
2
xx
~Ó˚ üyö •Î˚
1, −
Îy xyüÓ˚y xyˆÏà
2 ξ
ܲ¡õö˜Ï¢°#Ó˚ ˆ«˛ˆÏe
ˆ˛õˆÏÎ˚!åÈ–
~áyˆÏö xyüÓ˚y ~üö ò%!ê˛ §Ó˚° ˆòy°Ü˛ !öˆÏÎ˚ àîöyÓ˚ ܲyç ܲˆÏÓ˚!åÈ ˆÎñ ˆòy°Ü˛ ò%!ê˛Ó˚ ܲyÎ≈ܲÓ˚ ˜òâ≈ƒ ~ÓÇ
xÎ%!@¬ï˛ xÓfliyÎ˚ !öçfl∫ ܲ¡õyB˛ §üyö– !ܲv !˛õ[˛ ò%!ê˛Ó˚ û˛Ó˚ !û˛ß¨ •ÄÎ˚yÎ˚ Î%@¬ˆÏöÓ˚ ú˛ˆÏ° ~áyˆÏö
2 ω
~Ó˚
ˆÎ üyö ˛õyÄÎ˚y ˆàˆÏåÈñ ï˛y !ܲv ò%!ê˛ û˛ˆÏÓ˚Ó˚ í˛z˛õÓ˚•z !öû≈˛Ó˚¢#°– °«˛ƒ ܲÓ˚&öñ xyüÓ˚y Î!ò ò%!ê˛Ó˚ ܲyÎ≈ܲÓ˚ ˜òâ≈ƒ
!û˛ß¨ ôˆÏÓ˚ !öï˛yüñ ï˛y•ˆÏ° ò%!ê˛ ˆòy°ˆÏܲÓ˚ !öçfl∫ ܲ¡õyB˛Ä !û˛ß¨ •ï˛ ~ÓÇ
10 ωω=
öy •ˆÏÎ˚ ï˛y !ܲå%Èê˛y ç!ê˛°ï˛Ó˚
•ï˛– ò%!ê˛ !û˛ß¨ û˛Ó˚ ~ܲ•z ô &Óܲ !Ó!¢‹T !fl±Ç ÈÙÈ~ Î%_´ ܲˆÏÓ˚ ܲ!¡õï˛ Ü˛Ó˚ˆÏ°ñ ò%!ê˛Ó˚ çöƒ
0 ω
~Ó˚ üyö !û˛ß¨
•Î˚– ˆ§ˆÏ«˛ˆÏe àîöyÓ˚ ܲyç!ê˛ xyÓ˚Ä ç!ê˛° •ˆÏÎ˚ ˛õˆÏí˛¸– xyüÓ˚y ~áyˆÏö ~•z ç!ê˛° xyˆÏ°yã˛öy ˆÌˆÏܲ !ÓÓ˚ï˛
ˆÌˆÏܲ!åÈ–
5.6 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
Î%!@¬ï˛ ܲ¡õö ˛õòyÌ≈ !ÓòƒyÎ˚ ~ܲ!ê˛ =Ó˚&c˛õ)î≈ !Ó£ÏÎ˚ !•§yˆÏÓ ã˛ã≈˛y •ˆÏÎ˚ ÌyˆÏܲ– Ó‡ ˆ«˛ˆÏe Î%!@¬ï˛ ܲ¡õˆÏöÓ˚
í˛zòy•Ó˚î °«˛ƒ ܲÓ˚y ÎyÎ˚– ~ܲ!ê˛ Ü˛¡õö¢#° ï˛sf ˆÌˆÏܲ Î%@¬ˆÏöÓ˚ üyôƒˆÏü ܲ¡õö xöƒ ï˛ˆÏsf §MÈ˛y!Ó˚ï˛ •ˆÏï˛
˛õyˆÏÓ˚– ˆÎˆÏ•ï%˛ ~áyˆÏö ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ üï˛ ¢!_´ •hflÏyhsˇÓ˚ ˆÜ˛Ó° ~ܲü%á# öÎ˚ñ xï˛~Ó ~•z ܲ¡õˆÏöÓ˚
ˆ«˛ˆÏe ã˛y°Ü˛ Ä ã˛y!°ï˛ ï˛sf xy°yòyû˛yˆÏÓ !ã˛!•´ï˛ ܲÓ˚y §Ω˛Ó öÎ˚–
~•z ~ܲˆÏܲ xyüÓ˚y ò%’ôÓ˚ˆÏöÓ˚ Î%!@¬ï˛ ï˛ˆÏsfÓ˚ à!ï˛≤ÃÜ,˛!ï˛Ó˚ !ӈϟ’£Ïî ܲˆÏÓ˚!åÈ– ~Ó˚ ≤ÃÌü!ê˛ •° •y°Ü˛y
!fl±ÇˆÏÎ˚Ó˚ §y•yˆÏ΃ Î%!@¬ï˛ ò%!ê˛ §üyö û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ !fl±Ç û˛Ó˚ ï˛sf ~ÓÇ xöƒ!ê˛ §üyö ˜òˆÏâ≈ƒÓ˚ ò%•z!ê˛ §Ó˚°
ˆòy°ˆÏܲÓ˚ ~ܲ!ê˛ ï˛sfñ ÎyˆÏòÓ˚ !˛õ[˛ =!° •y°Ü˛y !fl±Ç ~Ó˚ myÓ˚y Î%!@¬ï˛–
ò%!ê˛ ï˛ˆÏsfÓ˚ ˆ«˛ˆÏe•z û˛Ó˚ Óy ˆòy°ˆÏܲÓ˚ !˛õ[˛=!°Ó˚ çöƒ xy°yòyû˛yˆÏÓ à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y ܲÓ˚y
•ˆÏÎ˚ˆÏåÈ– ~=!°Ó˚ §üyôyö ܲÓ˚ˆÏï˛ !àˆÏÎ˚ ˆòáy ˆàˆÏåÈ ˆÎñ û˛Ó˚ Óy !˛õˆÏ[˛Ó˚ !öˆÏò≈¢yˆÏB˛Ó˚ ~üö !ӈϢ£Ï ˜Ó˚!áܲ
xˆÏ˛õ«˛Ü˛ ˆöÄÎ˚y ÎyÎ˚ñ ˆÎ=!°Ó˚ üyôƒˆÏü xÓܲ° §ü#ܲÓ˚î=!° §yôyÓ˚î §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏî
˛õ!Ó˚îï˛ •Î˚– ~•z xˆÏ˛õ«˛Ü˛=!°ˆÏܲ xyüÓ˚y ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ öyˆÏü !ã˛!•´ï˛ ܲˆÏÓ˚!åÈ–
Î%!@¬ï˛ ï˛ˆÏsfÓ˚ à!ï˛Ó˚ !ӈϟ’£ÏˆÏî fl∫yû˛y!Óܲ !öˆÏò≈¢yˆÏB˛Ó˚ ÓƒÓ•yÓ˚ xyüyˆÏòÓ˚ ܲyˆÏåÈ xï˛ƒhsˇ í˛z˛õˆÏÎyà# •Î˚– xyüÓ˚y
ˆòˆÏá!åÈñ Îáö ï˛sf!ê˛ ~ܲ!ê˛ üye fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ ÓÓ˚yÓÓ˚ ܲ!¡õï˛ •Î˚ñ ï˛áö ï˛ˆÏsfÓ˚ ≤Ã!ï˛!ê˛ xÇ¢ ~ܲ•z
143
ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyˆÏܲ– ï˛ˆÏsfÓ˚ ~ çyï˛#Î˚ ܲ¡õöˆÏܲ xyüÓ˚y ~ܲ!ê˛ fl∫yû˛y!Óܲ ܲ¡õö˜Ï¢°# öyˆÏü
x!û˛!•ï˛ ܲ!Ó˚ ~ÓÇ ˙ ܲ¡õyB˛!ê˛ •Î˚ ~•z ܲ¡õö˜Ï¢°#Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛– xyÓyÓ˚ ~ܲ•z §ˆÏD ܲ¡õö˜Ï¢°#Ó˚
í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ° ˆÎ !Óê˛ Óy ܲ¡õˆÏöÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ü!í˛í˛zˆÏ°¢ˆÏöÓ˚ í˛zͲõ!_ •Î˚ñ ˆ§!ê˛Ä xyüÓ˚y ˆòáˆÏï˛
ˆ˛õˆÏÎ˚!åÈ–
~åÈyí˛¸y !ӈϢ£Ï ~ܲ!ê˛ Ü˛¡õö˜Ï¢°#ˆÏï˛ ï˛ˆÏsfÓ˚ xÇ¢=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyï˛ ~ÓÇ ï˛yˆÏòÓ˚ ܲ¡õˆÏöÓ˚ ò¢y
˛õyÌ≈ܲƒ §¡∫ˆÏ¶˛ xyüÓ˚y ~•z ~ܲˆÏܲ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ˆüyˆÏê˛Ó˚ í˛z˛õÓ˚ Ó°y ÎyÎ˚ñ ÎyˆÏï˛ xy˛õ!ö ~ܲ!ê˛ ç!ê˛°
ï˛ˆÏsfÓ˚ Î%!@¬ï˛ xyˆÏ®y°ö §¡∫ˆÏ¶˛ í˛zFã˛ï˛Ó˚ üyˆÏöÓ˚ !ӈϟ’£Ïî §•ˆÏç xö%ôyÓö ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚öñ ï˛yÓ˚ çöƒ xy˛õöyˆÏܲ
≤Ã›ï˛ Ü˛Ó˚yÓ˚ ܲyç!ê˛•z ~•z ~ܲˆÏܲ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. 5.4 !ã˛eyö%ÎyÎ˚# ò%!ê˛ û˛Ó˚ ܲ!¡õï˛ •ˆÏFåÈ– ï˛yˆÏòÓ˚ ܲ¡õˆÏöÓ˚ ¢ï≈˛=!° !öˆÏã˛ ˆòÄÎ˚y •° :
()()1112
00
0,00,0,tt
dxdxxAxu
dtdt ==
⎛⎞⎛⎞ ==== ⎜⎟⎜⎟ ⎝⎠⎝⎠
~áyö ˆÌˆÏܲ (5,12) ~ÓÇ (5,13) §ü#ܲÓ˚î ò%!ê˛Ó˚ §y•yˆÏ΃ ( )1x t ~ÓÇ
() 2 xt
~Ó˚ üyö !öî≈Î˚ ܲÓ˚&ö–
2. ~ܲ!ê˛ û˛Ó˚•#ö !fl±Ç ~Ó˚ ò%•z ≤ÃyˆÏhsˇ m1 ~ÓÇ m2 ò%!ê˛ û˛Ó˚ Î%_´ ܲÓ˚y
•°– !fl±Ç!ê˛Ó˚ ô Óܲ k ~ÓÇ û˛Ó˚ ò%!ê˛ !fl±Ç ~Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ܲ!¡õï˛ •ˆÏï˛
˛õyˆÏÓ˚– !ã˛e 5.10– ˆòáyö ˆÎñ ~•z ï˛sf!ê˛Ó˚ ܲ¡õyB˛ v •ˆÏ°
12
k vμ π =
ˆÎáyˆÏöñ 1 2
21m m
m mμ = +
3. ò%!ê˛ §üyö û˛Ó˚ˆÏܲ (m) ò%!ê˛ ~ܲ•z Ó˚ܲü !fl±ÇˆÏÎ˚Ó˚ S!fl±Ç ô Óܲ k) §y•yˆÏ΃ Î%_´
ܲˆÏÓ˚ 5.7 (a) !ã˛ˆÏe ˆÎüö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ˆ§•zû˛yˆÏÓ í˛zÕ‘¡∫û˛yˆÏÓ ~ܲ!ê˛ ò,ì˛¸ xÓ°¡∫ö
ˆÌˆÏܲ é%˛!°ˆÏÎ˚ ˆòÄÎ˚y •°– ~•zÓyÓ˚ §ü@ˇÃ ï˛sf!ê˛ˆÏܲ í˛zÕ‘¡∫ ï˛ˆÏ° ܲ!¡õï˛ Ü˛Ó˚y •°–
ˆòáyö ˆÎñ ~•z í˛zÕ‘¡∫ܲ¡õˆÏöÓ˚ ò%!ê˛ fl∫û˛yÓ Ü˛¡õyB˛ ˛õyÄÎ˚y §Ω˛Ó ~ÓÇ ~•z ˆÜ˛Ô!îܲ
ܲ¡õyB˛ ω •ˆÏ°ñ
ω
-Ó˚ ò%!ê˛ üyö !öˆÏã˛Ó˚ §¡õÜ≈˛!ê˛ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyˆÏÓ–
() 2352
km
ω=±
5.8 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ1 :
5.12 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õy•zñ
144
( ) ( ) ( )1 21 1 1 1 2 2 2
1 cos cos2 2x t A t A t+ ⎡ ⎤= = + + +⎣ ⎦
ξ ξ ω φ ω φ
5.13 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õy•z
()()() 122111222
1coscos 22 xtAtAt+⎡⎤ ==+−+ ⎣⎦ξξωφωφ ()() 1
111122221sinsin 2
dxAtAt
dt⎡⎤ =−+++ ⎣⎦ ωωφωωφ
~ÓÇñ
()() 211112222
1sinsin 2dx
AtAtdt
⎡⎤ =−+++ ⎣⎦ ωωφωωφ
≤Ãò_ ≤ÃyÌ!üܲ ¢ï≈˛=!° •ˆÏFåÈ
()()1212
00
0,0,0,0tt
dxdxxAxA
dtdt ==
⎛⎞⎛⎞ ==== ⎜⎟⎜⎟ ⎝⎠⎝⎠
~=!° ≤ÈÏÎ˚yà ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚
( )1 1 2 21 cos cos2
A A A+ =φ φ
( )1 1 2 21 cos cos2
A A A− =φ φ
- ( )1 1 1 2 2 21 sin sin 02
A A+ =ω φ ω φ
( )1 1 1 2 2 21 sin sin 02
A A+ =ω φ ω φ
~•z ã˛yÓ˚!ê˛ §¡õÜ≈˛ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ :
1 1 2 2 1 1 1 2 2 2cos 2 , cos 0, sin 0, sin 0A A A A A= = = =φ φ ω φ ω φ
ˆÎˆÏ•ï% ,
121 ,, AAω
~ÓÇ
2 ω
≤Ã!ï˛!ê˛•z ¢)öƒ •ˆÏï˛ ˛õyˆÏÓ˚ öy
xï˛~Óñ
120 φφ ==
§%ï˛Ó˚yÇñ
20 A=
~ÓÇñ
12 AA =
~áö ˆòáy ÎyÎ˚ ˆÎñ
11 cos xAt ω =
~ÓÇñ
21 cos xAt ω =
145
11 cos At ξω ∴=
~ÓÇ
20 ξ=
°«˛ƒ ܲˆÏÓ˚ ˆòá%ö ~ˆÏ«˛ˆÏe
12 xx =
§%ï˛Ó˚yÇñ
20 ξ=
~ÓÇ ï˛sf!ê˛ ˆÜ˛Ó°üye fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛
1 ξ
ÓÓ˚yÓÓ˚
ܲ!¡õï˛ •Î˚–
xö%˘¢#°ö# ÈÙÈxö%˘¢#°ö# ÈÙÈxö%˘¢#°ö# ÈÙÈxö%˘¢#°ö# ÈÙÈxö%˘¢#°ö# ÈÙÈ2 :
5.20 a ~ÓÇ 520 b §ü#ܲÓ˚î ò%!ê˛ ≤Ã̈Ïü ˆ°áy Îyܲ :
()2
22 10112 20 c
dxxxx
dtωω ++−= ()
222 20212 20 c
dxxxx
dtωω +−−=
§ü#ܲÓ˚î ò%!ê˛ ˆÎyà ܲˆÏÓ˚ ˛õy•z
()()2
212012 20 dxxxx
dtω +++=
....... (i)
ˆòáy ÎyˆÏFåÈ ˆÎñ
121 xxξ +=
ôÓ˚ˆÏ°
(i) §ü#ܲÓ˚î!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î •ˆÏÎ˚ ÎyÎ˚ :
2 2101 20
d
dt
ξωξ +=....... (ii)
§%ï˛Ó˚yÇñ
1 ξ
ˆÜ˛ ≤ÃÌü fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ó°y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
~ܲ•zû˛yˆÏÓñ 5.20 a ˆÌˆÏܲ 5.20 b §ü#ܲÓ˚î !ÓˆÏÎ˚yà ܲˆÏÓ˚ ˛õy•z
()()()2
221201212 220 c
dxxxxxx
dtωω −+−+−=
Óyñ
()()()2
2212012 220 c
dxxxx
dtωω −++−=
....... (iii)
~ÓyÓ˚
212 xx ξ=−
Ó§yˆÏ° (iii) §ü#ܲÓ˚î!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ xÓܲ° §ü#ܲÓ˚ˆÏî ˛õ!Ó˚îï˛ •ˆÏFåÈ :
()2
22 202 220 c
d
dt
ξωωξ ++=
....... (iv)
xÌ≈yÍ
2 ξ
- ˆÜ˛ !mï˛#Î˚ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ Ó°y ÎyÎ˚–
~áö (ii) Ä (iv) §ü#ܲÓ˚î ˆÌˆÏܲ Î%!@¬ï˛ ï˛ˆÏsfÓ˚ fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ˛õyÄÎ˚y ÎyÎ˚
10gl
ωω==
~ÓÇ,
2220
2 2c
gklm
ωωω =+=+
146
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ3 :
xyüÓ˚y ôˆÏÓ˚ !ö•z ˆÎ fl∫yû˛y!Óܲ !öˆÏò≈¢yB˛ 1ξ Ä
2 ξ
ˆÎáyˆÏöñ
112 xx ξ=+
~ÓÇñ
212 xx ξ=−
xÌ≈yÍñ
1212 x
+ =ξξ
~ÓÇñ
1222 x
− =ξξ
§%ï˛Ó˚yÇñ
()() 11221211 ,22
xx =+=− ξξξξ
à!ï˛¢!_´
2212
12
⎡⎤ =+ ⎣⎦ Kmxx
=
22
1212 1222
m⎡⎤ ⎛⎞⎛⎞ +− ⎢⎥ + ⎜⎟⎜⎟ ⎝⎠⎝⎠ ⎢⎥ ⎣⎦
ξξξξ
= ( ) ( )2 2 2 21 2 1 2
1 1 12 22 4 4
m m+ = +ξ ξ ξ ξ
!fli!ï˛¢!_´
()()2 221212
122lmg
Vxxkxxl
=++−
= 2 2
21 2 1 22
1 12 2 2 2
gm k
l
⎡ ⎤+ −⎛ ⎞ ⎛ ⎞⎛ ⎞ + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦
ξ ξ ξ ξ ξ
=
2222 1202
11222
mk⎡⎤ ++ ⎢⎥ ⎣⎦ξξ ωξ
=
22222201022
111442c mmm ++ ωξωξωξ
=
() 222220102
11244c mm ++ ωξωωξ
=
22220122
14
m⎡⎤ + ⎣⎦ ωξωξ
~áyˆÏöñ
220,c
gklm== ωω
,
222202c =+ ωωω
147
~áö xyüÓ˚y Î!ò
1 ξ
Ä
2 ξ
~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛
11 2m ηξ =
~ÓÇ 2 22mη ξ= ÓƒÓ•yÓ˚ ܲ!Ó˚ñ ï˛ˆÏÓ 1ξ Ä
2 ξ
~Ó˚ çyÎ˚àyÎ˚ ÎÌye´ˆÏü
1 2mη
Ä
2 2mη
Ó!§ˆÏÎ˚ ˛õy•z
à!ï˛¢!_´
() 2222212
1221 .42 l Kml
mm⎛⎞ =+=+ ⎝⎠ ηηη
~ÓÇñ !fli!ï˛¢!_´
() 2222222201220122
1221 ..42
Vmmm
⎛⎞ =+=+ ⎝⎠ ωηωηωηωη
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ1 ~Ó˚ í˛z_ˆÏÓ˚ ÓƒÓ•*ï˛ §¡õÜ≈˛=!°ˆÏï˛ !öˆÏã˛Ó˚ ¢ï≈˛=!° ≤ÈÏÎ˚yà ܲÓ˚y •ˆÏÓ– ~•z ¢ï≈˛=!° •°
()()1212
00
0,0,0,tt
dxdxxAxu
dtdt==
⎛⎞⎛⎞ === ⎜⎟⎜⎟ ⎝⎠⎝⎠
~áyö ˆÌˆÏܲ ˛õy•zñ
( )1 1 2 21 cos cos2
A A A+ =φ φ
( )1 1 2 21 cos cos 02
A A− =φ φ
( )1 1 1 2 2 21 sin sin 02
A A− + =ω φ ω φ
( )1 1 1 2 2 21 sin sin2
A A u− + =ω φ ω φ
≤ÃÌü ò%!ê˛ §¡õÜ≈˛ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ñ 1 1 2 2cos cosA A A= =φ φ ~ÓÇ ˆ¢£Ï ò%!ê˛ §¡õÜ≈˛ ˆÌˆÏܲ ˛õyÄÎ˚y
ÎyÎ˚ñ
111222 sin,sin AuAu =−= ωφωφ
∴
()()2
22 222 111111
cossinuAAAA =+=+ φφ
ω ()()2
22 222 222222
cossinuAAAA =+=+ φφ
ω
148
~åÈyí˛¸y
1212
tan,tan uuAA
=−= φφ ωω
~áö xy˛õ!ö 5.12 Ä 5.13 §ü#yܲÓ˚î ˆÌˆÏܲ
() 1 xt
Ä
() 2 xt
~Ó˚ Ó˚y!¢üy°y §•ˆÏç•z !°áˆÏï˛ ˛õyÓ˚ˆÏÓö–
2.5 10 (a) !ã˛ˆÏe û˛Ó˚ ò%!ê˛ §yüƒyÓfliyÎ˚ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö !fl±Ç!ê˛ ≤çy!Ó˚ï˛ Óy §B%˛!ã˛ï˛ •Î˚!ö–
5.10(b) !ã˛ˆÏe û˛Ó˚ ò%!ê˛ˆÏܲ §yüƒyÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ xÓfliyÎ˚ ˆòáy ÎyˆÏFåÈ– m1 û˛Ó˚!ê˛Ó˚ §yüƒÓfliy ˆÌˆÏܲ
!Óã%˛ƒ!ï˛ x1 ~ÓÇ m2 û˛Ó˚!ê˛Ó˚ !Óã%˛ƒ!ï˛ x2 – ôÓ˚y Îyܲñ x1<x2– !fl±Ç!ê˛
21 − xx
˛õ!Ó˚üyî ≤çy!Ó˚ï˛ •ˆÏÎ˚ˆÏåÈ–
~Ó˚ ú˛ˆÏ° !fl±Ç ÈÙÈ~ §,‹T ê˛yö F •ˆÏ°
F = k(x2 –x1)
~•z Ó° û˛Ó˚ ò%!ê˛Ó˚ IJõÓ˚ ˛õÓ˚flõÓ˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ !e´Î˚y¢#°– ï˛y•z m1 Ä m2 û˛ˆÏÓ˚Ó˚ àï˛#Î˚ xÓܲ° §ü#ܲÓ˚î
•ˆÏÓ ÎÌye´ˆÏü
() 1121 =− mxkxx
....... (i)
() 2221 =−− mxkxx
....... (ii)
(i) ˆÜ˛ m2 !òˆÏÎ˚ ~ÓÇ (ii) ˆÜ˛ m1 !òˆÏÎ˚ =î ܲˆÏÓ˚ (ii) ˆÌˆÏܲ (i)ÈÙÈˆÜ˛ !ÓˆÏÎ˚yà ܲÓ˚y •°–
()()() 1211221 1 −=−+− mmxxkmmxx
Óy
() 122121
12
+ ⎛⎞ −=−− ⎜⎟ ⎝⎠mm
xxkxxmm
≤ß¿ xö%ÎyÎ˚#ñ1 2
1 2
m mm m
μ=+
~áö 2 1x x x− = ôÓ˚ˆÏ°
21 xxx =−
∴
kxx =−μ
~!ê˛ ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚îñ ÎyÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛
k = ωμ
§%ï˛Ó˚yÇñ ï˛sf!ê˛Ó˚ ܲ¡õyB˛
12
kV=πμ
xy˛õ!ö !ö˘ÿ˛Î˚•z °«˛ƒ ܲÓ˚ˆÏåÈö ˆÎñ ~•z ܲ¡õyB˛ m1Óy m2 ~Ó˚ IJõÓ˚ §Ó˚y§!Ó˚ !öû≈˛Ó˚ öy ܲˆÏÓ˚
μ
~Ó˚
149
IJõÓ˚ !öû≈˛Ó˚ ܲÓ˚ˆÏåÈ–
12
12
mmmm
⎛⎞ =⎜⎟ + ⎝⎠ μ
ˆÜ˛ ï˛sf!ê˛Ó˚ §üyö#ï˛ û˛Ó˚ (reduced mass) Ó°y •Î˚– ~•z û˛Ó˚ m1
~ÓÇ m2 í˛zû˛ˆÏÎ˚Ó˚ ˆÌˆÏܲ•z ˆåÈyê˛–
3. !ã˛e 5.11(b) ˆï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ܲ#û˛yˆÏÓ í˛z˛õˆÏÓ˚Ó˚ û˛Ó˚ A ~ÓÇ !öˆÏã˛Ó˚ û˛Ó˚ B ˆÜ˛ ï˛yˆÏòÓ˚ §yüƒyÓfliyÓ˚
ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ í˛zÕ‘¡∫ !òˆÏܲ û˛Ó˚ ò%!ê˛Ó˚ ܲ¡õö ÷Ó˚& ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– û˛Ó˚ ò%!ê˛Ó˚ §Ó˚î ya Ä yb !öˆÏã˛Ó˚ !òˆÏܲ
ôöydܲ ӈϰ ôÓ˚y •ˆÏÎ˚ˆÏåÈ–
A û˛ˆÏÓ˚Ó˚ IJõÓ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° ( )a b aky k y y− + −
B û˛ˆÏÓ˚Ó˚ IJõÓ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° ( )b ak y y− −
~áyˆÏö ya ~ÓÇ yb t §üˆÏÎ˚ û˛Ó˚ ò%!ê˛Ó˚ §yüƒyÓfliyö ˆÌˆÏܲ !Óã%˛ƒ!ï˛ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ– û˛Ó˚ ò%!ê˛Ó˚ xÓܲ° à!ï˛
§ü#ܲÓ˚î ˆ°áy ÎyÎ˚
( ) ( ),= − + − = − −a a b amy ky k y y myb k yb ya
~áyö ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚
() 202 =−+ aab yyy ω
....... (i)
( )20= − −b b ay y yω ....... (ii)
ˆÎáyˆÏöñ20
km
ω =
fl∫û˛yÓ Ü˛¡õˆÏöÓ˚ ܲ¡õyB˛
ω
~ÓÇ ≤ÃyÌ!üܲ ò¢yˆÏܲyî
1 φ
Ä
2 φ
•ˆÏ°ñ ~•z §ü#ܲÓ˚î ò%!ê˛Ó˚ §üyôyö ˆ°áy
ÎyÎ˚ñ
()1 cos a yAt =+ ωφ ()2 cos b yBt =+ ωφ
ˆÎáyˆÏö A ~ÓÇ B ò%!ê˛ û˛ˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
ò%’ÓyÓ˚ xÓܲ°ˆÏöÓ˚ myÓ˚y ˛õyÄÎ˚y ÎyÎ˚
2aa yy =−ω
~ÓÇ 2
b by y= −ω
ay ~ÓÇ
b y
ÈÙÈÓ˚ üyö (i) ~ÓÇ (ii) Ó!§ˆÏÎ˚ ~ÓÇ §ü#ܲÓ˚î ò%!ê˛ˆÏܲ §y!çˆÏÎ˚ ˛õy•zñ
() 2220 2. aba yy −= ωωω () 2221
000 a yy =− ωωω
150
a
b
yy Ó˚y!¢Ó˚ ò%•z üyˆÏöÓ˚ §üï˛y ˆÌˆÏܲ ˛õyÄÎy ÎyÎ˚ñ
22200
22200
2−=−
ωωωωωω
Óy §Ó˚°#ܲÓ˚ˆÏîÓ˚ ˛õÓ˚ñ
422400 3.0 −+= ωωωω
~!ê˛ 2ω ~Ó˚ ~ܲ!ê˛ !mÈÙÈâyï˛ §ü#ܲÓ˚î –
2 ω
~Ó˚ ˆÎ ò%!ê˛ üyö ~áyö ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ ˆ§=!° •°ñ
()244
22 0000
39435 2
±−==±
ωωω ωω
Óy
() 352
km
±
~áyö ˆÌˆÏܲ ω ~Ó˚ ˆÎ ò%!ê˛ üyö ˛õyÄÎ˚y ÎyˆÏFåÈñ ï˛yÓ˚ üˆÏôƒ
()12
352km
⎡⎤ + ⎢⎥ ⎣⎦
o&ï˛ï˛Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛
~ÓÇ x˛õÓ˚!ê˛ xÌ≈yÍ,
()12
352km
⎡⎤ − ⎢⎥ ⎣⎦
ô#Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ §)!ã˛ï˛ ܲÓ˚ ÏåÈ–
151
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 6 ï˛Ó˚Dà!ïï˛Ó˚Dà!ïï˛Ó˚Dà!ïï˛Ó˚Dà!ïï˛Ó˚Dà!ï˛ (Wave Motion)
àë˛öàë˛öàë˛öàë˛öàë˛ö
6.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
6.2 ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•
6.3 ï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛ò
6.4 ï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚y
6.5 ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ
6.5.1 ï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢y ˛õyÌ≈ܲƒ
6.6 ~ܲüy!eܲ §yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î
6.7 ã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y fliyöyhsˇ!Ó˚ï˛ ¢!_´Ó˚ àîöy
6.7.1 ¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y
6.8 ï˛Ó˚ˆÏDÓ˚ ≤ÃyÓ°ƒ ~ÓÇ ¢∑ï˛Ó˚ˆÏDÓ˚ ≤ÃyÓ°ƒüy˛õܲ ˆflÒ°
6.9 í˛˛õ°yÓ˚ !e´Î˚y
6.10 §yÓ˚yÇ¢
6.11 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
6.12 í˛z_Ó˚üy°y
6.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
~ܲܲ ÈÙÈ5 ~ Î%!@¬ï˛ ܲ¡õö (coupled oscillation) §¡∫ˆÏ¶˛ xyˆÏ°yã˛öyÓ˚ §üÎ˚ xyüÓ˚y ˆòˆÏá!åÈ ˆÎñ Î%@¬ˆÏöÓ˚
üˆÏôƒ !òˆÏÎ˚ ܲ¡õö ܲ#û˛yˆÏÓ ~ܲ!ê˛ ï˛sf ˆÌˆÏܲ x˛õÓ˚ ~ܲ!ê˛ ï˛ˆÏsf §MÈ˛y!Ó˚ï˛ •Î˚– Ó›ï˛ Î%@¬ˆÏöÓ˚ ú˛ˆÏ° ~ܲ!ê˛
ï˛sf ˆÌˆÏܲ x˛õÓ˚!ê˛ˆÏï˛ ¢!_´Ó˚ •hflÏyhsˇÓ˚ âˆÏê˛– °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆÎñ ¢!_´Ó˚ ~•z •hflÏyhsˇÓ˚ ≤Ã!e´Î˚yÓ˚ §üÎ˚
ï˛sf=!° ï˛yˆÏòÓ˚ §yüƒyÓfliyÓ˚ ò%ÈÛ˛õyˆÏ¢ ܲ!¡õï˛ •Î˚ñ !ܲv ï˛yˆÏòÓ˚ §yüƒyÓfliyÓ˚ ˆÜ˛yöÄ fliyöyhsˇÓ˚ âˆÏê˛ öy– ˆÎüöñ
ˆÜ˛yöÄ ç°y¢ˆÏÎ˚ Î!ò ~ܲ!ê˛ ˛õÿÏÓ˚Ó˚ ê%˛Ü˛ˆÏÓ˚y ˆú˛°y ÎyÎ˚ñ ï˛y•ˆÏ° ˆòáy ÎyÎ˚ ˆÎñ !fliÓ˚ ç° ï˛ˆÏ° ˆì˛í˛z §,!‹T
•ˆÏÎ˚ ï˛y e´ü¢ åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸– ç°Ü˛îy=!° !ܲv §Ó≈e•z ï˛yˆÏòÓ˚ §yüƒyÓfliyˆÏöÓ˚ ò%’˛õyˆÏ¢ñ IJõÓ˚ÈÙÈ!öˆÏã˛ xyˆÏ®y!°ï˛
•ˆÏFåÈ– !ܲv ~Ó˚ ú˛ˆÏ° §,‹T ˆì˛í˛z!ê˛ ç°y¢ˆÏÎ˚Ó˚ §Ó≈e åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸– ç° ï˛ˆÏ°Ó˚ ~•z xyˆÏ°yí˛¸öˆÏܲ ï˛Ó˚D ~ÓÇ
~•z xyˆÏ°yí˛¸ˆÏöÓ˚ §MÈ˛y°ö ≤Ã!e´Î˚y!ê˛ˆÏܲ ï˛Ó˚Dà!ï˛ (wave motion) Ó°y •Î˚– Ó›ï˛ ~•z xyˆÏ°yí˛¸ˆÏöÓ˚ §üÎ˚
çˆÏ° û˛y§üyö ~ܲ!ê˛ •y°Ü˛y Ó›Ó˚ ê%˛Ü˛ˆÏÓ˚y °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ï˛y í˛zÕ‘¡∫ï˛ˆÏ° IJõÓ˚ ÈÙÈ!öˆÏã˛ xyˆÏ®y!°ï˛
•ˆÏFåÈ– ~ܲ•zû˛yˆÏÓ ¢∑ï˛Ó˚D ܲ!ë˛öñ ï˛Ó˚° Óy àƒy§#Î˚ üyôƒˆÏüÓ˚ üôƒ !òˆÏÎ˚ §MÈ˛y!Ó˚ï˛ •Î˚– üyôƒˆÏü ≤ÃÓ•üyö
ï˛Ó˚D Îáö üyôƒˆÏüÓ˚ ˆÜ˛yöÄ Ó›Ü˛îyˆÏܲ ܲ!¡õï˛ Ü˛ˆÏÓ˚ñ ï˛áö ˆ§•z ܲ¡õö ï˛yÓ˚ ˛õyˆÏ¢Ó˚ ӛܲîyÓ˚ §MÈ˛y!Ó˚ï˛
•Î˚– ~•zû˛yˆÏÓ xyˆÏ°yí˛¸ö üyôƒˆÏüÓ˚ !Ó!û˛ß¨ xLjϢ åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸ñ !ܲvñ Ó› û˛ˆÏÓ˚Ó˚ fliyöyhsˇÓ˚ âˆÏê˛ öy– ï˛ˆÏÓ ~
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çyï˛#Î˚ ï˛Ó˚D §MÈ˛yˆÏÓ˚Ó˚ çöƒ üyôƒü!ê˛ˆÏܲ !fli!ï˛fliy˛õܲ •ˆÏï˛ •ˆÏÓñ ï˛yÓ˚ çyí˛ƒôü≈Ä (inertia) ÌyܲˆÏï˛ •ˆÏÓ– ~çöƒ
¢∑ï˛Ó˚Dñ û)˛ÈÙÈܲ¡õï˛Ó˚D (seismic waves) ≤Ãû,˛!ï˛ˆÏܲ xyüÓ˚y !fli!ï˛fliy˛õܲ ï˛Ó˚D ӈϰ Ìy!ܲ–
xÓ¢ƒ ~•z ≤çˆÏD Ó°y òÓ˚ܲyÓ˚ ˆÎñ !fli!ï˛fliy˛õܲ ï˛Ó˚D åÈyí˛¸yÄ xyÓ˚Ä xöƒ ˆ◊î#Ó˚ ï˛Ó˚D Ó˚ˆÏÎ˚ˆÏåÈñ ÎyÓ˚
§MÈ˛yˆÏÓ˚Ó˚ çöƒ üyôƒˆÏüÓ˚ xyòˆÏ˛õ•z ≤ÈÏÎ˚yçö ˆö•z– xyüÓ˚y ˆÎ xyˆÏ°yˆÏï˛ ˆã˛yˆÏá ˆòáˆÏï˛ ˛õy•zñ xÌ≈yÍ ÎyˆÏܲ
xyüÓ˚y Ó!° ò,¢ƒüyö xyˆÏ°yܲñ ï˛y ~•z ˆ◊î#û%˛_´– ~ˆÏܲ Ó°y •Î˚ ï˛!í˛¸Fã%˛¡∫ܲ#Î˚ ï˛Ó˚D ~ÓÇ ï˛yÓ˚ §,!‹T ï˛Ìy
§MÈ˛yÓ˚ ˛õk˛!ï˛ !fli!ï˛fliy˛õܲ ï˛Ó˚D ˆÌˆÏܲ §¡õ)î≈ !û˛ß¨– ˆÎˆÏ•ï%˛ üyôƒˆÏüÓ˚ !fli!ï˛fliy˛õܲ ôˆÏü≈Ó˚ IJõÓ˚ ~•z ï˛Ó˚ˆÏDÓ˚
§MÈ˛yÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öyñ xï˛~Ó ~•z ï˛Ó˚D Ó› üyôƒü åÈyí˛¸yÄ §MÈ˛y!Ó˚ï˛ •ˆÏï˛ ˛õyˆÏÓ˚– Î!ò xy°yòyû˛yˆÏÓ í˛zˆÏÕ‘á
öy ܲÓ˚y •Î˚ñ ï˛y•ˆÏ° ~•z ~ܲˆÏܲ xyüÓ˚y xyüyˆÏòÓ˚ xyˆÏ°yã˛öyÎ˚ ï˛Ó˚D Ó°ˆÏï˛ !fli!ï˛fliy˛õܲ ï˛Ó˚ˆÏDÓ˚ ܲÌy ˆÓyé˛yÓñ
ÎyÓ˚ §MÈ˛yˆÏÓ˚Ó˚ çöƒ üyôƒü xÓ¢ƒ ≤ÈÏÎ˚yçö#Î˚– ï˛yåÈyí˛¸y ˆÜ˛yöÄ Îy!sfܲ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° §,‹T ~•z ï˛Ó˚DˆÏܲ xyüÓ˚y
Îy!sfܲ ï˛Ó˚D !•§yˆÏÓÄ í˛zˆÏÕ‘á ܲÓ˚Ó–
í˛ zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒ
~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ xy˛õ!ö ï˛Ó˚Dà!ï˛Ó˚ öyöy !òܲ §¡∫ˆÏ¶˛ xÓ!•ï˛ •ˆÏÓö– ~Ó˚ ú˛ˆÏ° xy˛õ!ö ˆÎ
ܲyç=!° ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆ§=!° •°ÈÙÙÙÈ
ï˛Ó˚Dà!ï˛ Ó°ˆÏï˛ Ü˛# ˆÓyé˛yÎ˚ ~ÓÇ Ü˛#û˛yˆÏÓ ï˛y §,!‹T •Î˚ñ ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!fli!ï˛fliy˛õܲ ï˛Ìy Îy!sfܲ ï˛Ó˚D Ä ï˛!í˛¸ÍFã%˛¡∫ܲ#Î˚ ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ Ä ˛õyÌ≈ܲƒ=!° Óƒyáƒy ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
ˆÜ˛yöÄ Îy!sfܲ ï˛Ó˚D ܲ#û˛yˆÏÓ üyôƒˆÏüÓ˚ §y•yˆÏ΃ ≤çyÓ˚°yû˛ ܲˆÏÓ˚ ~ÓÇ üyôƒˆÏüÓ˚ ôü≈yÓ°# ܲ#û˛yˆÏÓ
ï˛yÓ˚ à!ï˛ˆÏÓà !öÎ˚sfî ܲˆÏÓ˚ñ ï˛y Ó%!é˛ˆÏÎ˚ !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
ï˛Ó˚DˆÏܲ ày!î!ï˛Ü˛ Ó˚*ˆÏ˛õ ≤ÈÏÓ¢ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ ˆÌˆÏܲ ï˛yÓ˚ ˜Ó!¢‹Tƒ=!°
!ã˛ˆÏö !öˆÏï˛ ˛õyÓ˚ˆÏÓö–
~ܲüy!eܲñ !müy!eܲ Óy !eüy!eܲ ã˛°ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚D §ü#ܲÓ˚î ≤Ã!ï˛¤˛y ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢y ˛õyÌ≈ˆÏܲƒÓ˚ ï˛yͲõÎ≈ Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ã˛°ï˛Ó˚ˆÏDÓ˚ (progressive wave) myÓ˚y ˛õ!Ó˚Óy!•ï˛ «˛üï˛y Ä ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛yÓ˚ Ó˚y!¢üy°y=!°
≤Ã!ï˛!¤˛ï˛ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
¢ˆÏ∑Ó˚ í˛zͧ Ä ˆ◊yï˛yÓ˚ xyˆÏ˛õ!«˛Ü˛ à!ï˛Ó˚ ú˛ˆÏ° í˛zq(ï˛ í˛˛õ°yÓ˚ !e´Î˚y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ
!Ó!û˛ß¨ xÓfliyÎ˚ xy˛õyï˛ Ü˛¡õyˆÏB˛Ó˚ üyö §Ó˚y§!Ó˚ àîöy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
6.2 ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ§ü)•
ï˛Ó˚D ܲ#û˛yˆÏÓ §,!‹T •Î˚ñ ï˛y xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈö– xyüÓ˚y ~áyˆÏö ~•z âê˛öy!ê˛ Î%!@¬ï˛ ܲ¡õˆÏöÓ˚
ò,!‹TˆÏܲyî ˆÌˆÏܲ ˆòáyÓ˚ ˆã˛‹Ty ܲÓ˚Ó– !fli!ï˛fliy˛õܲ ï˛Ó˚ˆÏDÓ˚ öyüܲÓ˚ˆÏîÓ˚ ܲyÓ˚î!ê˛ ~•z xyˆÏ°yã˛öy ˆÌˆÏܲ xyÓ˚Ä
flõ‹T •ˆÏÎ˚ í˛zë˛ˆÏÓ– ~ܲ!ê˛ x!Ó!FåÈߨ üyôƒˆÏüÓ˚ ܲîy=!°Ó˚ Î%!@¬ï˛ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ï˛Ó˚D §,!‹T •Î˚– 6.1 (a) Ä
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(b) !ã˛ˆÏe ~ܲ!ê˛ !fl±Ç !òˆÏÎ˚ Î%!@¬ï˛ ï˛sf ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö N §ÇáƒÜ˛ §üyö û˛Ó˚ (m) !fl±Ç !òˆÏÎ˚ Î%_´
Ó˚ˆÏÎ˚ˆÏåÈ– ~•z û˛Ó˚=!°Ó˚ Óyü≤ÃyˆÏhsˇ xÓ!fliï˛ û˛Ó˚!ê˛ˆÏܲ ï˛yÓ˚ §yüƒyÓfliyö ˆÌˆÏܲ §yüyöƒ !Óã%˛ƒï˛ ܲˆÏÓ˚ ˆåȈÏí˛¸ !òˆÏ°
ܲ# ˆòáy ÎyˆÏÓ ÓˆÏ° xy˛õöyÓ˚ üˆÏö •Î˚⁄
xy˛õ!ö •Î˚ï˛ xö%üyö ܲÓ˚ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈöñ ≤ÃÌü û˛Ó˚!ê˛ ï˛yÓ˚ §yüƒyÓfliyˆÏöÓ˚ ò%’˛õyˆÏ¢ xyˆÏ®y!°ï˛ •ˆÏÓ– ˆ§•z
xyˆÏ®y°ö !ܲv ˆÜ˛Ó°üye ~ܲ!ê˛ û˛ˆÏÓ˚•z §#üyÓk˛ ÌyܲˆÏÓ öy– ï˛y §MÈ˛y!°ï˛ •ˆÏÓ ï˛yÓ˚ §!ߨ!•ï˛ û˛ˆÏÓ˚ ~ÓÇ
ˆ§•z û˛ˆÏÓ˚Ó˚ ܲ¡õö xyÓyÓ˚ ˛õÓ˚Óï≈˛# û˛ˆÏÓ˚ §MÈ˛y!°ï˛ •ˆÏÓ– e´ü¢ ˆòáy ÎyˆÏÓñ ~•zû˛yˆÏÓ ≤Ã!ï˛!ê˛ û˛Ó˚•z ~ˆÏܲÓ˚
˛õÓ˚ ~ܲ ï˛yˆÏòÓ˚ §yüƒyÓfliyˆÏöÓ˚ ò%’˛õyˆÏ¢ xyˆÏ®y!°ï˛ •ˆÏFåÈ– xÌ≈yÍ ˆòá%öñ ≤ÃÌü û˛Ó˚!ê˛ˆÏܲ ï˛yÓ˚ §yüƒyÓfliyö
ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ ï˛yÓ˚ üˆÏôƒ ˆÎ ¢!_´Ó˚ ˆÎyàyö ˆòÄÎ˚y •ˆÏÎ˚!åÈ° ˆ§•z ¢!_´ ~•z !fl±Ç Î%_´ û˛Ó˚=!°Ó˚ ~ܲ
≤Ãyhsˇ ˆÌˆÏܲ xöƒ ≤ÃyˆÏhsˇ åÈ!í˛¸ˆÏÎ˚ ˛õí˛¸°ñ !ܲv ˆÜ˛yöÄ û˛ˆÏÓ˚Ó˚•z §yüƒyÓfliyˆÏöÓ˚ fliyöã%˛ƒ!ï˛ âê˛° öy– ˆ§=!° ˆÜ˛Ó°
ï˛yˆÏòÓ˚ §yüƒyÓfliyˆÏöÓ˚ ò%’˛õyˆÏ¢ xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyˆÏܲ ~ÓÇ ˙ ï˛Ó˚ˆÏDÓ˚ §y•yˆÏ΃ üyôƒˆÏüÓ˚ ~ܲ !Ó®% ˆÌˆÏܲ
xöƒ !Ó®%ˆÏï˛ ¢!_´ ˛õ!Ó˚Óy!•ï˛ •Î˚– xy˛õ!ö Îáö ܲÌy ӈϰöñ ï˛áö xy˛õöyÓ˚ ü%ˆÏáÓ˚ §yüˆÏöÓ˚ ÓyÎ˚% hflψÏÓ˚
xyˆÏ°yí˛¸ö §,!‹T •Î˚– ˆ§•z xyˆÏ°yí˛¸ö ˛õÓ˚˛õÓ˚ ÓyÎ˚%hflÏÓ˚=!°ˆÏï˛ §MÈ˛y!Ó˚ï˛ •ˆÏÎ˚ ˆ◊yï˛yÓ˚ ܲyˆÏöÓ˚ ˛õò≈yÎ˚ ܲ¡õˆÏöÓ˚
§,!‹T ܲˆÏÓ˚– ˆ◊yï˛y xy˛õöyÓ˚ ܲÌy ÷öˆÏï˛ ˛õyöñ !ܲv xy˛õöyÓ˚ ü%ˆÏáÓ˚ §Ç°@¿ Óyï˛yˆÏ§ ˆÜ˛yöÄ ÓyÎ˚% ˆflÀyï˛ §,!‹T
•Î˚ öy–
ï˛Ó˚D üyôƒˆÏüÓ˚ üˆÏôƒ !òˆÏÎ˚ ˆÎ à!ï˛ˆÏÓà ≤çyÓ˚°yû˛ ܲˆÏÓ˚ñ ï˛yˆÏܲ Ó°y •Î˚ ï˛Ó˚DˆÏÓà(wave velocity)–
~•z à!ï˛ˆÏÓˆÏàÓ˚ üyö üyôƒˆÏüÓ˚ !fli!ï˛fliy˛õܲï˛y Ä xöƒyöƒ ˜Ó!¢ˆÏ‹TƒÓ˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– !ӈϢ£Ïû˛yˆÏÓ í˛zˆÏÕ‘á
ܲÓ˚y ≤ÈÏÎ˚yçö ˆÎñ ï˛Ó˚D ˆÓà üyôƒˆÏüÓ˚ ܲîy=!°Ó˚ xyˆÏ°yí˛¸ˆÏöÓ˚ à!ï˛ˆÏÓˆÏàÓ˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– ˆÎüöñ
ˆÜ˛yöÄ ˆçyÓ˚y° ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe üyôƒˆÏüÓ˚ ܲîy=!° x!ôܲï˛Ó˚ à!ï˛ˆÏÓà Ä !ÓhflÏyÓ˚ §• xyˆÏ®y!°ï˛ •ˆÏ°Ä
üyôƒˆÏüÓ˚ ˙ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà !ܲv xöƒ ˆÎ ˆÜ˛yöÄ §yôyÓ˚î ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ §üyö•z •Î˚–
ˆÎ ï˛Ó˚D üyôƒˆÏüÓ˚ ӛܲîy=!° §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •Î˚ñ xÌÓy ï˛!í˛¸ÍˆÏ«˛e Óy ˆã˛Ô¡∫ܲˆÏ«˛ˆÏeÓ˚
üï˛ §Ç!Ÿ’‹T ˆÜ˛yöÄ ˆû˛Ôï˛ Ó˚y!¢ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Äë˛yöyüy ܲˆÏÓ˚ñ ï˛yˆÏܲ §Ó˚° ˆòy°ï˛Ó˚D Óy §y•zöôü≈# ï˛Ó˚D
(sinusoidal wave) Ó°y •Î˚– ~•zÓ˚ܲü ï˛Ó˚D Îáö üyôƒˆÏüÓ˚ üˆÏôƒ !òˆÏÎ˚ ≤çyÓ˚°yû˛ ܲˆÏÓ˚ñ ï˛áö üyôƒˆÏüÓ˚
ˆÜ˛yöÄ ~ܲ!ê˛ Ü˛îyÓ˚ ܲ¡õö °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ §yüƒyÓfliyˆÏöÓ˚ ò%’˛õyˆÏ¢ ˆ§!ê˛Ó˚ ˆÎ ܲ¡õö âê˛ˆÏåÈñ ï˛yÓ˚
§üÎ˚È ò)Ó˚c §ü#ܲÓ˚î §y•zö (sine) ˆ°á!ã˛ˆÏeÓ˚ üï˛– ï˛Ó˚ˆÏDÓ˚ öyü!ê˛ ~áyö ˆÌˆÏܲ•z ~ˆÏ§ˆÏåÈ– ï˛Ó˚ˆÏDÓ˚ í˛zͧ
xÌÓy üyôƒˆÏüÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ Ó›Ü˛îy ≤Ã!ï˛ ˆ§ˆÏܲˆÏ[˛ Îï˛=!° ˛õ)î≈ ܲ¡õö §¡õߨ ܲˆÏÓ˚ Îy Îï˛=!° ï˛Ó˚D
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≤Ã!ï˛ ˆ§ˆÏܲˆÏ[˛ üyôƒˆÏüÓ˚ ˆÜ˛yöÄ !ö!ò≈‹T !Ó®%ˆÏܲ x!ï˛e´ü ܲˆÏÓ˚ñ ˆ§•z §Çáƒy!ê˛ˆÏܲ Ó°y •Î˚ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛
(frequency)– ≤Ã!ï˛!ê˛ ˛õ)î≈ ܲ¡õˆÏö ˆÎ §üÎ˚ °yˆÏà xÌÓy ˆÎ §üˆÏÎ˚ ~ܲ!ê˛ ˛õ)î≈ ï˛Ó˚D üyôƒˆÏüÓ˚ ˆÜ˛yöÄ !Ó®%ˆÏܲ
x!ï˛e´ü ܲˆÏÓ˚ ÎyÎ˚ñ ï˛yˆÏܲ xyüÓ˚y ï˛Ó˚ˆÏD ˛õÎ≈yÎ˚ܲy°(T) Ó!°– ~!ê˛ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyˆÏB˛Ó˚ !Ó˛õÓ˚#ï˛(reciprocal)
Ó˚y!¢– §yôyÓ˚îï˛ Ü˛¡õyB˛ˆÏܲ n Óy v S!öí˛zV !òˆÏÎ˚ §)!ã˛ï˛ ܲÓ˚y •Î˚– ~Ó˚ ~ܲܲ ˛õÎ≈yÎ˚/ˆ§ˆÏܲ[˛ (cycles per
second)– ï˛ˆÏÓ Ü˛¡õyˆÏB˛Ó˚ ~•z ~ܲܲ!ê˛ˆÏܲ xyüÓ˚y xyhsˇç≈y!ï˛Ü˛ ˛õk˛!ï˛ xö%ÎyÎ˚# !ÓK˛yö# •yÍ≈ˆÏçÓ˚ (Hertz)
öyüyö%§yˆÏÓ˚ Hz S•yÍ≈§‰V !ã˛•´ !òˆÏÎ˚ ≤Ãܲy¢ ܲ!Ó˚–
í˛z͈ϧÓ˚ ≤Ã!ï˛!ê˛ ˛õ)î≈ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° üyôƒˆÏüÓ˚ üˆÏôƒ !òˆÏÎ˚ ï˛Ó˚DyÜ,˛!ï˛ ~ܲ!ê˛ !ö!ò≈‹T ò)Ó˚ˆÏc ~!àˆÏÎ˚ ÎyÎ˚–
~•z ò)Ó˚c xhsˇÓ˚ ï˛Ó˚ˆÏDÓ˚ ò¢yÓ˚ ˛õ%öÓ˚yÓ,!_ âˆÏê˛– ~•z ò)Ó˚c!ê˛ˆÏܲ Ó°y •Î˚ ï˛Ó˚D˜Ïòâ≈ƒ (wavelength)– §yôyÓ˚îï˛
λ (lamda) !ã˛•´ !òˆÏÎ˚ ~•z ò)Ó˚cˆÏܲ §)!ã˛ï˛ ܲÓ˚y •Î˚– ~Ó˚ ~ܲܲ xÓ¢ƒ•z ˜òˆÏâ≈ƒÓ˚ ~ܲܲ–
xyˆÏàÓ˚ xyˆÏ°yã˛öy ˆÌˆÏܲ xy˛õ!ö !öÿ˛Î˚ Ó%é˛ˆÏï˛•z ˆ˛õˆÏÓ˚ˆÏåÈö ˆÎñ í˛z͈ϧÓ˚ ܲ¡õyB˛ Î!ò n •Î˚ñ xyÓ˚ ≤Ã!ï˛!ê˛
ܲ¡õˆÏöÓ˚ ú˛ˆÏ° Î!ò xyˆÏ°yí˛¸ö λ õ!Ó˚üyî x@ˇÃ§Ó˚ •Î˚ñ ï˛y•ˆÏ° n §ÇáƒÜ˛ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ï˛Ó˚D!ê˛ nλ ò)Ó˚ˆÏc
x@ˇÃ§Ó˚ •ˆÏÓ– xÌ≈yÍñ ~•z ü%•)ˆÏï≈˛ ~áyˆÏö ~ܲ!ê˛ §%Ó˚¢°yܲyˆÏܲ xyâyï˛ Ü˛Ó˚ˆÏ° ~áyöܲyÓ˚ Óyï˛yˆÏ§Ó˚ ˚ ˚ ˚ ˚ ܲîyÎ˚
ˆÎ ܲ¡õö §,!‹T •ˆÏÓñ ï˛y !ë˛Ü˛ ~ܲ ˆ§ˆÏܲˆÏ[˛ ÓyˆÏò nλ ò)Ó˚ˆÏc ˆ˛õÑÔåȈÏÓ– !ܲv ï˛Ó˚D ~ܲ ˆ§ˆÏܲˆÏu˛ ˆÎ ò)Ó˚c
x!ï˛e´ü ܲˆÏÓ˚ ï˛y •ˆÏFåÈ ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà v–
§%ï˛Ó˚yÇñ ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà
v = nλ ...... 6.1
xÌ≈yÍñ ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà= ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ × ï˛Ó˚D˜Ïòâ≈ƒ
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î!ê˛ ï˛Ó˚D˜Ïòâ≈ƒñ ܲ¡õyB˛ Ä ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓˆÏàÓ˚ üˆÏôƒ ˆÎ §¡õÜ≈˛ §)!ã˛ï˛ ܲˆÏÓ˚ˆÏåÈ ï˛yÓ˚ IJõÓ˚
ò%ÈÙÈ~ܲ!ê˛ xö%¢#°ö# ܲˆÏÓ˚ ˆòáy Îyܲ–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ1 : ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 340ms–1 •ˆÏ° 400Hz ܲ¡õyB˛ !Ó!¢‹T ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ
!öî≈Î˚ ܲÓ˚&ö– 300!ê˛ Ü˛¡õö §¡õߨ •ˆÏï˛ ˆÎ §üÎ˚ °yˆÏàñ ï˛yÓ˚ üˆÏôƒ ¢∑ Ü˛ï˛ ò)Ó˚ x@ˇÃ§Ó˚ •ˆÏÓ⁄
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ2 : !ö!ò≈‹T ܲ¡õyB˛!Ó!¢‹T ˆÜ˛yöÄ Ó› ܲ!¡õï˛ •ˆÏ° A üyôƒˆÏü 10cm ˜òˆÏâ≈ƒÓ˚ ~ÓÇ B
üyôƒˆÏü 15 cm ˜òˆÏâ≈ƒÓ˚ ï˛Ó˚D í˛zͲõߨ ܲˆÏÓ˚– A üyôƒˆÏü ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 90cms-1 •ˆÏ°ñ B üyôƒˆÏü ï˛yÓ˚
à!ï˛ˆÏÓà !öî≈Î˚ ܲÓ˚&ö–
~ ˛õÎ≈hsˇ xyüÓ˚y §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ í˛zͧ ~ÓÇ §Ó˚° ˆòy°ï˛Ó˚ˆÏDÓ˚ ܲÌy•z xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ–
ï˛Ó˚D§,!‹TÓ˚ çöƒ ˆÜ˛Ó° í˛z͈ϧÓ˚ §Ó˚° ˆòy°à!ï˛ åÈyí˛¸y xyÓ˚Ä ç!ê˛° ≤ÃÜ,˛!ï˛Ó˚ ܲ¡õöÄ òyÎ˚# •ˆÏï˛ ˛õyˆÏÓ˚– ˆÎüö
!Ó!û˛ß¨ ÓyòƒÎˆÏsf ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ ܲ¡õö ˆÓ¢ ç!ê˛° •ˆÏï˛ ˆòáy ÎyÎ˚– ˆ§ï˛yÓ˚ñ §ˆÏÓ˚yòñ !˛õÎ˚yˆÏöyñ !àê˛yÓ˚ ≤Ãû,˛!ï˛
ΈÏsf ¢∑ §,!‹TÓ˚ çöƒ ï˛yˆÏÓ˚Ó˚ ˛õÎ≈yÓ,_ ܲ¡õö òyÎ˚# •ˆÏ°Ä ˆÜ˛yöÄ ÎˆÏsf•z ˙ ܲ¡õö §Ó˚° ˆòy°à!ï˛ˆÏï˛ âˆÏê˛
öy– ˙ Îsf=!°ˆÏï˛ §,‹T ¢∑ï˛Ó˚ˆÏDÓ˚ xyÜ,˛!ï˛=!°Ä ˆÓ¢ ç!ê˛° •Î˚– 6.2 !ã˛ˆÏe ~•zÓ˚ܲü ܲˆÏÎ˚ܲ!ê˛ ¢∑ï˛Ó˚ˆÏDÓ˚
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§üÎ˚ §Ó˚î ˆ°á!ã˛e ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ï˛ˆÏÓ ˛õÓ˚#«˛yàyˆÏÓ˚ ÓƒÓ•*ï˛ §%Ó˚¢°yܲyÓ˚ ÓyåÈ=!° §Ó˚° ˆòy°à!ï˛ˆÏï˛
xyˆÏ®y!°ï˛ •Î˚ñ ÎyÓ˚ ú˛ˆÏ° §%Ó˚¢°yܲy ˆÌˆÏܲ !öɧ,ï˛ ¢∑ §Ó˚° ˆòy°ï˛Ó˚ˆÏDÓ˚ xyܲyˆÏÓ˚ !ÓhflÏyÓ˚ °yû˛ ܲˆÏÓ˚–
6.3 ï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛òï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛òï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛òï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛òï˛Ó˚ˆÏDÓ˚ ≤ÃܲyÓ˚ˆÏû˛ò
!Ó!û˛ß¨ ò,!‹TˆÏܲyî ˆÌˆÏܲ xyüÓ˚y ï˛Ó˚ˆÏDÓ˚ ˆ◊î#!Óû˛yà ܲÓ˚ˆÏï˛ ˛õy!Ó˚– ˆÎüö ôÓ˚&öñ ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ üyôƒˆÏüÓ˚
≤ÃÜ,˛!ï˛ ˆÌˆÏܲ xyüÓ˚y ~ܲüy!eܲ (one dimensional Óy I.D), !müy!eܲ (two dimensional Óy 2-D) ~ÓÇ
!eüy!eܲ (three dimensional Óy 3-D) ï˛Ó˚D ˆ˛õˆÏï˛ ˛õy!Ó˚– ~ܲ!ê˛ ò!í˛¸ Óy ï˛yˆÏÓ˚Ó˚ Óy òˆÏu˛Ó˚ üˆÏôƒ !òˆÏÎ˚
Îáö ï˛Ó˚D x@ˇÃ§Ó˚ •Î˚ñ ˆ§•z ï˛Ó˚DˆÏܲ xyüÓ˚y ~ܲüy!eܲ ï˛Ó˚D Ó!°– ܲyÓ˚îñ ~•z ï˛Ó˚D ~ܲ!ê˛ ˆÓ˚áy ÓÓ˚yÓÓ˚
ˆÜ˛Ó°‰ !ӈϢ£Ï ~ܲ!ê˛ !òˆÏܲ•z ~!àˆÏÎ˚ ã˛°ˆÏï˛ §«˛ü– xyÓyÓ˚ ç°y¢ˆÏÎ˚ !ì˛° ˆú˛°ˆÏ° ç°ï˛ˆÏ°Ó˚ í˛z˛õÓ˚ ˆÎ ï˛Ó˚D
§,!‹T •Î˚ñ ï˛y !müy!eܲ– ~ˆÏ«˛ˆÏe ï˛Ó˚D ~ܲ!ê˛ ï˛ˆÏ° ≤çyÓ˚°yû˛ ܲˆÏÓ˚– ~•z ôÓ˚ˆÏöÓ˚ ï˛Ó˚DˆÏܲ ˛õ,¤˛ï˛Ó˚D (surface
wave) Ó°y ÎyÎ˚– xyÓyÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ ¢∑ Óy xyˆÏ°yܲ ï˛Ó˚D Îáö §Ó!òˆÏܲ åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸ñ
ï˛áö ˆ§•z ï˛Ó˚DˆÏܲ xyüÓ˚y Ó!° !eüy!eܲ ï˛Ó˚D– xyüÓ˚y ~áyˆÏö xyüyˆÏòÓ˚ xyˆÏ°yã˛öy ~ܲüy!eܲ ï˛Ó˚ˆÏD•z
§#üyÓk˛ Ó˚yáˆÏÓy–
§y•zöôü≈#
§%Ó¢°yܲy
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ï˛Ó˚D §,!‹TÓ˚ çöƒ üyôƒˆÏüÓ˚ ˆÎ ܲ¡õö •Î˚ñ ï˛yÓ˚ ≤ÃÜ,˛!ï˛Ó˚ IJõÓ˚ !û˛!_ ܲˆÏÓ˚Ä ï˛Ó˚ˆÏDÓ˚ ˆ◊î# !Óû˛yà ܲÓ˚y
ÎyÎ˚– ï˛Ó˚D ˆÎ !òˆÏܲ x@ˇÃ§Ó˚ •Î˚ üyôƒˆÏüÓ˚ ܲîy=!° Î!ò ï˛yÓ˚ °¡∫ x!û˛ü%ˆÏá ܲ!¡õï˛ •Î˚ñ ï˛y•ˆÏ° ˆ§•z ï˛Ó˚DˆÏܲ
Ó°y •Î˚ xö%≤Ãfli ï˛Ó˚D (transverse wave)– xy˛õöyÓ˚y !öÿ˛Î˚•z !Ó!û˛ß¨ ê˛yö ˆòÄÎ˚y ï˛yˆÏÓ˚Ó˚ SˆÎüö !àê˛yÓ˚ñ
§ˆÏÓ˚yòñ ˆ§ï˛yÓ˚ •zï˛ƒy!òV ï˛yˆÏÓ˚Ó˚ ܲ¡õö °«˛ ܲˆÏÓ˚ˆÏåÈö– ~ˆÏ«˛ˆÏe ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ x!û˛ü%á ~ÓÇ ï˛Ó˚ˆÏDÓ˚
≤çyˆÏÓ˚Ó˚ x!û˛ü%á ˛õÓ˚flõÓ˚ °¡∫– §%ï˛Ó˚yÇñ ï˛yˆÏÓ˚ ˆÎ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •Î˚ ï˛y xö%≤Ãfli– !ã˛e 6.3 -~ ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ Ü˛#û˛yˆÏÓ ~ܲ!ê˛ !fl±ÇÈÙÈû˛Ó˚ ÓƒÓfliyˆÏï˛ xö%≤Ãfli ï˛Ó˚D §,!‹T ܲÓ˚y ÎyÎ˚–
˙ ~ܲ•z !ã˛ˆÏe [!ã˛e 6.3 (b)] ~ܲ•z ôˆÏÓ˚öÓ˚ !fl±Ç ÈÙÈû˛Ó˚ ÓƒÓfliyÓ˚ §y•yˆÏ΃ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ܲ#û˛yˆÏÓ xö%˜Ïòâ≈ƒ
ï˛Ó˚D (longitudinal wave) §,!‹T •ˆÏï˛ ˛õyˆÏÓ˚– xyÓyÓ˚ Î!ò ˆÜ˛yöÄ ï˛Ó˚ˆÏD üyôƒˆÏüÓ˚ ܲ¡õö¢#° ܲîy=!°Ó˚
ܲ¡õˆÏöÓ˚ x!û˛ü%á ~ÓÇ ï˛Ó˚D ≤çyˆÏÓ˚Ó˚ x!û˛ü%á x!û˛ß¨ •Î˚ñ ï˛y•ˆÏ° ˆ§•z ï˛Ó˚DˆÏܲ xyüÓ˚y xö%˜Ïòâ≈ƒ ï˛Ó˚D
Ó!°– 6.3 (b) !ã˛ˆÏe xyˆÏàÓ˚ !fl±Ç ÈÙÈû˛Ó˚ ÓƒÓfliy!ê˛ˆÏï˛ Ü˛#û˛yˆÏÓ xö%˜Ïòâ≈ƒ ï˛Ó˚D ã˛°ˆÏï˛ ˛õyˆÏÓ˚ñ ï˛y ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
~áyˆÏö ~ܲ!ê˛ û˛Ó˚ˆÏܲ !fl±Ç ~Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ï˛yÓ˚ §yüƒ xÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ ˆåȈÏí˛¸ !òˆÏ° !fl±Ç û˛Ó˚
¢,C° ÓÓ˚yÓÓ˚ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ §,!‹T •ˆÏÓ–
ï˛Ó˚D Óy àƒy§#Î˚ üyôƒˆÏü ˆÜ˛Ó° xö%˜Ïòâ≈ƒ ï˛Ó˚D•z §MÈ˛y!Ó˚ï˛ •ˆÏï˛ ˛õyˆÏÓ˚ ˆÜ˛ööyñ xö%≤Ãfli ï˛Ó˚D §,!‹T ܲÓ˚ˆÏï˛
üyôƒˆÏüÓ˚ ~ܲ!ê˛ hflÏÓ˚ˆÏܲ ˛õˆÏÓ˚Ó˚ hflψÏÓ˚Ó˚ í˛z˛õÓ˚ Ü,˛hsˇö ˛õ#í˛¸ö (shearing stress)≤ÈÏÎ˚yà ܲÓ˚ˆÏï˛ •Î˚ñ Îy ≤ÃÓy•#
üyôƒˆÏü §Ω˛Ó •Î˚ öy– ܲ!ë˛ö üyôƒˆÏü xÓ¢ƒ xö%˜Ïòâ≈ƒ Ä xö%≤Ãfli ÈÙÙÙÈò%•z ≤ÃܲyÓ˚ ï˛Ó˚D•z ã˛°ˆÏï˛ ˛õyˆÏÓ˚–
xyÓ˚ ~ܲ!ê˛ !ӣψÏÎ˚ xy˛õöyÓ˚ üˆÏö ≤ß¿ çyày fl∫yû˛y!Óܲ– ˆ§!ê˛ •°ñ ï˛!í˛¸Í ã%˛¡∫ܲ#Î˚ ï˛Ó˚DˆÏܲ ˆÜ˛yö çy!ï˛Ó˚
ï˛Ó˚D Ó°y ÎyÎ˚– ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ ï˛Ó˚D Îy!sfܲ ï˛Ó˚D öÎ˚ñ ˆÜ˛yöÄ Ó›Ü˛îyÓ˚ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ~•z ï˛Ó˚ˆÏDÓ˚ í˛zqÓ
•Î˚ öy– ~•z ï˛Ó˚ˆÏDÓ˚ ˛õˆÏÌ ≤Ã!ï˛!ê˛ !Ó®%ˆÏï˛ ï˛Ó˚Dà!ï˛Ó˚ °¡∫ x!û˛ü%ˆÏá ï˛!í˛¸ÍˆÏ«˛e Ä ˆã˛Ô¡∫ܲˆÏ«˛ˆÏeÓ˚ ï˛#Ó ï˛y
xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyˆÏܲ– ~çöƒ ~•z ï˛Ó˚DˆÏܲ xyüÓ˚y xö%≤Ãfli ï˛Ó˚D Ó!°–
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xyÓyÓ˚ §¡õ)î≈ ~ܲ!ê˛ !û˛ß¨ !ÓˆÏÓã˛öy ˆÌˆÏܲ Îy!sfܲ ï˛Ó˚ˆÏDÓ˚ ˆ◊î# !Óû˛yà ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– ˆÎüöñ xyüÓ˚y
ܲyˆÏö ˆÎ ˆÜ˛yöÄ Ü˛¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏï˛ ˛õy•z öy– !öˆÏã˛Ó˚ !òˆÏܲ 20Hz xyÓ˚ IJõˆÏÓ˚Ó˚ !òˆÏܲ 20kHz S!ܲˆÏ°y
•yÍ≈§‰V ܲ¡õyˆÏB˛Ó˚ ¢∑ ˛õÎ≈hsˇ xyüÓ˚y ܲyˆÏö ÷öˆÏï˛ ˛õy•z– Î!ò ˆÜ˛yö ¢∑ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ 20Hz-~Ó˚ !öˆÏã˛
•Î˚ñ ï˛y•ˆÏ° ˆ§•z ¢∑ˆÏܲ Ó°y •Î˚ xÓ¢∑ (infrasonic wave)– xöƒ!òˆÏܲñ ˆÜ˛yöÄ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ Î!ò
20kHz ~Ó˚ ˆÌˆÏܲ ˆÓ!¢ •Î˚ñ ï˛y•ˆÏ° ï˛yˆÏܲ xyüÓ˚y Ó!° x!ï˛¢∑ (ultrasonic wave)– xy˛õöyÓ˚ •Î˚ï˛
çyöy xyˆÏåÈñ !ã˛!ܲͧy !ÓK˛yˆÏö Ä xöƒe x!ï˛¢∑ˆÏܲ öyöy ܲyˆÏç °yàyˆÏöy •Î˚– ~•z §¡õˆÏÜ≈˛ myò¢ ~ܲˆÏܲ
!Óhfl,Ïï˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÓ–
ï˛yåÈyí˛¸y ˆÜ˛yöÄ !eüy!eܲ ï˛Ó˚D Îáö !ÓhflÏyÓ˚ °yû˛ ܲˆÏÓ˚ñ ï˛áö üyôƒˆÏüÓ˚ §üò¢yÎ˚ ܲ!¡õï˛ ˛õy¢y˛õy!¢
!Ó®%=!° Î!ò ~ܲ!ê˛ §üï˛ˆÏ° ÌyˆÏܲñ ï˛ˆÏÓ ï˛yˆÏܲ §üï˛° ï˛Ó˚D (plane wave)Ó°y •Î˚– xyÓ˚ Î!ò ˙ !Ó®%=!°
ˆÜ˛yöÄ ˆày°ˆÏܲÏÓ˚ (sphere) ï˛ˆÏ° ÌyˆÏܲñ ï˛ˆÏÓ ˆ§•z ï˛Ó˚DˆÏܲ xyüÓ˚y ˆày°#Î˚ ï˛Ó˚D (spherical wave)Ó!°–
xÓ¢ƒ ~•z ~ܲˆÏܲ xyüyˆÏòÓ˚ xyˆÏ°yã˛öy ~ܲüy!eܲ (one dimensional) ï˛Ó˚ˆÏD•z §#üyÓk˛ ÌyܲˆÏÓ ~ÓÇ
~ˆÏ«˛ˆÏe §üò¢yÎ˚ ܲ!¡õï˛ ˆÜ˛yö §!ߨ!•ï˛ !Ó®% öy ÌyܲyÎ˚ñ ~áyˆÏö í˛z˛õˆÏÓ˚Ó˚ ˆ◊î# !Óû˛yà!ê˛ Ü˛yÎ≈ܲÓ˚# •ˆÏÓ
öy–
6.4 ï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚yï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚yï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚yï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚yï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ˆÏîÓ˚ ≤Ã!e´Î˚y
ï˛Ó˚D !ë˛Ü˛ ܲ#û˛yˆÏÓ §MÈ˛y!Ó˚ï˛ •Î˚ñ ï˛y ˆÓyé˛yˆÏöyÓ˚ çöƒ
xyüÓ˚y ܲˆÏÎ˚ܲ!ê˛ !ã˛ˆÏeÓ˚ §y•yˆÏ΃ ˆöÓ– ≤ÃÌü !ã˛ˆÏe S!ãe 6.4)
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈñ ~ܲ!ê˛ °¡∫y ò!í˛¸Ó˚ ~ܲ ≤Ãyhsˇ ˆòÄÎ˚yˆÏ° ˆÓшÏô
xöƒ ≤Ãyhsˇ •yˆÏï˛Ó˚ ü%ˆÏë˛yÎ˚ ôˆÏÓ˚ í˛z˛õÓ˚ ÈÙÈ!öˆÏã˛ éÑ˛yܲyˆÏ° ò!í˛¸Ó˚
üˆÏôƒ !òˆÏÎ˚ ~ܲ!ê˛ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •ˆÏÓ– ~!ê˛ xy˛õ!ö !öˆÏç
~ܲÓyÓ˚ ܲˆÏÓ˚ ˆòá%ö– °«˛ƒ ܲÓ˚&ö xy˛õöyÓ˚ •yï˛ Îáö ï˛yÓ˚
fl∫yû˛y!Óܲ xÓfliyˆÏöÓ˚ ò%’!òˆÏܲ ˛õÎ≈yÓ,_(periodic) à!ï˛ˆÏï˛
Äë˛yöyüy ܲÓ˚ˆÏåÈñ ò!í˛¸Ó˚ !Ó!û˛ß¨ xÇ¢=!°Ä ~ܲ•z Ó˚ܲü
xyã˛Ó˚î ܲÓ˚ˆÏåÈ– xyˆÏ°yí˛¸ö!ê˛ §yü!@ˇÃܲû˛yˆÏÓ §y•zö ï˛Ó˚ˆÏDÓ˚
xyܲyˆÏÓ˚ ˆòÄÎ˚yˆÏ°Ó˚ !òˆÏܲ ~!àˆÏÎ˚ ÎyˆÏFåÈñ Îï˛«˛î öy ˆ§áyö
ˆÌˆÏܲ ï˛yÓ˚ ≤Ã!ï˛ú˛°ö âˆÏê˛– 6.4 !ã˛ˆÏe •yï˛!ê˛ í˛z˛õÓ˚ ÈÙÈ!öã˛
ܲÓ˚ˆÏ° ܲ#û˛yˆÏÓ ò!í˛¸ˆÏï˛ ï˛Ó˚D í˛zͲõߨ •ˆÏÓñ ò!í˛¸!ê˛Ó˚ ≤ÃÌü
ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ öÎ˚!ê˛ !Ó®%Ó˚ §Ó˚î Ä à!ï˛Ó˚ üyôƒˆÏü ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲˆÏÓ˚ ˆòá%öñ ≤Ã!ï˛!ê˛ !Ó®% ˛õÓ˚Óï≈˛# !Ó®%Ó˚ §ˆÏD
Î%!@¬ï˛ •ÄÎ˚yÎ˚ ˆ§!ê˛Ó˚ §Ó˚î ˛õÓ˚Óï≈˛# !Ó®%Ó˚ à!ï˛ˆÏܲ ≤Ãû˛y!Óï˛ Ü˛Ó˚ˆÏåÈ– ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚°yˆÏû˛Ó˚ !Ó!û˛ß¨ hflÏÓ˚=!°
6.5 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– °«˛ƒ ܲÓ˚&öñ §üˆÏÎ˚Ó˚ §ˆÏD ï˛Ó˚ˆÏDÓ˚ §ü@ˇÃ xyÜ,˛!ï˛ Ü˛#û˛yˆÏÓ x@ˇÃ§Ó˚ •ˆÏFåÈ–
158
6.5 ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ
~Ó˚ xyˆÏàÓ˚ xö%ˆÏFåȈÏò xy˛õ!ö ï˛Ó˚ˆÏDÓ˚ í˛zͲõ!_ ܲ#û˛yˆÏÓ •Î˚ ~ÓÇ ï˛Ó˚D ܲ#û˛yˆÏÓ x@ˇÃ§Ó˚ •Î˚ ˆ§ §¡∫ˆÏ¶˛
!ܲå%Èê˛y ôyÓ˚îy °yû˛ ܲˆÏÓ˚ˆÏåÈö– !ܲv ˛õòyÌ≈!ÓòƒyÓ˚ xyˆÏ°yã˛öyÓ˚ çöƒ ï˛Ó˚DˆÏܲ ày!î!ï˛Ü˛û˛yˆÏÓ ≤Ãܲy¢ ܲÓ˚yÄ
xï˛ƒhsˇ ≤ÈÏÎ˚yçö#Î˚– ~•z ˛õÎ≈yˆÏÎ˚Ó˚ ≤ÃÌü ܲˆÏÎ˚ܲ!ê˛ ~ܲˆÏܲ §Ó˚° ˆòy°à!ï˛ˆÏܲ ày!î!ï˛Ü˛û˛yˆÏÓ ≤Ãܲy¢ ܲˆÏÓ˚ ï˛yÓ˚
§¡∫ˆÏ¶˛ öyöy xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö xyüÓ˚y §Ó˚° ˆòy°ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ˆ§•z xyˆÏ°yã˛öyÓ˚•z §ÇˆÏÎyçö
ܲÓ˚Ó–
xyˆÏ°yã˛öyÓ˚ §%!ÓôyÓ˚ çöƒ ~ܲ!ê˛ ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ܲÌy !ÓˆÏÓã˛öy ܲÓ˚&ö– ôÓ˚&öñ ï˛Ó˚D!ê˛ x x«˛ x!û˛ü%ˆÏá
x@ˇÃ§Ó˚ •ˆÏFåÈ t §üˆÏÎ˚ ~ÓÇ üyôƒˆÏüÓ˚ ˆÎ ˆÜ˛yöÄ Ü˛îyÓ˚ §yüƒyÓfliyö ˆÌˆÏܲ §Ó˚î ψ (x,t)– xy˛õ!ö Î!ò ÷ô%
x = 0 !Ó®%ˆÏï˛ ˆÎ ܲîy!ê˛ xyˆÏåÈ ï˛yÓ˚ °«˛ƒ Ó˚yˆÏáöñ ï˛ˆÏÓ xy˛õ!ö ˆ§!ê˛Ó˚ §üÎ˚ §Ó˚î §ü#ܲÓ˚î ˛õyˆÏÓö
()() 0,sin tAwt =+ ψφ
........ 6.2
ˆÎáyˆÏö A = ܲîy!ê˛Ó˚˛ §Ó˚° ˆòy°à!ï˛Ó˚ !ÓhflÏyÓ˚ ~ÓÇ
ω=
˙ §Ó˚° ˆòy°à!ï˛Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ñ φ
ˆÜ˛yî!ê˛ ï˛Ó˚ˆÏDÓ˚ ≤ÃyÓ˚!Ω˛Ü˛ ò¢yˆÏܲyî– í˛z˛õÎ%_´ ü%•)ˆÏï≈˛ §üˆÏÎ˚Ó˚ àîöy ÷Ó˚& ܲÓ˚ˆÏ° φ ܲyîˆÏܲ ¢)öƒ ôÓ˚y ÎyÎ˚–
6.2 §ü#ܲÓ˚î!ê˛Ó˚ Ó˚*˛õ ï˛áö •ˆÏÓñ
ψ (0,t) = A sin
t ω
........ 6.3
159
~áö ≤ß¿ ~•z ˆÎñ t §üˆÏÎ˚
xx =Δ
!Ó®%ˆÏï˛ §Ó˚î Ü˛ï˛ •ˆÏÓñ xÌ≈yÍ
() , yxt Δ
ˆÜ˛ ܲ#û˛yˆÏÓ ˆ°áy ÎyˆÏÓ–
t §üˆÏÎ˚ x x= Δ !Ó®%ˆÏï˛ ˆÎ §Ó˚î •ˆÏÓ ï˛y !öÿ˛Î˚•z
xt
vΔ −
§üˆÏÎ˚ x = 0 !Ó®%ˆÏï˛ §Ó˚ˆÏîÓ˚ §üyö •ˆÏÓ–
ˆÜ˛ö öy ˙ §Ó˚î x
vΔ §üÎ˚ ˛õˆÏÓ˚ xÌ≈yÍñ t §üˆÏÎ˚
. xvx
vΔ=Δ
ò)Ó˚c x!ï˛e´ü ܲˆÏÓ˚
xx =Δ
!Ó®%ˆÏï˛ ˆ˛õÑÔåȈÏÓ–
() ,0,sin xxxtytAt
vv∆∆ ⎛⎞⎛⎞ ∴∆=−=− ⎝⎠⎝⎠ ω
ˆÎˆÏ•ï%˛ Δx •zFåÈyüï˛ ˆÓˆÏåÈ ˆöÄÎ˚y ~ܲ!ê˛ ò)Ó˚cñ xï˛~Ó xyüÓ˚y §yôyÓ˚îû˛yˆÏÓ !°áˆÏï˛ ˛õy!Ó˚ñ
( ), sin xx t A t
v⎛ ⎞= −⎝ ⎠ψ ω ........ 6.4
6.4 §ü#ܲÓ˚î!ê˛ §Ó˚° ˆòy°ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚&˛õ– ~!ê˛ˆÏܲ xÓ¢ƒ öyöy û˛yˆÏÓ ˆ°áy ÎyÎ˚ ˆÎüö:
( ), sin 2 xx t A v t
v⎛ ⎞= −⎝ ⎠ψ π ∴ 2 ,vω π= ˆÎáyˆÏöñ v = ܲ¡õyB˛ ........ 6.5(a)
=
() 2 sin Avtx − πλ
∴
vvλ =
........ 6.5(b)
= sin2txA
T⎛⎞ − ⎝⎠ πλ
∴ 1, vvTT
λ == ........ 6.5(c)
=
() sin Atkx ω−
ˆÎáyˆÏö 2
k = πλ S§MÈ˛yÓ˚î ô ÓܲV ........ 6.5(d)
§ü#ܲÓ˚î 6.5 -~ ò)Ó˚c x Ä §üÎ˚ t ~Ó˚ §ˆÏD ï˛Ó˚ˆÏDÓ˚ §Ó˚î ˛õ!Ó˚!fli!ï˛ !ӈϟ’£Ïî ܲÓ˚y Îyܲ– y ~Ó˚ ˆÎ
˛õ!Ó˚Óï≈˛ö §)!ã˛ï˛ •ˆÏÎ˚ˆÏåÈñ ï˛y ˆ°á!ã˛ˆÏeÓ˚ §y•yˆÏ΃ ˆòáyˆÏöy ˆÎˆÏï˛ ˛õyˆÏÓ˚– xy˛õ!ö Î!ò ï˛Ó˚D ˛õˆÏÌÓ˚ !ö!ò≈‹T
!Ó®%ˆÏï˛ñ xÌ≈yÍ x ~Ó˚ !ö!ò≈‹T üyˆÏöÓ˚ çöƒ
tψ−
~ÓÇ ˆÜ˛yöÄ ~ܲ !ö!ò≈‹T ü)•)ï≈˛ xÌ≈yÍñ t ~Ó˚ !ö!ò≈‹T üyˆÏöÓ˚
çöƒ
xψ−
ˆ°á!ã˛ˆÏeÓ˚ xB˛ö ܲˆÏÓ˚öñ ï˛y•ˆÏ° xy˛õ!ö ÎÌye´ˆÏü 6.5 (a) Ä (b) ~Ó˚ üï˛ ò%!ê˛ ˆ°á!ã˛e ˛õyˆÏÓö–
tψ−
ˆ°á!ã˛ˆÏe ï˛Ó˚ˆÏDÓ˚ ~ܲ ˛õÎ≈yÎ˚ܲy° T xhsˇÓ˚ ~ܲ•z ò¢y !ú˛ˆÏÓ˚ xy§ˆÏÓ– xÌ≈yÍñ §Ó˚ˆÏîÓ˚ ˛õÓ˚˛õÓ˚ ò%•z à!Ó˚¤˛
Óy ò%•z °!⤲ üyˆÏöÓ˚ üˆÏôƒ §üˆÏÎ˚Ó˚ ÓƒÓôyö •ˆÏÓ T– xöƒ!òˆÏܲñ
xψ−
ˆ°á!ã˛ˆÏe ï˛Ó˚D ˛õˆÏÌÓ˚ ˛õÓ˚˛õÓ˚ ˆÎ
ò%•z !Ó®%ˆÏï˛ ~ܲ•z ò¢y ˆòáˆÏï˛ ˛õyÄÎ˚y ÎyˆÏÓñ ï˛yˆÏòÓ˚ üˆÏôƒ ò)Ó˚c •ˆÏÓ ï˛Ó˚D˜Ïòâ≈ƒ λ- ~Ó˚ §üyö–
xyüÓ˚y ~ ˛õÎ≈hsˇ ~ܲüy!eܲ ï˛Ó˚D!ê˛ ˆÜ˛Ó° x xˆÏ«˛Ó˚ ôöydܲ !òˆÏܲ•z §MÈ˛y!Ó˚ï˛ •ˆÏFåÈ ÓˆÏ° ôˆÏÓ˚ !öˆÏÎ˚!åÈ–
!ܲv ~•z ~ܲ•z ï˛Ó˚ˆÏDÓ˚ §MÈ˛yÓ˚ x xˆÏ«˛Ó˚ îydܲ !òˆÏÜ˛Ä âê˛ˆÏï˛ ˛õyˆÏÓ˚– xÌ≈yÍñ x x«˛ x!û˛ü%ˆÏá ï˛Ó˚ˆÏDÓ˚
ˆÓà v ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ -v •ˆÏï˛ ˛õyˆÏÓ˚– xÌ≈yÍ
6.4 §ü#ܲÓ˚ˆÏî v ~Ó˚ çyÎ˚àyÎ˚ -v Ó§yˆÏ° ˛õyÄÎ˚y ÎyˆÏÓ–
160
() ,sinxxtAt
v⎛⎞ =+ ⎝⎠ ψω
........ 6.6
~áö 6.5 (a) –(d) §ü#ܲÓ˚î=!°Ó˚ §üï%˛°ƒ §ü#ܲÓ˚î=!° ˆ°áy ÎyˆÏÓ:
( ), sin 2 xx t A v t
v⎛ ⎞= +⎝ ⎠ψ π ........ 6.7 (a)
= ( )2sinA vt x+πλ ........ 6.7(b)
=
sin2txA
T⎛⎞ + ⎝⎠ πλ
........ 6.7 (c)
=
() sin Atkx + ω
........ 6.7 (d)
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ñ 6.4 ˆÌˆÏܲ 6.7 §ü#ܲÓ˚î=!°ˆÏï˛ §y•zö xˆÏ˛õ«˛ˆÏܲÓ˚ ˛õ!Ó˚ÓˆÏï≈˛ ˆÜ˛y§y•zö
xˆÏ˛õ«˛Ü˛ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚ !ܲ öy– ~ !Ó£ÏÎ˚!ê˛ §¡õ)î≈•z ≤ÃyÌ!üܲ §#üy ¢ˆÏï≈˛Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– 6.4 §ü#ܲÓ˚î
( ), sin xx t A t
v⎛ ⎞= −⎝ ⎠ψ ω
ˆÌˆÏܲ xy˛õ!ö ˛õyˆÏÓö ( )0, 0 0x tψ = = = ~ÓÇ
() 0,0 dxxtA
dtω ===
xÌ≈yÍñ ≤ÃyÌ!üܲ xÓfliyÎ˚ §Ó˚ˆÏîÓ˚ üyö ¢)öƒ !ܲvñ üyôƒˆÏüÓ˚ ӛܲîyÓ˚ ˆÓà §ˆÏÓ≈yFã˛– Î!ò
( ), cos xx t A t
v⎛ ⎞= −⎝ ⎠ψ ω ˆ°áy •Î˚ ï˛ˆÏÓñ
( )0, 0x t Aψ = = =
~ÓÇ,
() 0,00 dxxt
dt===
xÌ≈yÍñ ~ˆÏ«˛ˆÏe ≤ÃyÌ!üܲ xÓfliyÎ˚ §Ó˚î §ˆÏÓ≈yFã˛ñ !ܲv ӛܲîyÓ˚ ˆÓà ¢)öƒ–
§yôyÓ˚îû˛yˆÏÓñ Î!ò
sinxAt
v⎡⎤ ⎛⎞ =−+ ⎢⎥ ⎝⎠ ⎣⎦
ψωφ
ˆ°áy •Î˚ñ ï˛ˆÏÓ
()0 0,0sin xtA ==== ψφψ
~ÓÇñ
()0 0,0cos dxxtAu
dt==== ωφ
ôÓ˚&ö–
161
ψ0 ~ÓÇ u0 Î!ò !ö!ò≈‹T ÌyˆÏܲñ ï˛ˆÏÓ A ~ÓÇ φ ò%•z•z çyöy ÎyˆÏÓ ˆÜ˛ööyñ
12 2
2 020
uA
⎛ ⎞= +⎜ ⎟⎝ ⎠
ψω ........ 6.8 (a)
~ÓÇñ 0
0
tanu
= ωψφ ........ 6.8 (b)
ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ ≤ÃܲyˆÏ¢Ó˚ xyÓ˚ ~ܲ!ê˛ ˛õk˛!ï˛Ä ≤ÃyÎ˚•z ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– xy˛õ!ö çyˆÏöö ˆÎñ
( )cos sin 1je j j= + = −θ θ θ – §%ï˛Ó˚yÇ, ( )t kxjAe ω φ− + Ó˚y!¢!ê˛Ó˚ ÓyhflÏÓ xÇ¢
() cos Atkx −+ ωφ
~ÓÇ Ü˛y“!öܲ xÇ¢
() sin jAwtkx −+φ
– ~çöƒ
()() ,tkx j xtAe−+ =ωφ ψ
– .............. 6.9
§¡õÜ≈˛!ê˛ ï˛Ó˚ˆÏDÓ˚ §Ó˚î §ü#ܲÓ˚î !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ôˆÏÓ˚ !öˆÏï˛ •Î˚ ˆÎñ í˛yö!òˆÏܲÓ˚ Ó˚y!¢Ó˚ ÓyhflÏÓ
xÇ¢•z ï˛Ó˚ˆÏDÓ˚ §Ó˚î !öˆÏò≈¢ ܲÓ˚ˆÏåÈ–
!müy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!müy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!müy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!müy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!müy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y
~ ˛õÎ≈hsˇ xyüÓ˚y ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ܲÌy•z !ÓˆÏÓã˛öy ܲÓ˚°yü– !müy!eܲ Óy !eüy!eܲ üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚
ày!î!ï˛Ü˛ Ó˚*˛õ ˆÜ˛üö •ˆÏÓ⁄ xy§%öñ ~ÓyÓ˚ xyüÓ˚y ˆò!á ~ܲüy!eܲ
ï˛Ó˚ˆÏDÓ˚ çöƒ xyüÓ˚y ˆÎ ày!î!ï˛Ü˛ Ó˚*˛õ ≤Ã!ï˛¤˛y ܲˆÏÓ˚!åÈ S§ü#ܲÓ˚î 6.4),
ˆ§!ê˛ ~§Ó ˆ«˛ˆÏeÓ˚ çöƒ ˛õ!Ó˚Ó!ï≈˛ï˛ ܲÓ˚y ÎyÎ˚ !ܲ öy–
≤Ã̈Ïü !müy!eܲ ï˛Ó˚ˆÏDÓ˚ ܲÌy•z !ÓˆÏÓã˛öy ܲÓ˚y Îyܲ– ôÓ˚&ö x–y
ï˛ˆÏ° ~ܲ!ê˛ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •ˆÏFåÈ ÎyÓ˚ §üò¢yÓ˚ !Ó®%=!° ~ܲ
§Ó˚°ˆÏÓ˚áyÎ˚ xÓ!fliï˛– ~!ê˛ˆÏܲ !eüy!eܲ §üï˛° ï˛Ó˚ˆÏDÓ˚ !müy!eܲ
≤Ã!ï˛Ó˚*˛õ ӈϰ û˛yÓˆÏï˛ ˛õyˆÏÓ˚ö– §üò¢yÓ˚ §Ó˚°ˆÏÓ˚áy=!°ˆÏܲ xyüÓ˚y
ï˛Ó˚Dü)á (wave front) ~ÓÇ ˆ§=!°Ó˚ §ˆÏD °¡∫ §Ó˚°ˆÏÓ˚áyˆÏܲ xyüÓ˚y
ï˛Ó˚D °¡∫ (wave normal) Ó!°– 6.6 !ã˛ˆÏe F1,F2 ≤Ãû, !ï˛
§Ó˚°ˆÏÓ˚áy=!° ï˛Ó˚Dü%á ~ÓÇ ON Óy ï˛yÓ˚ §üyhsˇÓ˚y° ˆÎ ˆÜ˛yöÄ
§Ó˚°ˆÏÓ˚áy ï˛Ó˚D °¡∫–
~áö xyüÓ˚y r fliyöyˆÏB˛ P !Ó®%ˆÏï˛ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ!ê˛
!°áˆÏï˛ ã˛y•z– ôÓ˚y Îyܲñ ü)°!Ó®% O ˆï˛ 6.3 §ü#ܲÓ˚î!ê˛ ï˛Ó˚ˆÏDÓ˚
§ü#ܲÓ˚îˆÏܲ ˆÓyé˛yÎ˚– xÌ≈yÍñ
() 0,sin tAt ψω =
162
Î!ò ÷ô% ON §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ï˛Ó˚D!ê˛Ó˚ §MÈ˛yÓ˚î °«˛ƒ ܲÓ˚y ÎyÎ˚ñ ï˛ˆÏÓ Q !Ó®%ˆÏï˛ 6.4 §ü#ܲÓ˚î xö%§yˆÏÓ˚
ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ!ê˛ ˆ°áy ÎyˆÏÓ
( ) 0, sin QOQ t A t
v⎛ ⎞= −⎝ ⎠ψ ω
ˆÜ˛ööy O !Ó®%ˆÏï˛ ï˛Ó˚Dü%ˆÏáÓ˚ ~ܲ!ê˛ «%˛o xÇ¢ˆÏܲ xö%§Ó˚î ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆ§!ê˛•z !ܲå%È«˛î ˛õˆÏÓ˚ Q
!Ó®%ˆÏï˛ í˛z˛õ!fliï˛ •ˆÏÎ˚ˆÏåÈ– P Ä Q !Ó®% ~ܲ•z ï˛Ó˚Dü%ˆÏáÓ˚ í˛z˛õÓ˚ xÓ!fliï˛ •ÄÎ˚yÎ˚ñ ~•z !Ó®% ò%!ê˛ˆÏï˛Ä ï˛Ó˚ˆÏDÓ˚
ò¢y ~ܲ•zñ §%ï˛Ó˚yÇ P !Ó®%ˆÏï˛Ä ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î ~ܲ•z •ˆÏÓ– !ܲv cos ,= =OQ OP r nθ ˆÎáyˆÏö
n
=
ï˛Ó˚D °¡∫ x!û˛ü%á# ~ܲܲ ˆû˛QÓ˚ ~ÓÇ θ ~áyˆÏö
r
Ä
n
~Ó˚ xhsˇû%≈˛_´ ˆÜ˛yî– ~ÓyÓ˚ xyüÓ˚y P !Ó®%ˆÏï˛
!müy!eܲ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ!ê˛ !°áˆÏï˛ ˛õy!Ó˚ :
(). ,sinrnrtAt
v⎛⎞ =− ⎝⎠ ψω
........ 6.10
~áö x˛õ!ö x ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ r n !°ˆÏá 6.5 (a – b), 6.6 Óy 6.7 (a – b) §ü#ܲÓ˚î=!°Ó˚ !müy!eܲ
≤Ã!ï˛Ó˚*˛õ !öˆÏç•z !°áˆÏï˛ ˛õyÓ˚ˆÏÓö–
!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°y
!müy!eܲ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe xyüÓ˚y ˆÎ Î%!_´ xö%§Ó˚î ܲˆÏÓ˚!åÈñ
!eüy!eܲ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÄ ˆ§!ê˛ ≤ÈÏÎyçƒ •ˆÏÓ– ABC §üï˛°!ê˛
ï˛Ó˚Dü%áñ ON ï˛Ó˚D°¡∫– OP Î!ò
r
•Î˚ñ xÌ≈yÍ P !Ó®%Ó˚ fliyöyB˛
ˆû˛QÓ˚ Î!ò
r
•Î˚ ï˛ˆÏÓñ OQ = ˆ§áyˆÏöñ
rn
ˆ§áyˆÏö
n
xy ÏàÓ˚
üï˛ ï˛Ó˚D°¡∫ ON ÓÓ˚yÓÓ˚ ~ܲܲ ˆû˛QÓ˚– §%ï˛Ó˚yÇñ ~ˆÏ«˛ˆÏeˆ 6.10
§ü#ܲÓ˚îñ xÌ≈yÍ
(). ,sinrnrtAt
v⎛⎞ =− ⎝⎠ ψω
§ü#ܲÓ˚î!ê˛•z •ˆÏÓ !eüy!eܲ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ–
6.5.1 ï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢yÈÙÈ˛õyÌ≈ܲƒï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢yÈÙÈ˛õyÌ≈ܲƒï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢yÈÙÈ˛õyÌ≈ܲƒï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢yÈÙÈ˛õyÌ≈ܲƒï˛Ó˚ˆÏDÓ˚ ò¢y Ä ò¢yÈÙÈ˛õyÌ≈ܲƒ
ã˛°ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*ˆÏ˛õÓ˚ §ˆÏD xy˛õ!ö !ܲå%Èê˛y ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö– ï˛Ó˚ˆÏDÓ˚ Ó˚y!¢üy°yÓ˚ §y•yˆÏ΃ ~áö
xy˛õ!ö §•ˆÏç•z ï˛Ó˚D ˛õˆÏÌÓ˚ ò%•z !û˛ß¨ !Ó®% xÌÓy ~ܲ•z !Ó®%ˆÏï˛ ò%•z !û˛ß¨ §üˆÏÎ˚Ó˚ üˆÏôƒ ò¢y ˛õyÌ≈ܲƒ !öî≈Î˚
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– 6.5 (b) §ü#ܲÓ˚ˆÏî xyüÓ˚y ˆ˛õˆÏÎ˚!åÈ :
( ) ( )2, sinx t A vt x= −πψ λ
163
=
sin, Aδ
ˆ§áyˆÏö
δ
ò¢yˆÏܲyî–
°«˛ƒ ܲÓ˚&öñ ~áyˆÏö t §üˆÏÎ˚ ü)°!Ó®% ˆÌˆÏܲ x ò)Ó˚ˆÏc ò¢yˆÏܲyî
() 2vtx =− π δλ
!ӈϢ£Ïû˛yˆÏÓ ˆáÎ˚y° Ó˚yáy òÓ˚ܲyÓ˚ ˆÎñ ~•z ò¢yÓ˚ ò%!ê˛ xÇ¢
2vt
πλ
§üˆÏÎ˚Ó˚ §ˆÏD ~ÓÇ
2x
πλ
ò)Ó˚ˆÏcÓ˚
§ˆÏD ˛õ!Ó˚Óï≈˛ö¢#°– ôÓ˚y Îyܲñ ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ˛õˆÏÌ P Ä Q !Ó®%Ó˚ ò)Ó˚c ü)°!Ó®% ˆÌˆÏܲ ÎÌye´ˆÏü
1 x
~ÓÇ x2 S!ã˛e 6.8) ~ܲ!ê˛ !ӈϢ£Ï ü%•)ˆÏï≈˛ xÌ≈yÍñ ‘t’ ~Ó˚ ~ܲ!ê˛ !ӈϢ£Ï üyˆÏöÓ˚ çöƒ P !Ó®%ˆÏï˛ ï˛Ó˚ˆÏDÓ˚
ò¢y
122 txt
T=− ππ δλ
˙ ~ܲ•z ü%•)ˆÏï≈˛
Q !Ó®%ˆÏï˛ ï˛Ó˚ˆÏDÓ˚ ò¢y
22
2 2x tT
=−π π δλ
˙ !ӈϢ£Ï ü%•)ˆÏï≈˛ P Ä Q !Ó®%Ó˚ üˆÏôƒ ò¢y ÈÙÈ˛õyÌ≈ܲƒ
() 12212
xxπ δδλ −=−
∴ 2πδ λΔ = × !Ó®% ò%!ê˛Ó˚ ˛õÌ ˛õyÌ≈ܲƒ ............6.11
°«˛ƒ ܲÓ˚&ö ˆÎñ ~•z ò¢y ˛õyÌ≈ܲƒ §üˆÏÎ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#° öÎ˚–
ò%!ê˛ !Ó®%Ó˚ üˆÏôƒ ˛˛õÌ ˛õyÌ≈ܲƒ
2λ
-~Ó˚ ˆÜ˛yöÄ Î%@¬ =!îï˛Ü˛ (even multiple) •ˆÏ°Ä ò¢y ˛õyÌ≈ܲƒ =
2.222
ssπλπ λ=
~áyˆÏö s ~ܲ!ê˛ ˛õ)î≈ §Çáƒy– ò¢ ˛õyÌ≈ܲƒ 2sπ •ÄÎ˚yÓ˚ xÌ≈ñ ò¢yÓ˚ ˛õyÌ≈ܲƒ
2π
~Ó˚ xá[˛ =!îï˛Ü˛– ~Ó˚
ú˛ˆÏ° ˙ ò%•z !Ó®%ˆÏï˛ ï˛Ó˚D!ê˛ §üò¢y §¡õߨ •ˆÏÓ–
xöƒ!òˆÏܲñ Î!ò ˛õÌ ˛õyÌ≈ܲƒ
2λ
Ó˚ xÎ%@¬ =!îï˛Ü˛ (odd multiple)
() 212
xλ +
•Î˚ñ ˆÎáyˆÏö s = xáu˛
164
§Çáƒyñ ï˛y•ˆÏ° ò%•z !Ó®%Ó˚ üˆÏôƒ ò¢yÓ˚ ˛õyÌ≈ܲƒ •ˆÏÓ ( )2 . 2 1 (2 1)2
s sπ λ+ = + πλ – §%ï˛Ó˚yÇñ ˙ !Ó®% ò%!ê˛ˆÏï˛
ï˛Ó˚ˆÏDÓ˚ ò¢yÎ˚
π
≤ÈÏû˛ò ÌyܲˆÏÓ xÌ≈yÍñ ï˛Ó˚D!ê˛ !Ó˛õÓ˚#ï˛ ò¢yÎ˚ ÌyܲˆÏÓ–
6.5 ~ÓÇ 6.5.1 xö%ˆÏFåȈÏò xy˛õ!ö Îy ˛õí˛¸ˆÏ°ö ï˛y xyÓ˚Ä û˛y°û˛yˆÏÓ ˆÓyé˛yÓyÓ˚ çöƒ xy˛õ!ö ~•z xö%¢#°ö#
ò%!ê˛Ó˚ í˛z_Ó˚ ˆòÄÎ˚yÓ˚ ˆã˛‹Ty ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 3 : ˆÜ˛yöÄ ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î
[] 8sin4.000.02 ytx =− π
~áyˆÏö t ~Ó˚ üyö ˆ§ˆÏܲ[˛ ~ÓÇ x, y ~Ó˚ üyö cm -~ ˆòÄÎ˚y xyˆÏåÈ– ï˛Ó˚D!ê˛Ó˚ !ÓhflÏyÓ˚ñ ܲ¡õyB˛ Ä à!ï˛ˆÏÓà
!öî≈Î˚ ܲÓ˚&ö– ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ x!û˛ü%ˆÏá ò%!ê˛ !Ó®%Ó˚ ò)Ó˚c 20 cm •ˆÏ°ñ ˆÎ ˆÜ˛yöÄ §üˆÏÎ˚ !Ó®% ò%!ê˛ˆÏï˛ ï˛Ó˚ˆÏDÓ˚
ò¢y ˛õyÌ≈ܲƒ !öî≈Î˚ ܲÓ˚&ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -4 : ~ܲ!ê˛ ã˛°ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ 0.03m, ܲ¡õyB˛ 550Hz ~ÓÇ à!ï˛ˆÏÓà 1330ms− – ï˛Ó˚D!ê˛ Î!ò
x xˆÏ«˛Ó˚ ôöydܲ !òˆÏܲÓ˚ x!û˛ü%ˆÏá §MÈ˛y!Ó˚ï˛ •Î˚ñ ï˛y•ˆÏ° ã˛°ï˛Ó˚D!ê˛Ó˚ §Ó˚î §ü#ܲÓ˚î!ê˛ !°á%ö–
6.6 §yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î§yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î§yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î§yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î§yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î
~ܲüy!eܲ §ü#ܲÓ˚î~ܲüy!eܲ §ü#ܲÓ˚î~ܲüy!eܲ §ü#ܲÓ˚î~ܲüy!eܲ §ü#ܲÓ˚î~ܲüy!eܲ §ü#ܲÓ˚î :6.5 ~ÓÇ 6.5.1 xö%ˆÏFåȈÏò xyüÓ˚y ã˛°ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ xÌ≈yÍñ ï˛yÓ˚ §Ó˚î §ü#ܲÓ˚î §¡∫ˆÏ¶˛ çyöˆÏï˛
ˆ˛õˆÏÓ˚!åÈ– ~•z ˛õÎ≈yˆÏÎ˚Ó˚ xyˆÏàÓ˚ ~ܲܲ=!° ˛õí˛¸yÓ˚ §üÎ˚ xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈöñ §Ó˚° ˆòy°à!ï˛ñ
xÓü!®ï˛ ˆòy°à!ï˛ Óy ≤ÈÏîy!òï˛ Ü˛¡õˆÏöÓ˚ ˆ«˛ˆÏeÄ xyüÓ˚y ≤Ã̈Ïü ≤Ãò_ ˆû˛Ôï˛ ¢ï≈˛ xö%§Ó˚î ܲˆÏÓ˚ ~ܲ!ê˛
xÓܲ° §ü#ܲÓ˚î ≤Ã!ï˛¤˛y ܲˆÏÓ˚!åÈ– ~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö !•ˆÏ§ˆÏÓ xyüÓ˚y ӛܲîyÓ˚ ܲ¡õˆÏöÓ˚ §üÎ˚ÈÙÈò)Ó˚c
§¡õÜ≈˛ ˆ˛õˆÏÎ˚!åÈ– ~•z §üyôyˆÏö xÓ¢ƒ ~ܲy!ôܲ x!ö!ò≈‹T ô &Óܲ ÌyˆÏܲ– ï˛ˆÏÓ ≤ÃyÌ!üܲ ¢ï≈˛yÓ°# ≤ÈÏÎ˚yà ܲˆÏÓ˚
xyüÓ˚y ˙ ô &Óܲ=!°Ó˚ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ ˆ˛õˆÏÓ˚!åÈ– !ܲv ~áyˆÏö xyüÓ˚y ~ܲ!ê˛ !Ó˛õÓ˚#ï˛ ˛õk˛!ï˛ˆÏï˛ ï˛Ó˚ˆÏDÓ˚
xÓܲ° §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y ܲÓ˚Ó–
6.5 (b) §ü#ܲÓ˚ˆÏî xyüÓ˚y ˆ˛õˆÏÎ˚!åÈ°yüñ
()() 2 ,sin xtAvtx =− π ψλ
°«˛ƒ ܲÓ˚&öñ ~áyˆÏö ψ ~ܲ•z §ˆÏD x Ä t ~Ó˚ xˆÏ˛õ«˛Ü˛– §%ï˛Ó˚yÇñ ~áyˆÏö ψ ~Ó˚ xyÇ!¢Ü˛ xÓܲ°ö ܲÓ˚y
ÎyˆÏÓ– í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y ~•z xÓܲ°=!° ˛õy•z :
() 22 .cos Avtxx∂=−− ∂ψππ
λλ
2
2
2 22x
Atx FHGIKJ sin()
...... 6.12(a)
( )2 2cosvA vt x
tψ π π
λ λ∂ = −∂
165
() 22 cos∂=−
∂v Avtx
t
ψππλλ()
2 2
222 cos ∂⎛⎞ =−− ⎜⎟ ∂⎝⎠
vAvtx
t
ψππλλ
........ 6.12(b)
6.11 (a) Ä (b) ˆÌˆÏܲ §•ˆÏç•z ˛õyÄÎ˚y ÎyÎ˚
2 2
2 2 21
x v t
ψ ψ∂ ∂=∂ ∂ ........ 6.13
~!ê˛ ~ܲy!ôܲ ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚D §ü#ܲÓ˚î– °«˛ƒ ܲÓ˚&öñ ~•z §ü#ܲÓ˚ˆÏî v2 ÌyܲˆÏ°Äñ v ˆö•z– §%ï˛Ó˚yÇñ ~!ê˛Ó˚
§üyôyö v Ä ÈÙÈv í˛zû˛ˆÏÎ˚Ó˚ çöƒ•z §üû˛yˆÏÓ ≤ÈÏÎyçƒ •ˆÏÓ– ~üö !ܲ x -~Ó˚ ôöydܲ !òˆÏܲ Ä îydܲ !òˆÏܲ
§MÈ˛y!Ó˚ï˛ ò%•z ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ö âˆÏê˛ fliyö%ï˛Ó˚D §,!‹T ܲÓ˚ˆÏ° ï˛yÓ˚ ˆ«˛ˆÏeÄ ~•z ï˛Ó˚D §ü#ܲÓ˚î ≤ÈÏÎyçƒ
ÌyܲˆÏÓ–
!müy!eܲ Ä !eüy!eܲ §ü#ܲÓ˚î !müy!eܲ Ä !eüy!eܲ §ü#ܲÓ˚î !müy!eܲ Ä !eüy!eܲ §ü#ܲÓ˚î !müy!eܲ Ä !eüy!eܲ §ü#ܲÓ˚î !müy!eܲ Ä !eüy!eܲ §ü#ܲÓ˚î :
~ܲüy!eܲ ï˛Ó˚D §ü#ܲÓ˚î!ê˛ xyüÓ˚y ˆÎ ˛õk˛!ï˛ˆÏï˛ ˆ˛õˆÏÎ˚!åÈñ ˆ§•z ~ܲ•z ˛õk˛!ï˛ˆÏï˛ !müy!eܲ Ä !eüy!eܲ
ï˛Ó˚D §ü#ܲÓ˚î ˛õyÄÎ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– ï˛ˆÏÓ ~ˆÏ«˛ˆÏe xyüyˆÏòÓ˚ 6.10 §ü#ܲÓ˚î!ê˛ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– ~•z
§ü#ܲÓ˚î!ê˛ •° :
(). ,sinrnrtAt
v⎛⎞ =− ⎝⎠ ψω
sin . nA t r
v⎛ ⎞= −⎜ ⎟⎝ ⎠
ωω
~áyˆÏö n
vω ˆû˛QÓ˚!ê˛Ó˚ üyö
2vωπ
λ =
~ÓÇ !òܲ ï˛Ó˚D °¡∫ x!û˛ü%á#– ~•z ˆû˛QÓ˚ˆÏܲ xyüÓ˚y §MÈ˛yÓ˚î ˆû˛QÓ˚
(propagation vector) Ó!° ~ÓÇ k Ó˚y!¢ !òˆÏÎ˚ x!û˛!•ï˛ ܲ!Ó˚– §%ï˛Ó˚yÇñ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ :
()() .sin. rtAtkr =− ψω
= ( )sin x y zA t k x k y k z− − −ω
ˆÎáy Ïö ,x yk k Ä kz •°
k
ˆû˛QˆÏÓ˚Ó˚ ܲyˆÏê≈˛ç#Î˚ í˛z˛õyÇ¢ ~ÓÇ (x,y,z) •° ˛õ!Ó˚°!«˛ï˛ !Ó®%Ó˚ fliyöyB˛–
!müy!eܲ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe xÓ¢ƒ
z k
Óy
z
Óy
z kZ
ÙÙÙÈˆÜ˛yöÄ Ó˚y!¢•z ÌyܲˆÏÓ öy–
~áö Î!ò xy˛õ!ö
() .rt ψ
ˆÜ˛ t,x,y,z ≤Ãû,˛!ï˛Ó˚ §yˆÏ˛õˆÏ«˛ xyÇ!¢Ü˛ xÓܲ°ö ܲˆÏÓ˚öñ ï˛ˆÏÓ ˛õyˆÏÓöñ
166
( )cos x y zA t k x k y k zt
ψ ω ω∂ = − − −∂
()2
222sinxyz Atkxkykzt
ψωωωψ ∂=−−−−=−∂
....... 6.14 (a)
~ܲ•zû˛yˆÏÓ xy˛õ!ö ˆòáyˆÏï˛ ˛õyˆÏÓ˚öñ
222222
222 ,,, zz kkkxyz∂∂∂ =−=−=−∂∂∂
ψψψ ψψψ
xÌ≈yÍñ
()222
2222222kyz kkkk
xyz∂∂∂ ++=−++=−∂∂∂
ψψψψ
....... 6.14 (b)
6.12 (a) Ä (b) §¡õÜ≈˛=!°ˆÏܲ ~ܲe ܲˆÏÓ˚ ˛õyˆÏÓöñ
22222
22222k
xyzt∂∂∂∂ ++=∂∂∂∂
ψψψψω
....... 6.15
ÓÑy !òˆÏܲÓ˚ Ó˚y!¢!ê˛ §yôyÓ˚îï˛ 2ψ∇ Ó˚&ˆÏ˛õ ˆ°áy •Î˚– ~åÈyí˛¸y
()2 2
2
22222111 /2. k
vvv
ππ λ ωλ⎛⎞ === ⎜⎟ ⎝⎠§%ï˛Ó˚yÇ 6.15 §ü#ܲÓ˚î!ê˛ˆÏܲ ˆ°áy ÎyÎ˚
22
2210vt
ψ ψ∂ ∇−=∂
....... 6.16
~•z §ü#ܲÓ˚î!ê˛ !müy!eܲ Ä !eüy!eܲ ÈÙÙÙÈò%•z ˆ◊î#Ó˚ ï˛Ó˚ˆÏDÓ˚•z ï˛Ó˚D §ü#ܲÓ˚î– !müy!eܲ ï˛Ó˚ˆÏDÓ˚ çöƒ
() 2,, xyt ψ ∇
ˆÜ˛
2 2
2 2x y
⎛ ⎞∂ ∂+⎜ ⎟⎝ ∂ ∂ ⎠ψ ψ
ӈϰ ôÓ˚ˆÏï˛ •ˆÏÓ–
6.7 ã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y «˛üï˛yÓ˚ ˛õ!Ó˚Ó•îã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y «˛üï˛yÓ˚ ˛õ!Ó˚Ó•îã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y «˛üï˛yÓ˚ ˛õ!Ó˚Ó•îã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y «˛üï˛yÓ˚ ˛õ!Ó˚Ó•îã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y «˛üï˛yÓ˚ ˛õ!Ó˚Ó•î
~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏï˛ !àˆÏÎ˚ xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ ã˛°ï˛Ó˚ˆÏDÓ˚ §y•yˆÏ΃ üyôƒˆÏüÓ˚ ~ܲ !Ó®% ˆÌˆÏܲ
xöƒ !Ó®%ˆÏï˛ ¢!_´ ˛õ!Ó˚Óy!•ï˛ •Î˚– ~áö ≤ß¿ñ ~•z ˛õ!Ó˚Ó•î ܲ#û˛yˆÏÓ âˆÏê˛– ôÓ˚&öñ ¢ˆÏ∑Ó˚ ~ܲ!ê˛ í˛zͧ ÓyÎ˚%ˆÏï˛
ܲ!¡õï˛ •ˆÏFåÈ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° xö%˜Ïòâ≈ƒ ¢∑ï˛Ó˚D í˛zͲõߨ •ˆÏFåÈ– üyôƒˆÏüÓ˚ í˛zͧ §Ç°@¿ hflψÏÓ˚ ≤Ã̈Ïü ˆÎ ܲ¡õö
÷Ó˚& •ˆÏÓñ ï˛y ˛õÓ˚˛õÓ˚ §Ç°@¿ hflÏÓ˚=!°ˆÏï˛ •hflÏyhsˇ!Ó˚ï˛ •ˆÏÓ ~ÓÇ e´ü¢ ï˛Ó˚D!ê˛ §MÈ˛y!Ó˚ï˛ •ˆÏÎ˚ åÈ!í˛¸ˆÏÎ˚ ˛õí˛¸ˆÏÓ–
~Ó˚ ˆÌˆÏܲ•z ˆÓyôy ÎyÎ˚ ˆÎñ í˛zͧ ˆÌˆÏܲ í˛zͲõߨ ¢!_´ ï˛Ó˚ˆÏDÓ˚ üyôƒˆÏü ã˛ï%˛!ò≈ˆÏܲ åÈ!í˛¸ˆÏÎ˚ ÎyÎ˚–
167
≤Ã̈Ïü xyüÓ˚y ï˛Ó˚ˆÏDÓ˚ üyôƒˆÏüÓ˚ à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ ˛õ!Ó˚üyî !öî≈Î˚ ܲÓ˚Ó– ôÓ˚y Îyܲñ ~ܲ!ê˛ §üï˛°
¢∑ï˛Ó˚D x x«˛ ÓÓ˚yÓÓ˚ §MÈ˛y!Ó˚ï˛ •ˆÏFåÈ– xyüÓ˚y xyˆÏàÓ˚ üï˛ ~Ó˚ §üÎ˚ §Ó˚î §ü#ܲÓ˚î !°áˆÏï˛ ˛õy!Ó˚ :
()() ,sin yxtAtkx =− ω
~áö x fliyöyˆÏB˛ üyôƒˆÏüÓ˚ ~ܲ!ê˛ ˛õyï˛°y hflÏÓ˚ !ÓˆÏÓã˛öy ܲÓ˚&öñ ˆÎ!ê˛ ï˛Ó˚Dü%ˆÏáÓ˚ §üyhsˇÓ˚y°ñ ÎyÓ˚ ˆÓô xΔ
Ä ≤ÃfliˆÏFåȈÏòÓ˚ ˆ«˛eú˛°
α
– üyôƒˆÏüÓ˚ âöc p Ó˚y!¢!ê˛ˆÏܲ xyüÓ˚y ˆüyê˛yü%!ê˛ ô Óܲ ӈϰ ôˆÏÓ˚ ˆöÓ– ˆ§ˆÏ«˛ˆÏe
üyôƒˆÏüÓ˚ hflÏÓ˚!ê˛Ó˚ û˛Ó˚
mpx ∆=∆ α
~ÓÇ ˆ§!ê˛Ó˚ à!ï˛¢!_´
212
yKm
t∂⎛⎞ =Δ⎜⎟ ∂ ⎝⎠
=
() 222 1.cos2
pxAtkx αωω Δ−
, ÎyÓ˚ üyö ¢)öƒ ˆÌˆÏܲ 2 21
2A ap xω Δ ~Ó˚ üˆÏôƒ xyˆÏ®y!°ï˛ •Î˚–
ܲ¡õˆÏöÓ˚ ~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yˆÏÎ˚
() 2 costkx ω−
Ó˚y!¢!ê˛Ó˚ §üˆÏÎ˚Ó˚ §yˆÏ˛õˆÏ«˛ àí˛¸ üyö •Î˚ 12
– §%ï˛Ó˚yÇñ ˛õ)î≈
˛õÎ≈yˆÏÎ˚ à!ï˛¢!_´ K ~Ó˚ àí˛¸ üyö
()22 14
KAapx ω =Δ ....... 6.17
~ÓyÓ˚ xyüyˆÏòÓ˚ ˆòáˆÏï˛ •ˆÏÓñ üyôƒˆÏüÓ˚ ˙ hflÏÓ˚!ê˛Ó˚ !fli!ï˛¢!_´ ܲï˛⁄ ~çöƒ ˙ hflψÏÓ˚Ó˚ í˛z˛õÓ˚ ˛˛≤ÃÎ%_´ Ó°
ˆ§!ê˛Ó˚ IJõÓ˚ ܲï˛ê˛y ܲyÎ≈ ܲˆÏÓ˚Ä ï˛y !öî≈Î˚ ܲÓ˚y òÓ˚ܲyÓ˚– ~•z Ó° xÓ¢ƒ•z hflÏÓ˚!ê˛Ó˚ û˛Ó˚ Ä cÓ˚ˆÏîÓ˚ =îú˛ˆÏ°Ó˚
§üyöñ xÌ≈yÍ Ó°ñ
2
2y
Fmt
∂ =Δ∂
=
() 2 .sin apxAwtkx −∆− ω
= 2.ap x yω− Δ
°«˛ƒ ܲÓ˚&öñ y ôöydܲ •ˆÏ° F öydܲñ xÌ≈yÍ Ó°!ê˛ ≤Ãï˛ƒyöÎ˚ܲ– ~•z ӈϰÓ˚ !ÓÓ˚&ˆÏm y ˛õ!Ó˚üyî §Ó˚î
âê˛yˆÏï˛ ˆÎ ܲyÎ≈ ܲÓ˚ˆÏï˛ •Î˚ ï˛yÓ˚ üyö ;
2
0
.y
Vapxydy =∆ ∫ω
= 2 21 .2
Δap x yω
168
= ( )2 2 21 sin2
ap xA t kxω ωΔ −
°«˛ƒ ܲˆÏÓ˚ ˆòá%öñ K ~Ó˚ üï˛ V ~Ó˚ üyöÄ ˘¢)öƒ ˆÌˆÏܲ ω Δ2 212
A ap x ~Ó˚ üˆÏôƒ Äë˛yöyüy ܲˆÏÓ˚– ܲ¡õˆÏöÓ˚
~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yˆÏÎ˚
() 2 sintkx ω−
Ó˚y!¢Ó˚ àí˛¸ üyö 12
– §%ï˛Ó˚yÇñ xyüÓ˚y !fli!ï˛¢!_´Ó˚Ä àí˛¸ üyö ˆ˛õˆÏï˛
˛õy!Ó˚ ;
()2
2 1. 22A
Vapx =∆ ω
= 2 214
A ap xω Δ ....... 6.18
6.17 Ä 6.18 §ü#ܲÓ˚î ò%!ê˛Ó˚ ï%˛°öy ܲÓ˚ˆÏ° ˆòáˆÏÓö (K) =(V) – xÌ≈yÍñ !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ üyö
àˆÏí˛¸ §üyö–
ˆüyê˛ ¢!_´ xÌ≈yÍñ !fli!ï˛¢!_´ Ä à!ï˛¢!_´Ó˚ ˆÎyàú˛° :
E = K + V
=
()() 2222 1cossin2
Aapxtkxtkx ⎡⎤ ∆−+− ⎣⎦ ωωω
= 2 212
A ap xω Δ ....... 6.19
x Δ
ˆÓˆÏôÓ˚ hflÏÓ˚!ê˛ Î!ò
t Δ
§üˆÏÎ˚ ï˛Ó˚D˛õˆÏÌÓ˚ ~ܲ!ê˛ !Ó®% x!ï˛e´ü ܲˆÏÓ˚ ÎyÎ˚ñ ï˛ˆÏÓ
α
ˆ«˛eú˛ˆÏ°Ó˚
≤ÃfliˆÏFåȈÏòÓ˚ üôƒ !òˆÏÎ˚ ¢!_´ ˛õ!Ó˚Ó•ˆÏîÓ˚ •yÓ˚ñ xÌ≈yÍ ˛õ!Ó˚Óy!•ï˛ «˛üï˛y :
22 12
ExPAwap
tt∆ == ∆∆
22 12
PAapv =ω
ñ ˆÎáyˆÏö x
vt
∆= ∆ ñ ï˛Ó˚ˆÏDÓ˚ ˆÓà ....... 6.20
6.19 §¡õÜ≈˛!ê˛ ˆÌˆÏܲ xyüÓ˚y ï˛Ó˚ˆÏDÓ˚ ¢!_´Ó˚ âöc xÌ≈yÍñ ~ܲܲ xyÎ˚ï˛ö !˛õå%È ¢!_´Ó˚ üyö ˆ˛õˆÏï˛ ˛õy!Ó˚–
xyüÓ˚y ˆÎ hflÏÓ˚!ê˛Ó˚ ܲÌy !ÓˆÏÓã˛öy ܲˆÏÓ˚!åÈ ï˛yÓ˚ xyÎ˚ï˛ö
x αΔ
– §%ï˛Ó˚yÇñ ï˛Ó˚ˆÏDÓ˚ ¢!_´Ó˚ âöcñ
22 1 /2
ExA =∆= εαωρ
....... 6.21
°«˛ƒ ܲÓ˚&öñ ~•z Ó˚y!¢!ê˛Ó˚ üyö A ~ÓÇ ω, í˛zû˛ˆÏÎ˚Ó˚ ÓˆÏà≈Ó˚ ~ÓÇ üyôƒˆÏüÓ˚ âöc p ~Ó˚ §üyö%˛õyï˛#–
169
6.8 ¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y (Intensity)
ˆÜ˛yöÄ ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ Îáö ¢∑ï˛Ó˚D !öɧ,ï˛ •Î˚ñ ï˛áö ï˛Ó˚D˛õˆÏÌÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ !Ó®%ˆÏï˛ ~ܲܲ
§üˆÏÎ˚ ï˛Ó˚ˆÏDÓ˚ x!û˛ü%ˆÏáÓ˚ §ˆÏD °¡∫ ~ܲܲ ˆ«˛eú˛ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ˆÎ ï˛Ó˚D¢!_´ ≤ÃÓy!•ï˛ •Î˚ñ ï˛yˆÏܲ ˙
¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ӈϰ–
6.20 §ü#ܲÓ˚ˆÏî xyüÓ˚y ï˛Ó˚ˆÏD ˛õ!ÓÓy!•ï˛ «˛üï˛yÓ˚ (P) ˆÎ Ó˚y!¢üy°y ˆ˛õˆÏÎ˚!åÈñ ï˛yˆÏܲ ˆÎ °¡∫ ˆ«˛eú˛ˆÏ°Ó˚
üôƒ !òˆÏÎ˚ ˙ ¢!_´ ˛õ!Ó˚Óy!•ï˛ •ˆÏFåÈ xÌ≈yÍ, α !òˆÏÎ˚ û˛yà ܲÓ˚ˆÏ° ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ˛õyÄÎ˚y ÎyˆÏÓ– ï˛#Ó ï˛yˆÏܲ
I !òˆÏÎ˚ !öˆÏò≈¢ ܲÓ˚ˆÏ°
22 12
PIApv
aω ==
....... 6.22
°«˛ƒ ܲÓ˚&ö I ~Ó˚ üyö ˆÜ˛yö‰ ˆÜ˛yö‰ !ӣψÏÎ˚Ó˚ IJõÓ˚ ܲ#û˛yˆÏÓ !öû≈˛Ó˚ ܲˆÏÓ˚– üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚ ˆÜ˛yöÄ ˆ¢y£Ïî
öy âê˛ˆÏ° §üï˛° ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲñ ˆÎˆÏ•ï%˛ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ω ~ÓÇ §ü§c üyôƒˆÏü P
~ÓÇ v !ö!ò≈‹T– xï˛~Óñ §üï˛° ï˛Ó˚ˆÏD ï˛#Ó ï˛y ï˛Ó˚ˆÏDÓ˚ ˛õˆÏÌ §Ó≈e §üyö ÌyˆÏܲ–
ï˛#Ó ï˛yÓ˚ ~ܲܲ ï˛#Ó ï˛yÓ˚ ~ܲܲ ï˛#Ó ï˛yÓ˚ ~ܲܲ ï˛#Ó ï˛yÓ˚ ~ܲܲ ï˛#Ó ï˛yÓ˚ ~ܲܲ : SI ˛õk˛!ï˛ˆÏï˛ ï˛#Ó ï˛yÓ˚ ~ܲܲ 1 2Js m− − Óy
2 Wm−
– 6.22 §ü#ܲÓ˚ˆÏî I ~Ó˚ Ó˚y!¢üy°yÎ˚
í˛yö!òˆÏܲ ˆÎ Ó˚y!¢=!° Ó˚ˆÏÎ˚ˆÏåÈ SI ˛õk˛!ï˛ˆÏï˛ ï˛yˆÏòÓ˚ ~ܲܲ=!° Ó!§ˆÏÎ˚ ˆòáy ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~Ó˚ ~ܲܲ
12 Jsm−−
Óy
2 Wm−
SˆÎˆÏ•ï%˛
1 WattJs− =
) ˛õyÄÎ˚y ÎyˆÏFåÈ !ܲöy– ~ ܲyç!ê˛ xy˛õöyÓ˚ çöƒ•z Ó˚yáy •°–
ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y : ~ ˛õÎ≈hsˇ xyüÓ˚y §üï˛° ï˛Ó˚ˆÏDÓ˚ ܲÌy !ÓˆÏÓã˛öy ܲÓ˚ˆÏ°Ä ~ÓyÓ˚ ˆày°#Î˚
ï˛Ó˚ˆÏDÓ˚ ܲÌy xyüyˆÏòÓ˚ û˛yÓˆÏï˛ •ˆÏÓ– ˆày°#Î˚ ï˛Ó˚D ~ܲ!ê˛ !Ó®%ˆÏï˛ í˛zq(ï˛ •Î˚ ~ÓÇ ï˛Ó˚D x@ˇÃ§Ó˚ •ÄÎ˚yÓ˚
§ˆÏD §ˆÏD ˆ§!ê˛Ó˚ ˆày°Ü˛yÜ,˛!ï˛ ï˛Ó˚Dü%ˆÏáÓ˚ ˆ«˛eú˛° Ó,!k˛ ˛õyÎ˚– ¢ˆÏ∑Ó˚ ~ܲ!ê˛ !Ó®% í˛zͧˆÏܲ ˆÜ˛ˆÏw ˆÓ˚ˆÏá
r Óƒy§yˆÏô≈Ó˚ ~ܲ!ê˛ ˆày°Ü˛ ܲ“öy ܲÓ˚&ö ~ÓÇ ôˆÏÓ˚ !öö ¢∑ í˛zͧ S ˆÌˆÏܲ !Óܲ#î≈ •ˆÏÎ˚ ˙ ˆày°ˆÏܲÓ˚ ï˛ˆÏ°Ó˚
§Ó≈e §üyö ï˛#Ó ï˛yÎ˚ ˆ˛õÑÔˆÏåȈÏåÈ– Î!ò ~ܲܲ §üˆÏÎ˚ !Óܲ#î≈ ¢!_´ E •Î˚ñ ï˛y•ˆÏ° r Óƒy§yˆÏô≈Ó˚ ˆày°ˆÏܲÓ˚ üôƒ
!òˆÏÎ˚ ~•z ˛õ!Ó˚üyî ¢!_´ ~ܲܲ §üˆÏÎ˚ !öà≈ï˛ •ˆÏÓ– §%ï˛Ó˚yÇñ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
() 2 4 Err =π
....... 6.23
ˆÎáyˆÏö I(r) •ˆÏFåÈ ˙ ˆày°ˆÏܲÓ˚ ï˛ˆÏ° ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y– ~!ê˛ Óƒy§yô≈ r ~Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– ˆày°ˆÏܲÓ˚
Óƒy§yô≈ Ó,!k˛ ˆ˛õˆÏ° xÌ≈yÍñ ˆày°ˆÏܲÓ˚ ï˛° í˛zͧ ˆÌˆÏܲ xyÓ˚Ä ò)ˆÏÓ˚ §ˆÏÓ˚ ˆàˆÏ°Ä I(r)r2 Ó˚y!¢!ê˛Ó˚ üyö
x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏÓñ ܲyÓ˚î í˛zͧ!ê˛ ~ܲ!ê˛ !ö!ò≈‹T •yˆÏÓ˚•z ¢!_´ §Ó˚ÓÓ˚y• ܲˆÏÓ˚– ܲyˆÏç•zñ í˛zͧ ˆÌˆÏܲ ò)Ó˚c
r ~Ó˚ Ó,!k˛Ó˚ §ˆÏD §ˆÏD ï˛#Ó ï˛y I !öÿ˛Î˚•z 2Ir
~Ó˚ §üyö%˛õyˆÏï˛ Ü˛üˆÏï˛ ÌyܲˆÏÓ– 6.22 §ü#ܲÓ˚ˆÏî ˆòáy ˆàˆÏåÈñ
ï˛#Ó ï˛y ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õyï˛#–
21
Ir∝
~ÓÇ
2 IA ∝
- ~•z ò%•z §¡õÜ≈˛ ˆÌˆÏܲ Ó°y ÎyÎ˚ñ
1A
r∝
xÌ≈yÍ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ í˛zͧ ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚ ÓƒhflÏyö%˛õyˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚–
k ~ܲ!ê˛ ô Óܲ •ˆÏ° ˆ°áy ÎyÎ˚ñ
170
2k
Ir
=
~•z §¡õÜ≈!ê˛ˆÏܲ ÓƒhflÏyö%˛õy!ï˛Ü˛ Óà≈ §)e (inverse -square law) Ó°y •Î˚–
ˆÎˆÏ•ï%˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ò)Ó˚ˆÏcÓ˚ ÓˆÏà≈Ó˚ ÓƒhflÏyö%˛õyˆÏï˛ •…y§ ˛õyÎ˚ñ ï˛y•z í˛zͧ ˆÌˆÏܲ ò)ˆÏÓ˚ §ˆÏÓ˚ ˆÎˆÏï˛ ÷Ó˚& ܲÓ˚ˆÏ°
ï˛yÓ˚ ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö e´ü¢ «˛#î ˆ¢yöyÎ˚– xyÓ˚ í˛z͈ϧÓ˚ «˛üï˛y xö%ÎyÎ˚# !ܲå%È ò)Ó˚ ˆàˆÏ° ¢∑ xyÓ˚ ˆ¢yöy•z
ÎyÎ˚ öy– ò%•z !üê˛yÓ˚ ò)Ó˚ ˆÌˆÏܲ •Î˚ï˛ xy˛õ!ö xyüyÓ˚ fl∫yû˛y!Óܲ ܲZ˛fl∫Ó˚ flõ‹T ÷öˆÏï˛ ˛õyˆÏÓöñ !ܲv ò¢ !üê˛yÓ˚
ò)Ó˚ ˆÌˆÏܲ ˆ§•z ¢∑ xy˛õöyÓ˚ ܲyˆÏåÈ xflõ‹T üˆÏö •ˆÏÓ– xyÓyÓ˚ ò¢ !üê˛yÓ˚ ò)ˆÏÓ˚ ÓyçyˆÏöy ˆê˛˛õ ˆÓ˚ܲí≈˛yˆÏÓ˚Ó˚
xyÄÎ˚yç ˆ˛õÑÔˆÏåÈ ÎyˆÏÓ xy˛õöyÓ˚ ܲyˆÏö– °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ xyÄÎ˚yç ˆï˛y ã˛!Õ‘¢ !üê˛yÓ˚ ò)Ó˚ ˆÌˆÏÜ˛Ä Ü˛î≈˛õ#í˛¸y
§,!‹T ܲˆÏÓ˚– ~ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ñ ~ܲ ~ܲ!ê˛ í˛z͈ϧÓ˚ «˛üï˛y ~ܲ ~ܲ üyˆÏöÓ˚ ~ÓÇ ˆ§=!° ~ܲ•z ò)Ó˚ˆÏc
!Ó!û˛ß¨ ï˛#Ó ï˛yÓ˚ ¢∑ §,!‹T ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– ¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y üy˛õyÓ˚ !Ó!û˛ß¨ í˛z˛õyÎ˚ §¡∫ˆÏ¶˛ xy˛õ!ö ~•z ˛õÎ≈yˆÏÎ˚Ó˚
ò¢ü ~ܲˆÏܲ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
xy˛õöyÓ˚ üˆÏö ≤ß¿ çyàˆÏï˛ ˛õyˆÏÓ˚ñ §Ó≈!ö¡¨ ˆÜ˛yö‰ ï˛#Ó ï˛yÓ˚ ¢∑ xyüÓ˚y ÷öˆÏï˛ ˛õy•z⁄ xy˛õ!ö çyöˆÏ° xyÿ˛Î≈
•ˆÏÓö ˆÎñ üye
122 10Wm −−
ï˛#Ó ï˛yÓ˚ ¢∑Ä Ü˛yö ôÓ˚ˆÏï˛ ˛õyˆÏÓ˚– Îáö ~Ó˚ܲü ï˛#Ó ï˛yÓ˚ ¢∑ xyüyˆÏòÓ˚ ܲî≈˛õê˛ˆÏ•
ˆ˛õÑÔåÈÎ˚ñ ï˛áö Óyï˛yˆÏ§ ï˛yÓ˚ !ÓhflÏyÓ˚ ܲï˛ê˛y •Î˚ ï˛y xyüÓ˚y ܲ“öyÄ Ü˛Ó˚ˆÏï˛ ˛õyÓ˚Ó öy– xö%¢#°ö# 5 ˆÌˆÏܲ xy˛õ!ö
~ §¡õˆÏÜ≈˛ ~ܲê˛y ôyÓ˚îy ˛õyˆÏÓö–
xyüyˆÏòÓ˚ ܲyˆÏöÓ˚ xyÿ˛Î≈ «˛üï˛yÓ˚ ~áyˆÏö•z ˆ¢£Ï öÎ˚– ~ܲ!òˆÏܲ ܲyö ˆÎüö
122 10Wm −−
≤ÃyӈϰƒÓ˚ ¢∑
÷öˆÏï˛ ˛õyÎ˚ñ xöƒ!òˆÏܲ ï˛yÓ˚ ˆò¢ ˆÜ˛y!ê˛ (108) =î ï˛#Ó ï˛yÓ˚ xÌ≈yÍ,
42 10Wm −−
˛õ!Ó˚üyî ï˛#Ó ï˛yÓ˚ ¢∑Ä
xyüyˆÏòÓ˚ ܲyö §•ƒ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– Ó›ï˛ñ xyüÓ˚y ˆÎ §Ó ¢∑ ã˛yÓ˚!òˆÏܲ ÷öˆÏï˛ ˛õy•zñ ï˛yÓ˚ üˆÏôƒ !Ó!û˛ß¨ ï˛#Ó ï˛yÓ˚
¢∑ Ó˚ˆÏÎ˚ˆÏåÈ– ï˛ˆÏÓ ~ §¡∫ˆÏ¶˛ xy˛õ!ö ~•z ˛õyë˛e´ˆÏü ò¢ü ~ܲˆÏܲ xyÓ˚Ä !Ó¢òû˛yˆÏÓ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -5 : Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓà 340ms–1 ~ÓÇ Óyï˛yˆÏ§Ó˚ âöc 1.29kgm–3 •ˆÏ°ñ 1kHz ¢ˆÏ∑Ó˚
çöƒ ö)ƒöï˛ü ◊ÓîˆÏÎyàƒ xÌ≈yÍ 10–12Wm–2 ï˛#Ó ï˛yÎ˚ ¢∑ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ Ü˛ï˛ •ˆÏÓ⁄
6.9 í˛˛õ°yÓ˚ !e´Î˚yí˛˛õ°yÓ˚ !e´Î˚yí˛˛õ°yÓ˚ !e´Î˚yí˛˛õ°yÓ˚ !e´Î˚yí˛˛õ°yÓ˚ !e´Î˚y (Doppler effect)
ï˛Ó˚Dà!ï˛Ó˚ §ˆÏD §Ç!Ÿ’‹T ~ܲ!ê˛ x!ï˛ ˛õ!Ó˚!ã˛ï˛ âê˛öy •ˆÏFåÈ í˛˛õ°yÓ˚ !e´Î˚y (Doppler effect)– ôÓ˚&öñ
xy˛õ!ö ˆÓ˚ˆÏ°Ó˚ ≤’ƒyê˛ú˛ˆÏü≈ òÑy!í˛¸ˆÏÎ˚ xyˆÏåÈö– ~ܲ!ê˛ ˆê˛Δö ˆfiê˛¢ˆÏöÓ˚ !òˆÏܲ xy§ˆÏåÈñ ï˛yÓ˚ ~!OˆÏöÓ˚ ÓyÑ!¢ ÓyçˆÏåÈ–
ˆê˛Δö!ê˛ ˆÎüö ≤’ƒyê˛ú˛ˆÏü≈ ì%˛Ü˛°ñ ˆï˛üö•z o&ï˛ à!ï˛ˆÏï˛ ˆÓ!Ó˚ˆÏÎ˚ ~°– ~Ó˚ܲü ˆ«˛ˆÏe xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ
ܲˆÏÓ˚ˆÏåÈö ˆÎñ xy=Î˚yö ˆê˛ΔˆÏöÓ˚ ÓÑy!¢Ó˚ xyÄÎ˚yç xy˛õöyÓ˚ ܲyˆÏåÈ Îï˛ê˛y ï˛#«¯˛ ˆ¢yöyÎ˚ñ ˆ§•z ˆê˛Δö Îáö xy˛õöyˆÏܲ
x!ï˛e´ü ܲˆÏÓ˚ ò)ˆÏÓ˚ ã˛ˆÏ° ˆÎˆÏï˛ ÌyˆÏܲñ ˙ ~ܲ•z xyÄÎ˚yç !ܲv ˆ§Ó˚ܲü ï˛#«¯˛ ˆ¢yöyÎ˚ öy– ÓÓ˚Ç ï˛y ˆÓ¢ áy!öܲê˛y
áyˆÏò ˆöˆÏü ÎyÎ˚– ~ê˛y !ܲ !öåÈܲ•z ˆ¢yöyÓ˚ û%˛°⁄ öyñ ˆáÎ˚y° ܲÓ˚ˆÏ° ˆòáˆÏÓö ¢ˆÏ∑Ó˚ ˆÎ ˆÜ˛yöÄ í˛zͧ Îáö
xy˛õöyÓ˚ !òˆÏܲ å%ȈÏê˛ xyˆÏ§ñ ï˛áö ï˛yÓ˚ ï˛#«¯˛ï˛y ˆÓ!¢– xyÓ˚ Îáö ˆ§!ê˛ ò)ˆÏÓ˚ ã˛ˆÏ° ˆÎˆÏï˛ ÌyˆÏܲñ ï˛áö ˆ◊yï˛yÓ˚
ܲyˆÏåÈ ï˛yÓ˚ ï˛#«¯˛ï˛y ܲü ӈϰ üˆÏö •Î˚–
¢∑ï˛Ó˚ˆÏDÓ˚ í˛zͧ ~ÓÇ ˆ◊yï˛yÓ˚ Óy @ˇÃy•Ü˛ ΈÏsfÓ˚ üˆÏôƒ Î!ò xyˆÏ˛õ!«˛Ü˛ à!ï˛ ÌyˆÏܲñ ï˛y•ˆÏ° í˛z͈ϧÓ˚
ܲ¡õyˆÏB˛Ó˚ xy˛õyï˛ ˛õ!Ó˚Óï≈˛ö âˆÏê˛– ~•z âê˛öy!ê˛ˆÏܲ Ó°y •Î˚ í˛˛õ°yÓ˚ !e´Î˚y– xyüÓ˚y ~ÓyÓ˚ ¢∑ï˛Ó˚ˆÏDÓ˚
171
ܲ¡õyˆÏB˛Ó˚ ~•z xy˛õyï˛ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ˛õ!Ó˚üyî !öî≈Î˚ ܲÓ˚Ó–
≤ÃÌü ˆ«˛e≤ÃÌü ˆ«˛e≤ÃÌü ˆ«˛e≤ÃÌü ˆ«˛e≤ÃÌü ˆ«˛e : Îáö í˛zͧ !fliÓ˚ñ ˆ◊yï˛y í˛zͧ x!û˛ü%ˆÏá xÌÓy !Ó˛õÓ˚#ï˛ ü%ˆÏá ã˛°üyö–
ôÓ˚y Îyܲñ ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ ' 'n ܲ¡õyˆÏB˛Ó˚ ¢∑ !öà≈ï˛
•ˆÏFåÈ ~ÓÇ ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà v
∴ ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ λ =
vn
ˆ◊yï˛y !fliÓ˚ ÌyܲˆÏ° ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ n §ÇáƒÜ˛ ï˛Ó˚D ˆ◊yï˛yˆÏܲ x!ï˛e´ü ܲÓ˚ˆÏÓ ~ÓÇ í˛z͈ϧÓ˚ ܲ¡õyB˛ ïÑ˛yÓ˚
ܲyˆÏåÈ ‘n’ •z üˆÏö •ˆÏÓ– ~ÓyÓ˚ ôÓ˚y Îyܲñ ˆ◊yï˛y u à!ï˛ˆÏÓà !öˆÏÎ˚ í˛z͈ϧÓ˚ !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏï˛ ÷Ó˚& ܲÓ˚ˆÏ°ö–
ï˛yÓ˚ ܲyˆÏö !ë˛Ü˛ ~ܲ ˆ§ˆÏܲˆÏ[˛ ˆÎ ܲ!ê˛ ï˛Ó˚D ˆ˛õÑÔåȈÏÓñ ï˛yÓ˚ ܲyˆÏåÈ ˆ§ê˛y•z ܲ¡õyB˛ ӈϰ üˆÏö •ˆÏÓ– 6.9 !ã˛ˆÏe
~•z xÓfliy!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
ôÓ˚&ö ˆ◊yï˛y t = 0 §üˆÏÎ˚ 6.9 !ã˛ˆÏeÓ˚ L !Ó®%ˆÏï˛ !åȈϰö– ~ܲ ˆ§ˆÏܲu˛ ˛õˆÏÓ˚ !ï˛!ö Îáö L’ !Ó®%ˆÏï˛
ˆ˛õÑÔåȈϰöñ ï˛áö ¢∑ ï˛Ó˚ˆÏDÓ˚ L !Ó®% v ò)Ó˚c x!ï˛e´ü ܲˆÏÓ˚ F’ xÓfliyˆÏö ˆ˛õÑÔåȈÏÓ– ~áyˆÏö LL'= ˆ◊yï˛yÓ˚
ˆÓà u ~ÓÇ LF'= ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓà v– ~•z ~ܲ ˆ§ˆÏܲu˛ L'F' ò)Ó˚ˆÏcÓ˚ §Ó ܲ!ê˛ ï˛Ó˚D•z ˆ◊yï˛yˆÏܲ x!ï˛e´ü
ܲÓ˚ˆÏÓ– ~•z ï˛Ó˚D=!°Ó˚ §Çáƒy•z ˆ◊yï˛yÓ˚ ܲyˆÏåÈ ¢ˆÏ∑Ó˚ xy˛õyï˛ Ü˛¡õyB˛ n', ÎyÓ˚ üyö
'''' '1 LFLFLLvuvuunnn
vv+++⎛⎞ =====+ ⎝⎠ λλλ
..... 6.24
~•z xy˛õyï˛ Ü˛¡õyB˛ n' flõ‹Tï˛•z n ~Ó˚ ˆã˛ˆÏÎ˚ !ܲå%Èê˛y ˆÓ!¢–
~Ó˚ !Ó˛õÓ˚#ï˛ âê˛öy âê˛ˆÏÓ Îáö ˆ◊yï˛y !fliÓ˚ í˛zͧ ˆÌˆÏܲ u
ˆÓˆÏà ò)ˆÏÓ˚ §ˆÏÓ˚ ÎyˆÏÓö (!ã˛e 6.10)– ~ˆÏ«˛ˆÏe ~ܲ ˆ§ˆÏܲ[˛
§üˆÏÎ˚ ˆ◊yï˛y L ˆÌˆÏܲ L' !Ó®%ˆÏï˛ ˆ˛õÑÔåȈÏÓö ~ÓÇ ˙ §üˆÏÎ˚
L'F' ò)Ó˚ˆÏcÓ˚ üˆÏôƒÜ˛yÓ˚ §Ó ï˛Ó˚D•z ˆ◊yï˛yˆÏܲ ˆ˛õ!Ó˚ˆÏÎ˚ ÎyˆÏÓ–
~•z ï˛Ó˚D=!°Ó˚ §Çáƒy
' ' ' '' 1L F LF LL v u v u un n n
v v− − − ⎛ ⎞= = = = = −⎝ ⎠λ λ λ ..... 6.25
~!ê˛•z •ˆÏÓ ˆ◊yï˛yÓ˚ ܲyˆÏåÈ ¢ˆÏ∑Ó˚ xy˛õyï˛ Ü˛¡õyˆÏB˛Ó˚ üyö– 6.25 §¡õÜ≈˛!ê˛ xÓ¢ƒ 6.24 §¡õˆÏÜ≈˛ u-~Ó˚
˛õ!Ó˚ÓˆÏï≈˛ -u !°ˆÏá•z ˛õyÄÎ˚y ˆÎï˛– °«˛ƒ ܲÓ˚&öñ ~ˆÏ«˛ˆÏe n' ~Ó˚ üyö n xˆÏ˛õ«˛y ܲü–
!mï˛#Î˚ ˆ«˛e!mï˛#Î˚ ˆ«˛e!mï˛#Î˚ ˆ«˛e!mï˛#Î˚ ˆ«˛e!mï˛#Î˚ ˆ«˛e : Îáö ˆ◊yï˛y !fliÓ˚ñ ¢ˆÏ∑Ó˚ í˛zͧ ˆ˘◊yï˛yÓ˚ x!û˛ü%ˆÏá Óy !Ó˛õÓ˚#ï˛ü%ˆÏá ã˛°üyö–
≤Ã̈Ïü xyüÓ˚y ¢ˆÏ∑Ó˚ í˛zͧ ˆ◊yï˛y x!û˛ü%ˆÏá w ˆÓˆÏà à!ï˛¢#° ӈϰ ܲ“öy ܲÓ˚Ó– ôÓ˚&öñ í˛zͧ Î!ò !fliÓ˚
172
Ìyܲï˛ñ ï˛ˆÏÓ t=0 §üˆÏÎ˚ ˆÎ ï˛Ó˚D!ê˛ !öà≈ï˛ •ï˛ñ ˆ§!ê˛ 1 ˆ§ˆÏܲ[˛ ˛õˆÏÓ˚ xÌ≈yÍñ t=1s §üˆÏÎ˚ v ò)Ó˚c x!ï˛e´ü
ܲˆÏÓ˚ F !Ó®%ˆÏï˛ ˆ˛õÑÔåÈyï˛ S!ã˛e 6.11) – ò)Ó˚c SF = v ~ÓÇ ~•z ò)Ó˚ˆÏcÓ˚
üˆÏôƒ ï˛Ó˚ˆÏDÓ˚ §Çáƒy ¢ˆÏ∑Ó˚ ܲ¡õyB˛ n ~Ó˚ §üyö– ~áö Î!ò í˛zͧ!ê˛ ˙
1 ˆ§ˆÏܲu˛ §üˆÏÎ˚ S ˆÌˆÏܲ S' !Ó®%ˆÏï˛ ~ˆÏ§ ˆ˛õÑÔåÈÎ˚ñ ˆÎáyˆÏö SS'=w, ï˛ˆÏÓ
˙ §üˆÏÎ˚Ó˚ üˆÏôƒ ˆÎ n §ÇáƒÜ˛ ï˛Ó˚D í˛zͧ ˆÌˆÏܲ !öà≈ï˛ •ˆÏÎ˚ˆÏåÈñ ˆ§=!°
§Ó•z S'F ò)Ó˚ˆÏcÓ˚ üˆÏôƒ ÌyܲˆÏÓ– !ܲv S'F=SF-SS'=v-w– xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ
~ˆÏ«˛ˆÏe í˛zÍˆÏ§Ó ˆÓà ï˛Ó˚D =!°ˆÏܲ §B%˛!ã˛ï˛ ܲÓ˚ˆÏåÈ ~ÓÇ ˆ§=!°Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ ܲˆÏü òÑyí˛¸yˆÏFåÈ
λ' = 1 1v w v w wn n v v− ⎛ ⎞ ⎛ ⎞= − = −⎝ ⎠ ⎝ ⎠λ – ~áö ˆ◊yï˛y ˆÎ ¢∑ ÷öˆÏÓö ï˛yÓ˚ ܲ¡õyB˛ •ˆÏÓñ
'' 11
v v nn
wwvv
= = =⎛ ⎞ −−⎝ ⎠
λ λ ..... 6.26
~ÓyÓ˚ ôÓ˚y Îyܲñ ¢ˆÏ∑Ó˚ í˛zͧ ˆ◊yï˛yÓ˚ ܲyåÈ ˆÌˆÏܲ u ˆÓˆÏà ò)ˆÏÓ˚ §ˆÏÓ˚ ÎyˆÏFåÈ S!ã˛e 6.12) – 1 ˆ§ˆÏܲ[˛
§üˆÏÎ˚ í˛zͧ Î!ò S !Ó®% ˆÌˆÏܲ S' !Ó®%ˆÏï˛ ˆ˛õÑÔåÈÎ˚ñ ï˛ˆÏÓ S'S'=u– ˙ §üˆÏÎ˚ !öà≈ï˛ n §ÇáƒÜ˛ ï˛Ó˚D ~áö
≤çy!Ó˚ï˛ •ˆÏÎ˚ FS' ò)Ó˚c ç%ˆÏí˛¸ ÌyܲˆÏÓ– !ܲvñ FS'=FS+SS' = v + u– §%ï˛Ó˚yÇñ ~áö ˛õ!Ó˚Ó!ï≈˛ï˛ ï˛Ó˚D˜Ïòâ≈ƒ
•ˆÏÓñ
' 1 1v w v w wn n v v+ ⎛ ⎞ ⎛ ⎞= = + = +⎝ ⎠ ⎝ ⎠λ λ
§%ï˛Ó˚yÇñ ˆ◊yï˛y ˆÎ ܲ¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏÓö ï˛y •°ñ
'' 11
v v nn
wwvu
= = =⎛ ⎞ ++⎝ ⎠
λ λ ....... 6.27
~ˆÏ«˛ˆÏeÄ 6.26 §¡õÜ≈˛!ê˛ˆÏï˛ u ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ -u Ó§yˆÏ° 6.27 §¡õÜ≈˛!ê˛ ˛õyÄÎ˚y ˆÎï˛–
ï,˛ï˛#Î˚ ˆ«˛e ï,˛ï˛#Î˚ ˆ«˛e ï,˛ï˛#Î˚ ˆ«˛e ï,˛ï˛#Î˚ ˆ«˛e ï,˛ï˛#Î˚ ˆ«˛e : Îáö ˆ◊yï˛y Ä í˛zͧ í˛zû˛Î˚•z
˛õÓ˚flõÓ˚ §ÇˆÏÎyàܲyÓ˚# §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ã˛°üyö–
ôÓ˚y Îyܲñ í˛z͈ϧÓ˚ ˆÓà ˆ◊yï˛yÓ˚ !òˆÏܲ w ~ÓÇ
ˆ◊yï˛yÓ˚ ˆÓà í˛zͧ x!û˛ü%ˆÏá u S!ã˛e 6.13)–
~ˆÏ«˛ˆÏe ¢ˆÏ∑Ó˚ í˛zͧ ~ܲ ˆ§ˆÏܲˆÏu˛ S ˆÌˆÏܲ S'
!Ó®%ˆÏï˛ ~ˆÏ§ ˆ˛õÑÔˆÏåȈÏåÈ ~ÓÇ ˙ ~ܲ•z §üˆÏÎ˚ ˆ◊yï˛y L ˆÌˆÏܲ L’ !Ó®%ˆÏï˛ §ˆÏÓ˚ ~ˆÏ§ˆÏåÈö– §%ï˛Ó˚yÇñ SS'=
í˛z͈ϧÓ˚ ˆÓà w ~ÓÇ LL' = ˆ◊yï˛yÓ˚ ˆÓà u– ¢ˆÏ∑Ó˚ ˆÎ ï˛Ó˚D!ê˛ ≤ÃyÌ!üܲ xÓfliyÎ˚ L !Ó®%ˆÏï˛ !åÈ°ñ ˆ§!ê˛
F1
173
~ܲ ˆ§ˆÏܲˆÏ[˛ F’ !Ó®%ˆÏï˛ ˆ˛õÑÔˆÏåȈÏåÈñ §%ï˛Ó˚yÇ LF' = v – ¢ˆÏ∑Ó˚ í˛zͧ!ê˛ ˆÎˆÏ•ï%˛ ˆ◊yï˛yÓ˚ !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏFåÈñ
xï˛~Ó ï˛Ó˚D˜Ïòâ≈ƒ λ ̈Ïܲ §B%˛!ã˛ï˛ •ˆÏÎ˚ 1 wv
⎛ ⎞= −′ ⎝ ⎠λ λ •ˆÏÓ– í˛z˛õÓ˚vñ ~ܲ ˆ§ˆÏܲˆÏ[˛ L'F' ˜òˆÏâ≈ƒÓ˚ üˆÏôƒ
xÓ!fliï˛ §Ó ï˛Ó˚D ˆ◊yï˛yˆÏܲ x!ï˛e´ü ܲˆÏÓ˚ ÎyˆÏÓ– §%ï˛Ó˚yÇñ ˆ◊yï˛y ˆÎ ܲ¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏÓö ï˛y •ˆÏÓñ
' ' ' '' .'
1
L F LF LL v u v v un
v wwv
+ + += = = = −⎛ ⎞+⎝ ⎠λ λ λλ
Óyñ ' v un n
v w+= − ........ 6.28
~•z §¡õÜ≈˛!ê˛ xy§ˆÏ° 6.23 Ä 6.25 §¡õÜ≈˛ ò%!ê˛Ó˚ !ü!°ï˛ Ó˚*˛õ– °«˛ƒ ܲˆÏÓ˚ ˆòá%öñ Î!ò ˆ◊yï˛y Ä í˛zͧ
~ܲ•z !òˆÏܲ ˆÓˆÏà ã˛°üyö •Î˚ xÌ≈yÍ Î!ò
wu =−
•Î˚ñ ï˛ˆÏÓ n2 = n– ~Ó˚ xÌ≈ ~•z ˆÎñ ˆ◊yï˛y Ä í˛zͧ üyôƒˆÏüÓ˚
§yˆÏ˛õˆÏ«˛ à!ï˛¢#° •ˆÏ°Ä Î!ò ï˛yˆÏòÓ˚ üˆÏôƒ ˆÜ˛yöÄ xyˆÏ˛õ!«˛Ü˛ à!ï˛ öy ÌyˆÏܲñ ï˛ˆÏÓ Ü˛¡õyˆÏB˛ ˆÜ˛yöÄ xy˛õyï˛
˛õ!Ó˚Óï≈˛ö âˆÏê˛ öy– ~!ê˛ xöƒû˛yˆÏÓÄ Óƒyáƒy ܲÓ˚y ÎyÎ˚– ˆ◊yï˛y Ä í˛z͈ϧÓ˚ üˆÏôƒ Îáö ÓƒÓôyö x˛õ!Ó˚Ó!ï≈˛ï˛
ÌyˆÏܲñ ï˛áö í˛zͧ ˆÌˆÏܲ !öà≈ï˛ ≤Ã!ï˛!ê˛ ï˛Ó˚D•z ˆ◊yï˛yÓ˚ ܲyˆÏåÈ ˆ˛õÑÔåÈÎ˚ ˆÜ˛ö öy í˛zͧ Ä ˆ◊yï˛yÓ˚ üôƒÓï≈˛#
üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚ ˆÜ˛yöÄ §MÈ˛Î˚ Óy «˛Î˚ âê˛ˆÏï˛ ˛õyˆÏÓ˚ öy– §%ï˛Ó˚yÇñ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyˆÏB˛Ó˚ ˆÜ˛yöÄ xy˛õyï˛ ˛õ!Ó˚Óï≈˛ö
âˆÏê˛ öy–
í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öy ˆÌˆÏܲ !ï˛ö!ê˛ !ö!ò≈‹T ˆ«˛ˆÏe í˛˛õ°yÓ˚ !e´Î˚y §¡∫ˆÏ¶˛ xy˛õ!ö çyöˆÏï˛ ˛õyÓ˚ˆÏ°ö– !ܲv
~ åÈyí˛¸yÄ í˛õy°Ó˚ !e´Î˚y §¡∫ˆÏ¶˛ !ܲå%È ≤ß¿ xy˛õöyÓ˚ üˆÏö çyày fl∫yû y!Óܲ– ~ÓyÓ˚ xyüÓ˚y ˆ§=!° §¡∫ˆÏ¶˛ çyöyÓ˚
ˆã˛‹T ܲÓ˚Ó–
SܲV xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ !ï˛ö!ê˛ ˆ«˛ˆÏe•z üyôƒüˆÏܲ !öÿ˛° ӈϰ ôˆÏÓ˚ ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈ–
üyôƒˆÏüÓ˚ Î!ò ˆÜ˛yöÄ à!ï˛ˆÏÓà ÌyˆÏܲ ï˛ˆÏÓ 6.23-27 §)e=!°Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö âê˛ˆÏÓ !ܲ⁄ ~Ó˚ í˛z_ˆÏÓ˚
Ó°y ÎyÎ˚ ˆÎñ ˆÎˆÏ•ï%˛ üyôƒü•z ¢∑ï˛Ó˚DˆÏܲ Ó•ö ܲˆÏÓ˚ñ xï˛~Ó üyôƒˆÏüÓ˚ ˆÓà ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ §ˆÏD
Ó#çày!î!ï˛Ü˛û˛yˆÏÓ Î%_´ •ˆÏÓ– xÌ≈yÍñ üyôƒˆÏüÓ˚ ˆÓà V Î!ò í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ !òˆÏܲ •Î˚ñ ï˛ˆÏÓ ¢ˆÏ∑Ó˚
ܲyÎ≈ܲÓ˚# ˆÓà •ˆÏÓ v + V– Î!ò üyôƒˆÏüÓ˚ ˆÓà ˆ◊yï˛y ˆÌˆÏܲ í˛z͈ϧÓ˚ !òˆÏܲ •Î˚ñ ï˛ˆÏÓ ˙ ˆÓà •ˆÏÓ v – V–
í zòy•Ó˚î !•§y ÏÓ ôÓ˚y Îyܲñ Óyï˛y ϧÓ˚ Óà V í zͧ Ì Ïܲ ◊yï˛yÓ˚ !ò Ïܲñ ◊yï˛yÓ˚ Óà í zͧ x!û˛ü% Ïá u ~ÓÇ í zÍ Ï§Ó˚
ˆÓà ˆ◊yï˛y x!û˛ü%ˆÏá w– ~áö xy˛õyï˛ Ü˛¡õyˆÏB˛Ó˚ üyö 6.28 §ü#ܲÓ˚ Ïî vÈ È~Ó˚ çyÎ˚àyÎ˚ v + V Ó!§ˆÏÎ˚ ˛õyÄÎ˚y ÎyˆÏÓ
'vVunn
vVw++ =+−
......... 6.29
°«˛ƒ ܲÓ˚&öñ ~áöÄ Î!ò ˆ◊yï˛y Ä í˛zͧ ~ܲ•z !òˆÏܲ ~ܲ•z ˆÓˆÏà ã˛°üyö •Î˚ñ xÌ≈yÍ Î!ò w=-u •Î˚ñ
ï˛ˆÏÓ n' ~Ó˚ üyö •ˆÏÓ
174
'vVunnn
vVu++ == ++
xÌ≈yÍñ ܲ¡õyˆÏB˛Ó˚ ˆÜ˛yöÄ xy˛õyï˛ ˛õ!Ó˚Óï≈˛ö ~ ˆ«˛ˆÏeÄ âê˛ˆÏÓ öy–
SáV ~ ˛õÎ≈hsˇ xyüÓ˚y ¢ˆÏ∑Ó˚ í˛zͧ Óy ˆ◊yï˛yÓ˚ ˆÓà ˛õÓ˚flõÓ˚ §ÇˆÏÎyàܲyÓ˚# §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ӈϰ ܲ“öy
ܲˆÏÓ˚!åÈ– í˛zͧ Óy ˆ◊yï˛yÓ˚ ˆÓà ˙ §Ó˚°ˆÏÓ˚áyÓ˚ §yˆÏ˛õˆÏ«˛
!ï˛Î≈ܲû˛yˆÏÓ ÌyܲˆÏ° xyˆÏàÓ˚ ≤Ãüy!îï˛ §)e=!° ≤ÈÏÎ˚yà ܲÓ˚y
ÎyˆÏÓ öy– !ã˛e 6.14 -ˆï˛ ~•z xÓfliy!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
~áyˆÏö ¢ˆÏ∑Ó˚ í˛zͧ S ~ÓÇ ˆ◊yï˛y L-~Ó˚ §ÇˆÏÎyàܲyÓ˚#
§Ó˚°ˆÏÓ˚áyÓ˚ §ˆÏD í˛z͈ϧÓ˚ ˆÓà
θ, w
ˆÜ˛yˆÏî ~ÓÇ ˆ◊yï˛yÓ˚
ˆÓà
φ, u
ˆÜ˛yˆÏî xyöï˛– ˆÓà ò%!ê˛Ó˚ xö%≤Ãfli í˛z˛õyÇ¢
sin wθ
Ä
sin uφ
¢∑ ï˛Ó˚ˆÏDÓ˚ í˛˛õ°yÓ˚ !e´Î˚yÎ˚ ˆÜ˛yöÄ û)˛!üܲy ˛õy°ö ܲˆÏÓ˚ öy– ˆÜ˛Ó°üye xö%˜Ïòâ≈ƒ í˛z˛õyÇ¢
cos wθ
~ÓÇ
cos uφ
ܲ¡õyˆÏB˛Ó˚ xy˛õyï˛ ˛õ!Ó˚Óï≈˛ö âê˛yÎ˚– §%ï˛Ó˚yÇñ 6.24 ˆÌˆÏܲ 6.28 §ü#ܲÓ˚î =!°ˆÏï˛
u ~Ó˚ çyÎ˚àyÎ˚
cos uφ
~ÓÇ
w−
~Ó˚ çyÎ˚àyÎ˚
cos wθ
Ó§yˆÏ°•z !ï˛Î≈ܲ à!ï˛Ó˚ ˆ«˛ˆÏe í˛z˛õÎ%_´ §)e ˛õyÄÎ˚y
ÎyˆÏÓ– ï˛ˆÏÓ üˆÏö Ó˚yáˆÏï˛ •ˆÏÓ ˆÎñ ~ˆÏ«˛ˆÏe í˛zͧ Ä ˆ◊yï˛yÓ˚ §ÇˆÏÎyàܲyÓ˚# ˆÓ˚áy!ê˛Ó˚ !òܲ §ï˛ï˛•z ˛õyŒê˛yˆÏï˛
ÌyˆÏܲ– §%ï˛Ó˚yÇñ u ~ÓÇ w ~Ó˚ üyö x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏ°Ä í˛˛õ°yÓ˚ !e´Î˚yÓ˚ ú˛ˆÏ° ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚
üyö §üyö ÌyˆÏܲ öy–
SàV ˆ◊yï˛y Óy í˛z͈ϧÓ˚ ˆÓà Îáö üyôƒˆÏü ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚ §üyö •Î˚ñ ï˛áö ܲï˛Ü˛=!° !ӈϢ£Ï âê˛öy
â Ïê˛–
ˆ◊yï˛y Î!ò í˛zͧ ˆÌˆÏܲ ¢ˆÏ∑Ó˚ §üyö ˆÓˆÏà ò)ˆÏÓ˚ §ˆÏÓ˚ ˆÎˆÏï˛ ÌyˆÏܲö xÌ≈yÍ Î!ò 6.25 §)ˆÏe u=v •Î˚ñ
ï˛ˆÏÓ xy˛õyï˛ Ü˛¡õyB˛ n' ~Ó˚ üyö •ˆÏÓ ¢)öƒ– xy§ˆÏ° ~ˆÏ«˛ˆÏe ˆÜ˛yöÄ ï˛Ó˚D•z ˆ◊yï˛yˆÏܲ ï˛yí˛¸y ܲˆÏÓ˚ x!ï˛e´ü
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓ öy– ú˛ˆÏ° ˆ◊yï˛yÓ˚ ܲyˆÏåÈ Ü˛¡õyˆÏB˛Ó˚ üyöÄ ¢)öƒ ӈϰ üˆÏö •ˆÏÓ–
xyÓyÓ˚ ˆòá%öñ Î!ò ˆ◊yï˛y !fliÓ˚ ÌyˆÏܲö ~ÓÇ í˛zͧ ¢ˆÏ∑Ó˚ §üyö ˆÓˆÏà ˆ◊yï˛yÓ˚ !òˆÏܲ ôy!Óï˛ •Î˚ñ ï˛ˆÏÓ
6.26 §)ˆÏe w = v Ó!§ˆÏÎ˚ ˛õyÄÎ˚y ÎyˆÏÓ
'' 1
nn
vv
=−
ÎyÓ˚ üyö x§#ü– ~ˆÏ«˛ˆÏe í˛zͧ ˆÌˆÏܲ !öà≈ï˛ §Ó ï˛Ó˚D §B%˛!ã˛ï˛ •ˆÏÎ˚ ~ܲ!ê˛ !Ó¢y° xyˆÏ°yí˛¸ö
!•§yˆÏÓ í˛z͈ϧÓ˚ §ˆÏD•z ã˛°ˆÏï˛ ÌyܲˆÏÓ– ~•z xyˆÏ°yí˛¸ö Îáö ˆ◊yï˛yˆÏܲ x!ï˛e´ü ܲÓ˚ˆÏÓñ ï˛áö ˆ◊yï˛y x§Çáƒ
ï˛Ó˚ˆÏDÓ˚ ~ܲ !ü!°ï˛ xyâyï˛ xö%û˛Ó ܲÓ˚ˆÏÓöñ Îy ܲï˛Ü˛ê˛y !ÓˆÏfl≥˛yÓ˚ˆÏîÓ˚ ¢ˆÏ∑Ó˚ üï˛ ˆ¢yöyˆÏÓ– ~•z ¢∑!ê˛ˆÏܲ
Ó°y •Î˚ ¢y∑ÈÙÈÓ%ü (sonic boom)– ¢ˆÏ∑y_Ó˚ ˆÓˆÏà í˛zí˛¸ˆÏï˛ §«˛ü (supersonic) !ÓüyˆÏöÓ˚ ˆÓà Îáö ¢ˆÏ∑Ó˚
ˆÓàˆÏܲ x!ï˛e´ü ܲˆÏÓ˚ñ ï˛áö ˆ§!ê˛ ¢y∑ÈÙÈÓ%ü ˜ï˛!Ó˚ ܲˆÏÓ˚ñ Îy üy!ê˛ ˆÌˆÏܲ ˆ¢yöy ÎyÎ˚–
í˛z͈ϧÓ˚ ˆÓà ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ ˆã˛ˆÏÎ˚ ˆÓ!¢Ä •ˆÏï˛ ˛õyˆÏÓ˚– °«˛ƒ ܲÓ˚öñ 6.26 §)ˆÏe Î!ò w>v •Î˚ñ ï˛ˆÏÓ xy˛õyï˛
175
ܲ¡õyˆÏB˛Ó˚ üyö îydܲ •Î˚– xy§ˆÏ° ~•z xÓfliyÎ˚ í˛zͧ ˆÌˆÏܲ !öà≈ï˛ ¢ˆÏ∑Ó˚ xyˆÏà•z í˛zͧ!ê˛ ˆ◊yï˛yˆÏܲ
x!ï˛e´ü ܲˆÏÓ˚ ~ÓÇ ˛õˆÏÓ˚ í˛zͲõߨ ï˛Ó˚D ˆ◊yï˛yÓ˚ ܲyˆÏåÈ xyˆÏà ˆ˛õÑÔåÈyÎ˚– ~•z xÓfliyÎ˚ í˛˛õ°yÓ˚ !e´Î˚y ≤ÈÏÎyçƒ
ÌyˆÏܲ öy–
SâV ¢∑ï˛Ó˚ˆÏDÓ˚ üï˛ xyˆÏ°yܲ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÄ í˛˛õ°yÓ˚ !e´Î˚y âê˛ˆÏï˛ ˆòáy ÎyÎ˚– ï˛ˆÏÓ xyˆÏ°yˆÏܲÓ˚ ˆÓà
¢ˆÏ∑Ó˚ ï%˛°öyÎ˚ xˆÏöܲ ˆÓ!¢ •ÄÎ˚yÎ˚ñ ܲ¡õyˆÏB˛Ó˚ xy˛õyï˛ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ üyö ~ˆÏ«˛ˆÏe xï˛ƒhsˇ ܲü •Î˚ ~ÓÇ xï˛ƒhsˇ
§)«¯˛ ΈÏsfÓ˚ §y•yˆÏ΃•z ï˛y ôÓ˚y ÎyÎ˚– xyüÓ˚y ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ˆÎ ôÓ˚ˆÏöÓ˚ àîöy ˛õk˛!ï˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚!åÈñ
xyˆÏ°yܲï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ï˛y ≤ÈÏÎyçƒ ÌyˆÏܲ öy– ˆÜ˛ö öy xyˆÏ°yˆÏܲÓ˚ ˆ«˛ˆÏe xyˆÏ˛õ!«˛Ü˛ï˛yÓ˚ !ӈϢ£Ï ï˛_¥ xö%ÎyÎ˚#
àîöy ܲÓ˚ˆÏï˛ •Î˚– xyˆÏ°yܲï˛Ó˚ˆÏD í˛˛õ°yÓ˚ !e´Î˚y §¡∫ˆÏ¶˛ ˛õˆÏÓ˚ xöƒ ˛õÎ≈yˆÏÎ˚ !ÓˆÏ˘¢£Ï xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÓ–
~áö í˛˛õ°yÓ˚ !e´Î˚y !Ó£ÏÎ˚ܲ !öˆÏã˛Ó˚ xö%¢#°ö# ò%!ê˛Ó˚ í˛z_Ó˚ !òˆÏï˛ xy˛õöyÓ˚ •Î˚ û˛y°•z °yàˆÏÓ–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ6 : 600 Hz ܲ¡õyB˛ !Ó!¢‹T ~ܲ!ê˛ ¢ˆÏ∑Ó˚ í˛zͧ !fliÓ˚ ˆ◊yï˛yÓ˚ !òˆÏܲ 40ms–1 à!ï˛ˆÏÓˆÏà x@ˇÃ§Ó˚
•ˆÏ° ˙ ܲ¡õyB˛ ˆ◊yï˛yÓ˚ ܲyˆÏåÈ Ü˛ï˛ ÓˆÏ° üˆÏö •ˆÏÓ⁄ Î!ò ~ܲ•z à!ï˛ˆÏÓˆÏà ˆ◊yï˛y x@ˇÃ§Ó˚ •Î˚ ~ÓÇ í˛zͧ
!fliÓ˚ ÌyˆÏܲñ ï˛y•ˆÏ°•z Óy ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛ Ü˛ï˛ •ˆÏÓ⁄ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 340ms–1–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 7 : ˆÓ˚°ÄˆÏÎ˚ ≤’ƒyê˛ú˛ˆÏü≈ òÑyí˛¸yˆÏöy ~ï˛ Óƒ!_´ ˆòáˆÏ°ö ˆÎñ 36kmph ˆÓˆÏà ~ܲ!ê˛ ˆê˛Δö ÓÑy!¢
Óy!çˆÏÎ˚ ˆfiê˛¢ˆÏö ≤ÈÏÓ¢ ܲÓ˚ˆÏåÈ– ˆê˛ö!ê˛ ~ܲ•z ˆÓˆÏà ÓÑy!¢ Óy!çˆÏÎ˚ ˆfiê˛¢ö ˆåȈÏí˛¸ ˆà°– ˙ Óƒ!_´ °«˛ƒ ܲÓ˚ˆÏ°ö
ˆÎñ ˆê˛Δö!ê˛ ˆfiê˛¢ˆÏö ≤ÈÏÓ¢ ܲÓ˚yÓ˚ Ä ˆÓ!Ó˚ˆÏÎ˚ ÎyÄÎ˚yÓ˚ §üÎ˚ !ï˛!ö ÓÑy!¢Ó˚ ˆÎ ܲ¡õyB˛ ÷ˆÏöˆÏåÈö ï˛yˆÏòÓ˚ üˆÏôƒ
˛õyÌ≈ܲƒ 17 Hz, Î!ò Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 340ms-1 •Î˚ñ ï˛y•ˆÏ° ÓÑy!¢Ó˚ ≤ÃÜ,˛ï˛ ܲ¡õyB˛ ܲï˛⁄
6.10 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
Îy!sfܲ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° §,‹T xyˆÏ°yí˛¸ö !fli!ï˛fliy˛õܲ üyôƒˆÏüÓ˚ üˆÏôƒ !òˆÏÎ˚ ã˛°ï˛Ó˚ˆÏDÓ˚ xyܲyˆÏÓ˚ ≤çyÓ˚°yû˛
ܲˆÏÓ˚– ◊ÓîˆÏÎyàƒ Ü˛¡õyˆÏB˛Ó˚ xö%˜Ïòâ≈ƒ !fli!ï˛fliy˛õܲ ï˛Ó˚D•z ¢∑ï˛Ó˚D– ã˛°ï˛Ó˚ˆÏDÓ˚ üyôƒˆÏü ¢!_´Ó˚ ˛õ!Ó˚Ó•î
â Ïê˛–
~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ !Ó!û˛ß¨ ˜Ó!¢‹Tƒ ï˛yÓ˚ ˆÓàñ ܲ¡õyB˛ Ä ï˛Ó˚D˜Ïòâ≈ƒ– ˆÜ˛yöÄ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ n Ä ï˛Ó˚D˜Ïòâ≈ƒ
λ •ˆÏ°ñ ï˛Ó˚ˆÏDÓ˚ ˆÓà
vnλ =
§¡õˆÏÜ≈˛Ó˚ myÓ˚y §)!ã˛ï˛ •Î˚–
ï˛Ó˚ˆÏDÓ˚ ˆÓà üyôƒˆÏü !fli!ï˛fliy˛õܲ ôü≈ Ä xöƒyöƒ ˜Ó!¢ˆÏ‹TƒÓ˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚–
ã˛!Ó˚eàï˛ ˜Ó!¢‹Tƒ ˆÌˆÏܲ ï˛Ó˚DˆÏܲ !Ó!û˛ß¨ ˆ◊î#ˆÏï˛ û˛yà ܲÓ˚y ÎyÎ˚– ˆÎüöñ xö%˜Ïòâ≈ƒ Ä xö%≤Ãfli ï˛Ó˚D
~ܲüy!eܲñ !müy!eܲ Óy !eüy!eܲ ï˛Ó˚Dñ §üï˛° Óy ˆày°#Î˚ ï˛Ó˚D– ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚°yˆÏû˛Ó˚ §üÎ˚ üyôƒˆÏüÓ˚
§y!Ó≈ܲ §Ó˚î âˆÏê˛ öyñ üyôƒˆÏüÓ˚ ܲîy=!° ï˛yˆÏòÓ˚ §yüƒyÓfliyˆÏöÓ˚ í˛zû˛Î˚ ˛õyˆÏ¢ xyˆÏ®y!°ï˛ •Î˚–
~ܲüy!eܲ ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î
()() 2 ,sin yxtAvtxπλ =±
!•§yˆÏÓ ˆ°áy ÎyÎ˚– ~•z §ü#ܲÓ˚ˆÏîÓ˚ xyÓ˚Ä
ܲˆÏÎ˚ܲ!ê˛ !Óܲ“ Ó˚*˛õ Ó˚ˆÏÎ˚ˆÏåÈ– ‘+’ Óy '-' !ã˛•´ xö%ÎyÎ˚# §ü#ܲÓ˚î!ê˛ x xˆÏ«˛Ó˚ ÎÌye´ˆÏü ôöydܲ Ä îydܲ
!òˆÏܲ §MÈ˛y!Ó˚ï˛ ï˛Ó˚D ≤Ãܲy¢ ܲˆÏÓ˚– ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î!ê˛ !Ó!û˛ß¨ !Óܲ“Ó˚*ˆÏ˛õÄ ˆ°áy ÎyÎ˚– §ü#ܲÓ˚î!ê˛
!müy!eܲ Ä !eüy!eܲ ï˛Ó˚ˆÏDÓ˚ çöƒ ˛õ!Ó˚Ó!ô≈ï˛Ó˚*ˆÏ˛õ ˆ°áy ÎyÎ˚–
176
x x«˛ ÓÓ˚yÓÓ˚ !ÓhflÏyÓ˚°yû˛ ܲÓ˚ˆÏåÈ ~üö ~ܲüy!eܲ ã˛°ï˛Ó˚ˆÏDÓ˚ §yôyÓ˚î ï˛Ó˚D §ü#ܲÓ˚î
22
2221
xvt∂∂ =∂∂
ψψ
!müy!eܲ Ä !eüy!eܲ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ï˛Ó˚D §ü#ܲÓ˚î!ê˛Ó˚ Ó˚*˛õ •Î˚
22
2210vt
∂ ∇−=∂
ψ ψ
ã˛°ï˛Ó˚ˆÏDÓ˚ myÓ˚y ¢!_´Ó˚ ˛õ!Ó˚Ó•ˆÏîÓ˚ •yÓ˚ 2 21
2P pA v= α ω –
ˆÎáyˆÏö
a=
ï˛Ó˚ˆÏDÓ˚ ≤ÃfliˆÏFåÈò ˆ«˛eú˛°ñ
p=
üyôƒˆÏüÓ˚ âöcñ A,
ω
Ä v ÎÌye´ˆÏü ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ñ
ˆÜ˛Ô!îܲ ܲ¡õyB˛ Ä ˆÓà–
~•z ¢!_´Ó˚ xˆÏô≈ܲ àí˛¸ !fli!ï˛¢!_´ Ä xˆÏô≈ܲ àí˛¸ à!ï˛¢!_´– ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y Ó°ˆÏï˛ ~ܲܲ °¡∫ ≤ÃfliˆÏFåȈÏòÓ˚
üôƒ !òˆÏÎ˚ ˛õ!Ó˚Óy!•ï˛ «˛üï˛y ˆÓyé˛yÎ˚– §üï˛° ï˛Ó˚ˆÏD ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ï˛#Ó ï˛y §Ó≈e §üyö ÌyˆÏܲñ !ܲv
í˛zͧ !Ó®% ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚ ÓˆÏà≈Ó˚ ÓƒhflÏyö%˛õyˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚–
í˛zͧ ~ÓÇ ˆ◊yï˛yÓ˚ üˆÏôƒ xyˆÏ˛õ!«˛Ü˛ à!ï˛ ÌyܲˆÏ°ñ í˛zͧ ˆÌˆÏܲ í˛zͲõߨ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ˆ◊yï˛yÓ˚ ܲyˆÏåÈ
˛õ!Ó˚Ó!ï≈˛ï˛ üˆÏö •Î˚– ~•z âê˛öy!ê˛ˆÏܲ í˛˛õ°yÓ˚ !e´Î˚y Ó°y •Î˚–
6.11 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ~ܲüy!eܲ Ä !eüy!eܲ ï˛Ó˚Dñ xö%˜Ïòâ≈ƒ Ä xö%≤Ãfli ï˛Ó˚D Ä §üï˛° Ä ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ üˆÏôƒ ˛õyÌ≈ܲƒ
Ó%!é˛ˆÏÎ˚ !òö–
2. ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ
() ,sinxxtAt
v⎛⎞ =− ⎝⎠ ψω
§ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y ܲÓ˚&ö– ˆÜ˛Ô!îܲ
ܲ¡õyB˛ Ä ω ï˛Ó˚DˆÏÓà v ~Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ ˛õÎ≈yÎ˚ܲy° T ~ÓÇ ï˛Ó˚D˜Ïòâ≈ƒ λ ~Ó˚ üyôƒˆÏü §ü#ܲÓ˚î!ê˛ !°á%ö–
3. ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ày!î!ï˛Ü˛ Ó˚*ˆÏ˛õÓ˚ §y•yˆÏ΃ ï˛Ó˚D §ü#ܲÓ˚î
2 2
2 2 21
x v t∂ ∂=∂ ∂
ψ ψ !ê˛ ≤Ã!ï˛¤˛y ܲÓ˚&ö– ˆòáyö
ˆÎñ fliyî% ï˛Ó˚ˆÏDÓ˚ §üÎ˚ §Ó˚î §¡õÜ≈˛!ê˛
()22 ,sinsin vtxxta =ππ ψλλ
ï˛Ó˚D §ü#ܲÓ˚îˆÏܲ !§k˛ ܲˆÏÓ˚–
4. ≤Ãüyî ܲÓ˚%ö ˆÎñ ~ܲüy!eܲ !fli!ï˛fliy˛õܲ ã˛°ï˛Ó˚ˆÏD à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ àí˛¸ üyö §üyö–
5. ≤Ãüyî ÜÓ˚&ö ˆÎñ !fli!ï˛fliy˛õܲ ã˛°ï˛Ó˚ˆÏDÓ˚ ¢!_´ ˛õ!Ó˚Ó•ˆÏîÓ˚ •yÓ˚ Ä ï˛#Ó ï˛y ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚
§üyö%˛õyï˛#–
177
6. ≤Ãüyî ÜÓ˚&ö ˆÎñ Îáö ¢ˆÏ∑Ó˚ í˛zͧ !öÿ˛° ˆ◊yï˛yÓ˚ !òˆÏܲ à!ï˛¢#° •Î˚ ~ÓÇ Îáö ˆ◊yï˛y !öÿ˛° í˛z͈ϧÓ˚
!òˆÏܲ à!ï˛¢#° •öñ ò%•z ˆ«˛ˆÏe•z ˆ◊yï˛y í˛z͈ϧÓ˚ ≤ÃÜ,˛ï˛ ܲ¡õyˆÏB˛Ó˚ ˆã˛ˆÏÎ˚ í˛zFã˛ï˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏï˛ ˛õyö–
7. (a) ÓyÎ˚%üyôƒˆÏü ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà
1 340ms−
•ˆÏ° 255Hz ܲ¡õyB˛ !Ó!¢‹T §%Ó˚¢°yܲy ˆÌˆÏܲ !öɧ,ï˛
ܲï˛=!° ¢∑ï˛Ó˚D 68m ò)Ó˚c ç%ˆÏí˛¸ ÌyܲˆÏÓ⁄
(b) çˆÏ° ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 1450ms–1 – xyüyˆÏòÓ˚˚ ܲyö §Ó≈!ö¡¨ Ä §ˆÏÓ≈yFã˛ ˆÎ ܲ¡õyB˛ ÷öˆÏï˛ ˛õyˆÏÓ˚ñ
ï˛y ÎÌye´ˆÏü 20Hz ~ÓÇ 20kHz– ~•z ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ=!° àîöy ܲÓ˚&ö–
8. ˆÜ˛yöÄ ~ܲ!ê˛ ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î
4sin10020
xyt ⎛⎞ =− ⎝⎠
π π
~áyˆÏö y ~ÓÇ x !üê˛yˆÏÓ˚ ~ÓÇ t
ˆ§ˆÏܲˆÏu˛ ≤Ãܲy¢ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ï˛Ó˚D!ê˛Ó˚ !ÓhflÏyÓ˚ñ ܲ¡õyB˛ ~ÓÇ ï˛Ó˚D˜Ïòâ≈ƒ !öî≈Î˚ ܲÓ˚&ö–
9. ~ܲ!ê˛ ˛õê˛Ü˛y ú˛yê˛yˆÏöy •ˆÏ° !ÓˆÏfl≥˛yÓ˚î ˆÌˆÏܲ lm ò)Ó˚ˆÏc ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •Î˚ 8 ×10-5 Wm-2– •Î˚ñ
ï˛ˆÏÓ !ÓˆÏfl≥˛yÓ˚ˆÏîÓ˚ !Ó®% ˆÌˆÏܲ ܲï˛ò)Ó˚ ˛õÎ≈hsˇ ˛õê˛Ü˛yÓ˚ ¢∑ ˆ¢yöy ÎyˆÏÓ⁄ .
10. ˆ◊yï˛y ¢ˆÏ∑Ó˚ í˛zͧ x!û˛ü%ˆÏá Ü˛ï˛ ˆÓˆÏà ã˛°ˆÏ° ïÑ˛yÓ˚ ܲyˆÏåÈ 270Hz ܲ¡õyˆÏB˛Ó˚ ¢∑ 279Hz
ܲ¡õyˆÏB˛Ó˚ ӈϰ üˆÏö •ˆÏÓ⁄ ôˆÏÓ˚ !ööñ ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 330ms–1–
6.12 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# 1 :
v nλ=∴ 340 = 400λ
∴ λ = 0.85m
ˆÎˆÏ•ï%˛ñ ~ܲ!ê˛ Ü˛¡õˆÏöÓ˚ ú˛ˆÏ° ¢∑ï˛Ó˚D λ õ!Ó˚üyî ò)Ó˚c x@ˇÃ§Ó˚ •Î˚ñ xï˛~Ó 300 ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ¢ˆÏ∑Ó˚
x!ï˛e´yhsˇ ò)Ó˚c = 300 ×0.85m = 255m
2. ôÓ˚&ö ܲ¡õyB˛ = n– ≤ß¿yö%§yˆÏÓ˚ñ
10, Acm λ=
15, Bcm λ=
1 90,? AB vcmsv − == AA vnλ =
~ÓÇ
BB vnλ =
∴
1510
BBB
AAA
vnvn
=== λλλλ
∴ 11590 13510Bv ms−= × =
178
3 ˆòÄÎ˚y xyˆÏåÈñ ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î
( )8sin 4.00 0.02y t x cmπ= −
§ü#ܲÓ˚î!ê˛ˆÏܲ §y!çˆÏÎ˚ ≤Ãyüyîƒ Ó˚*ˆÏ˛õ ˆ°áy ÎyÎ˚ñ
( )28sin 200.00100
y t x cmπ= −
ã˛°ï˛Ó˚ˆÏDÓ˚ ≤Ãyüyîƒ §ü#ܲÓ˚î ( )2siny A vt xπλ= − -~Ó˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚ñ
ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ A = 8cm
ï˛Ó˚D˜Ïòâ≈ƒ λ = 100cm = lm SˆÎˆÏ•ï%˛ ~áyˆÏö x ~ÓÇ ycm- ~ ˛õ!Ó˚üy˛õ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ)
à!ï˛ˆÏÓà v = 200cms–1 SˆÎˆÏ•ï%˛ §üÎ˚ t ~Ó˚ ~ܲܲ ~áyˆÏö ˆ§ˆÏܲ[˛V
= 2ms–1
∴ ܲ¡õyB˛
2 vnHz λ ==
ï˛Ó˚D!ê˛ x xˆÏ«˛Ó˚ ôöydܲ !òˆÏܲ §MÈ˛y!Ó˚ï˛ •ˆÏFåÈ– ~•z ï˛Ó˚ˆÏDÓ˚ ≤çyÓ˚°yˆÏû˛Ó˚ x!û˛ü%ˆÏá 20cm ò)Ó˚ˆÏc ò%!ê˛
!Ó®%Ó˚ ò¢y ˛õyÌ≈ܲƒ δ •ˆÏ°ñ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
2 220 20100
= × = ×radianπ πδ λ rad
=
0 2725
radπ=
4. ~áyˆÏö !ÓhflÏyÓ˚ A = 0.03m
ܲ¡õyB˛ 550 ,n Hz= ï˛Ó˚ˆÏDÓ˚ ˆÓà v = 330ms–1
ï˛Ó˚D˜Ïòâ≈ƒ
33030.65505
vmm
nλ====
ˆÎˆÏ•ï%˛ ï˛Ó˚D!ê˛ + x ~Ó˚ !òˆÏܲ §MÈ˛y!Ó˚ï˛ •ˆÏFåÈñ xï˛~Ó ˆ°áy ÎyÎ˚ñ
179
ˆÎ ï˛Ó˚ˆÏDÓ˚ §Ó˚î §ü#ܲÓ˚î ( )2siny A vt xπλ= −
Óyñ
() 2 .03sin3300.6
ytxmπ =−
= ( ).03sin 1100 3.33t x mπ −
~áyˆÏö x ~ÓÇ y !üê˛yˆÏÓ˚ ~ÓÇ t ˆ§ˆÏܲˆÏ[˛ ≤Ãܲy!¢ï˛ •ˆÏÎ˚ˆÏåÈ–
§ü#ܲÓ˚î!ê˛ ( ).03cos 1100 3.33y t x mπ= −
Óyñ ( ).03sin 1100 3.33y t x⎡ ⎤= − +⎣ ⎦π φ (
φ
= x!ö!ò≈‹T ò¢yˆÏܲyî)
~û˛yˆÏÓÄ ˆ°áy ÎyÎ˚–
5. xyüÓ˚y çy!ö ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚xyüÓ˚y çy!ö ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚xyüÓ˚y çy!ö ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚xyüÓ˚y çy!ö ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚xyüÓ˚y çy!ö ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚
ï˛#Ó ï˛y
22222 122
IAvAnv == ωρπρ
SˆÜ˛ööy
2 =n ωπ
)
~áyˆÏöñ
1000, nHz =
p =1.29Kgm–3
1 340,? vmsA − ==ˆÎˆÏ•ï%˛ §ÓˆÏã˛ˆÏÎ˚ ܲü ≤ÃyӈϰƒÓ˚ ˆÎ ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö ¢ˆÏ∑Ó˚ xö%û)˛!ï˛Ó˚ §,!‹T ܲˆÏÓ˚ ï˛y •ˆÏFåÈ 122, 10Wm −−
xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
()2 2212 210001.2934010 IA π− =××=
∴ ( )12
222
102 1000 340 1.29
A−
=× × ×π
∴ 111.074 10A m−= ×
°«˛ƒ ܲÓ˚&öñ ~•z !ÓhflÏyÓ˚ ˛õÓ˚üyî%Ó˚ xyܲyˆÏÓ˚Ó˚
() 10 10m −
ˆÌˆÏܲ ˆåÈyê˛– ~Ó˚ ˆÌˆÏܲ xyüyˆÏòÓ˚ ◊ÓîÎsf!ê˛Ó˚
ܲü≈ò«˛ï˛yÓ˚ !ܲå%Èê˛y öü%öy ˛õyÄÎ˚y ÎyÎ˚–
6. 600n Hz= , ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà
1 340 vms− =
!fliÓ˚ ˆ◊yï˛yÓ˚ !òˆÏܲ í˛z͈ϧÓ˚ à!ï˛ˆÏÓà Îáö
1 40 s vms− =
˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛ 1600 600 340 680 .
40 30011 340s
nn Hz
vv
×= = = =⎛ ⎞ −−⎜ ⎟⎝ ⎠
180
Îáö í˛zͧ !fliÓ˚ ~ÓÇ ˆ◊yï˛y 12 40v ms−= ˆÓˆÏà í˛z͈ϧÓ˚ !òˆÏܲ à!ï˛¢#°ñ ï˛áö ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛ n2
•ˆÏ°
12
40 600 3801 600 1 670.6340 340
vn n Hz
v×⎛ ⎞ ⎛ ⎞= + = + = =⎜ ⎟ ⎝ ⎠⎝ ⎠
°«˛ƒ ܲÓ˚&öñ ò%!ê˛ ˆ«˛ˆÏe ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛ ò%!ê˛ x§üyö–
7. ˆê˛ΔˆÏöÓ˚ à!ï˛ˆÏÓà = 136 1000363600
kmph ms−×=
= 110ms−
ôÓ˚y Îyܲñ ÓÑy!¢Ó˚ ≤ÃÜ,˛ï˛ ܲ¡õyB˛ n– ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà 340ms–1– Îáö ˆê˛Δö!ê˛ ˆfiê˛¢ˆÏö ì%˛Ü˛ˆÏåÈ ï˛áö ˆ◊yï˛y
!fliÓ˚ñ í˛zͧ ˆ◊yï˛yÓ˚ x!û˛ü%ˆÏá
1 10ms−
à!ï˛ˆÏÓˆÏà ôyÓüyö–
∴ ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛
1340
10330 1340
nnn==
−
Îáö ˆê˛Δö!ê˛ ˆfiê˛¢ö ˆÌˆÏܲ ˆÓ!Ó˚ˆÏÎ˚ ÎyˆÏFåÈñ ï˛áöÄ ˆ◊yï˛y !fliÓ˚ñ !ܲv í˛zͧ 10ms–1 ÈÙÈ~ ˆ◊yï˛yÓ˚ ˆÌˆÏܲ
ò)ˆÏÓ˚ §ˆÏÓ˚ ÎyˆÏFåÈ–
∴ ˛õ!Ó˚Ó!ï≈˛ï˛ ܲ¡õyB˛ 2340
10 3501340
n nn = =
+
≤ß¿yö%§yˆÏÓ˚ñ 1 2 17n n Hz− =
∴
11 34017330350
n⎛⎞ −= ⎜⎟ ⎝⎠
∴
3503301728920340
nHz×× == ×
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. 6.3 xö%ˆÏFåȈÏòÓ˚ §y•yˆÏ΃ xy˛õ!ö ≤ß¿!ê˛Ó˚ í˛z_Ó˚ !°áˆÏï˛ ˛õyÓ˚ˆÏÓö–
2. 6.5 xö%ˆÏFåȈÏòÓ˚ ~ !ӣψÏÎ˚ !Óhfl,Ïï˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
3. ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ xy˛õ!ö 6.6 xö%ˆÏFåȈÏò ˛õyˆÏÓö–
181
≤ÃÌü xÇ¢ : ~áyˆÏöñ
2 2
2222 .sinsin vtx
ax
∂⎛⎞ =−⎝⎠ ∂ψπππ
λλλ2 2
2222 .sinsin vvtx
ax
∂⎛⎞ =−⎝⎠ ∂ψπππ
λλλ
§%ï˛Ó˚yÇñ
22
2221
xvt∂∂ =∂∂
ψψ
4. 6.7 xLjϢ ~!ê˛ ≤Ãüyî ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5. 6.7 Ä 6.7.1 xLjϢ ≤ß¿!ê˛Ó˚ í˛z_Ó˚ ˛õyˆÏÓö–
6. 6.9 xLjϢ ~ !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
7. (a) ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà
1 340,255 vmsnHz − ==
340 4255 3
vm m
nλ = = =
§%ï˛Ó˚yÇñ ~ܲ!ê˛ ï˛Ó˚D 43
m ˜òâ≈ƒ!Ó!¢‹T xMÈ˛° òᰠܲˆÏÓ˚–
68m ò)Ó˚c x!ï˛e´ˆÏüÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ï˛Ó˚D §Çáƒy = 68λ
=
685143
=
≤ÈÏÎ˚yçö#Î˚ ï˛Ó˚D §Çáƒy = 51
(b) çˆÏ° ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà
1 1450 w Vms− =
20 Hz ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ çöƒ çˆÏ° ï˛Ó˚D˜Ïòâ≈ƒ λ114520
72 5= = . m
( )20 20000z zkH H ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ çöƒ çˆÏ°
ï˛Ó˚D˜Ïòâ≈ƒ
21450.07257.2520000
mcm λ===
ï˛Ó˚D˜Ïòâ≈ƒ 21450 .0725 7.2520000
m cmλ = = =
182
8. ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î 4sin 10020
⎛ ⎞= −⎜ ⎟⎝ ⎠x
y tππ
∴ ( )4sin 200020
= −y t xπ
= ( )24sin 200040
t x−π
~•z §ü#ܲÓ˚î!ê˛ˆÏܲ xyüyˆÏòÓ˚ ˛õ!Ó˚!ã˛ï˛ ( )2siny A vt xπλ= − -~Ó˚ §ˆÏD ï%˛°öy ܲˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚ñ ï˛Ó˚ˆÏDÓ˚
!ÓhflÏyÓ˚
4 Am =
ï˛Ó˚D˜Ïòâ≈ƒ λ = 40m
1 2000 vms− =
ܲ¡õyB˛
20005040
vnHz λ ===
9. ≤ß¿yö%§yˆÏÓ˚ ¢ˆÏ∑Ó˚ í˛zͧ S˛õê˛Ü˛y ú˛yê˛yˆÏöy !Ó®%V ˆÌˆÏܲ lm ò)ˆÏÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y I=8×10–5Wm–2 xyüÓ˚y
çy!öñ ◊ÓîˆÏÎyàƒ ö)ƒöï˛ü ï˛#Ó ï˛y 12 2
0 10I Wm− −= ~ÓÇ !Ó®% í˛zͧ ˆÌˆÏܲ í˛zͲõߨ ˆày°#Î˚ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe
ï˛#Ó ï˛y ò)Ó˚ˆÏcÓ˚ ÓˆÏà≈Ó˚ ÓƒhflÏyö%˛õyˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– Î!ò ô!Ó˚ ˆÎñ ~•z ˛õê˛Ü˛yÓ˚ ¢∑ í˛zͧ ˆÌˆÏܲ dm ò)Ó˚c
˛õÎ≈hsˇ ◊&!ï˛ˆÏàyã˛Ó˚ •ˆÏÓñ ï˛y•ˆÏ° ˆ°áy ÎyÎ˚
2
20 1I dI
=
∴ 5
212
8 1010
d−
−× = ∴
27 810 d=×
∴
3 9109 dmkm =×=
xï˛~Óñ ˙ ˛õê˛Ü˛yÓ˚ ¢∑ 9 km ò)Ó˚ ˛õÎ≈hsˇ ◊&!ï˛ˆÏàyã˛Ó˚ •ÄÎ˚y §Ω˛Ó– ~•z àîöyÎ˚ xÓ¢ƒ xyüÓ˚y ÓyÎ˚%üyôƒˆÏü
¢ˆÏ∑Ó˚ ˆ¢y£Ïî !ÓˆÏÓã˛öy ܲ!Ó˚!ö–
10. ≤ß¿yö%§yˆÏÓ˚ñ n'=279Hz, n = 270Hz, ¢ˆÏ∑Ó˚ ˆÓà 330ms–1
ôÓ˚y Îyܲñ ˆ◊yï˛yÓ˚ à!ï˛ˆÏÓà
,L v
ˆ°áy ÎyÎ˚ ˆÎñ
'1L vnn
v⎛⎞ =+ ⎜⎟ ⎝⎠
∴ 279 =270
1330
L v ⎛⎞ + ⎜⎟ ⎝⎠
∴
1 279133011270 L vms− ⎛⎞ =−= ⎜⎟ ⎝⎠
183
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 7 ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛
àë˛öàë˛öàë˛öàë˛öàë˛ö
7.1 ≤ÃhflyÓöy
í˛ zˆ Ïj¢ƒí˛ zˆ Ïj¢ƒí˛ zˆ Ïj¢ƒí˛ zˆ Ïj¢ƒí˛ zˆ Ïj¢ƒ
7.2 ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥
7.3 fliyî% ï˛Ó˚D
7.3.1 fliyî% ï˛Ó˚ˆÏD ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ Ó›Ü˛îyÓ˚ ˆÓà Ä !ÓÜ,˛!ï˛
7.3.2 fliyî% ï˛Ó˚ˆÏD §üˆÏü°
7.3.3 fliyî% ï˛Ó˚ˆÏDÓ˚ ôü≈
7.4 ï˛Ó˚DˆÏÓà Ä ò°àï˛ ˆÓà S§Çâ ˆÓàV
7.5 fl∫Ó˚ܲ¡õ
¢∑ ï˛Ó˚ˆÏDÓ˚ Óƒ!ï˛ã˛yÓ˚
7.6 §yÓ˚yÇ¢
7.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
7.8 í˛z_Ó˚üy°y
7.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
xy˛õöyÓ˚y ï˛Ó˚D §¡∫ˆÏ¶˛ Îy ˛õˆÏí˛¸ˆÏåÈö ï˛yˆÏï˛ ˆòˆÏáˆÏåÈö ˆÎñ
1. ï˛Ó˚ˆÏDÓ˚ !ӈϢ£Ïc •°ñ fliyö Ä Ü˛y° í˛zû˛ˆÏÎ˚Ó˚ §ˆÏD ï˛Ó˚Dy!Î˚ï˛ Ó˚y!¢!ê˛Ó˚ ˛õÎ≈yÓ,_ ˛õ!Ó˚Óï≈˛ö âˆÏê˛–
2. x@ˇÃàyü# ï˛Ó˚ˆÏD ~•z ˛õ!Ó˚Óï≈˛ö ˛õÓ˚flõÓ˚ ˜Ó˚!áܲû˛yˆÏÓ Î%_´ xÌ≈yÍ (x1,t
1) -~ ˆÎ ò¢yñ ~ܲ•z ò¢y Î!ò
§!ߨ!•ï˛ !Ó®%
() 1 xx +Δ
~
() 1tt +Δ
§üˆÏÎ˚ ˛õyÄÎ˚y ÎyÎ˚ñ ï˛ˆÏÓ ˆ§áyˆÏö
xt
ΔΔ
= ô Óܲ v, ˆÎáyˆÏö v =
ï˛Ó˚ˆÏDÓ˚ ˆÓà–
3. ~çöƒ §yôyÓ˚îû˛yˆÏÓ ï˛Ó˚Dy!Î˚ï˛ ˆû˛Ôï˛ Ó˚y!¢ S~!ê˛ §Ó˚îñ ã˛y˛õñ ï˛!í˛¸Í ã%˛¡∫ܲ#Î˚ ï˛#Ó ï˛y •zï˛ƒy!ò Îy !ܲå%È
•ˆÏï˛ ˛õyˆÏÓ˚V
() xvt ±
~•z Ó˚y!¢!ê˛Ó˚ ~ܲ!ê˛ ˛õÎ≈yÓ,_ xˆÏ˛õ«˛Ü˛ •ˆÏÓ– ï˛ˆÏÓ xyüyˆÏòÓ˚ xyˆÏ°yã˛öyÎ˚ §Ó˚°#ܲÓ˚ˆÏîÓ˚
çöƒ xyüÓ˚y ÷ô% §y•zö xˆÏ˛õ«˛Ü˛ !öˆÏÎ˚ !Óã˛yÓ˚ ܲˆÏÓ˚!åÈ– xÌ≈yÍ Ü˛¡õüyö ˆû˛Ôï˛ Ó˚y!¢!ê˛ Î!ò y •Î˚ñ
ï˛ˆÏÓñ ( ) ( )π ν ωλ= ± = ±2sin siny a x t a kx t
~áyˆÏö λ = ï˛Ó˚D ˜òâ≈ƒñ
ω
ˆÜ˛Ô!îܲ ܲ¡õyB˛ =
2v π
xyÓyÓ˚
2k
πλ =
ˆÎáyˆÏöñ ϑ = ܲ¡õyB˛
~ÓÇñ λ=v v
184
xyˆÏàÓ˚ xyˆÏ°yã˛öyÎ˚ ~ܲ!ê˛ !ö!ò≈‹T !òˆÏܲ §MÈ˛Ó˚î¢#°ñ !ö!ò≈‹T ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛üye ï˛Ó˚D ôˆÏÓ˚ ˆöÄÎ˚y
•ˆÏÎ˚ˆÏåÈñ !ܲv ≤ÃÜ,˛!ï˛ˆÏï˛ ï˛y ≤ÃyÎ˚•z âˆÏê˛ öy– í˛zòy•Ó˚î !•§yˆÏÓ Ó°y ÎyÎ˚ñ xyüyˆÏòÓ˚ ã˛y!Ó˚!òˆÏܲ Îáö ~ܲy!ôܲ
Óƒ!_´ ܲÌy Ó°ˆÏåÈöñ ï˛áö ˆ§•z ¢∑ï˛Ó˚D=!°Ó˚ í˛z˛õ!Ó˚˛õyï˛ âˆÏê˛–
~•z ~ܲˆÏܲ xy˛õöyÓ˚y ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ §¡∫ˆÏ¶˛ xÓàï˛ •ˆÏÓö– §Ó ˆÜ˛Ô!îܲ ܲ¡õyB˛ xÌ≈yÍ §üyö ω,
§ü ï˛Ó˚D ˜òâ≈ƒ xÌ≈yÍ §üyö §MÈ˛Ó˚î ô Óܲ (propagation constant) k ~ÓÇ §üyö !ÓhflÏyÓ˚ §¡õߨ ò%•z!ê˛
ï˛Ó˚D !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ ~ܲ•z §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ x@ˇÃ§Ó˚ •ˆÏÎ˚ Îáö ~ˆÏܲ x˛õˆÏÓ˚Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •Î˚ñ
ï˛áö fliyî% ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚– x˛õÓ˚!òˆÏܲ ܲ¡õyˆÏB˛Ó˚ §yüyöƒ ï˛ú˛yï˛ xyˆÏåÈ ~Ó˚ܲü ò%!ê˛ ¢∑ ï˛Ó˚D Îáö ~ˆÏܲ
x˛õˆÏÓ˚Ó˚ IJõÓ˚ ~ˆÏ§ ˛õˆÏí˛¸ñ ï˛áö ˜ï˛!Ó˚ •Î˚ fl∫Ó˚ܲ¡õ (beats)–
~ˆÏܲ x˛õÓ˚ ˆÌˆÏܲ §yüyöƒ ï˛ú˛yˆÏï˛Ó˚ ܲ¡õyˆÏB˛Ó˚ xˆÏöܲ=!° ï˛Ó˚D Îáö ˛õÓ˚flõˆÏÓ˚Ó˚ í˛z˛õÓ˚ ˛õˆÏí˛¸ñ ï˛áö
ˆÎ !ü!°ï˛ ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚ ï˛yˆÏܲ ӈϰ ï˛Ó˚D ò°–
ˆÜ˛yÎ˚yrê˛yü Ó°!ÓK˛yö ˛õí˛¸ˆÏï˛ ˆàˆÏ° ï˛Ó˚D òˆÏ°Ó˚ ôyÓ˚îy Ìyܲy á%Ó•z ≤ÈÏÎ˚yçö#Î˚–
í˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒ
~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° xy˛õ!öÈÙÙÙÈ
ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ö#!ï˛Ó˚ Óî≈öy !òˆÏï˛ §üÌ≈ö •ˆÏÓö–
fliyö% ï˛Ó˚D §,!‹TÓ˚ ü)° ôyÓ˚îy Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
fliyö% ï˛Ó˚ˆÏDÓ˚ üˆÏôƒ §%flõ® Ä !öflõ® !Ó®%Ó˚ xÓfliyö !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
fliyö% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ=!°Ó˚ §¡∫ˆÏ¶˛ ôyÓ˚îy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ï˛Ó˚Dò° ܲ#û˛yˆÏÓ ˜ï˛!Ó˚ •Î˚ñ ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ IJõÓ˚ ï˛Ó˚D ˆÓˆÏàÓ˚ !öû≈˛Ó˚¢#°ï˛yÓ˚ ôyÓ˚îy ˆÌˆÏܲ ò°#Î˚ ˆÓà (group velocity) !öî≈Î˚
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ò%•z!ê˛ í˛z˛õ!Ó˚˛õy!ï˛ï˛ fl∫ˆÏÓ˚Ó˚ ܲ¡õyB˛ çyöy ÌyܲˆÏ°ñ fl∫Ó˚ܲˆÏ¡õÓ˚ •yÓ˚ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
7.2 ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ü)°ï˛_¥
~•z ˛õÎ≈yˆÏÎ˚Ó˚ !mï˛#Î˚ ~ܲˆÏܲ xy˛õöyÓ˚y §Ó˚° ˆòy°à!ï˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ §¡∫ˆÏ¶˛ ˛õˆÏí˛¸ˆÏåÈö– ˆ§áyˆÏö ˆòáy ˆàˆÏåÈ
ˆÎñ Îáö ˆÜ˛yöÄ Ü˛îy Óy Ó›Ó˚ IJõÓ˚ ò%•z Óy ï˛ˆÏï˛y!ôܲ §Ó˚° ˆòy°à!ï˛ ~ܲ•z !òˆÏܲ !e´Î˚y ܲˆÏÓ˚ñ ï˛áö ˆÎ
ˆÜ˛yöÄ §üˆÏÎ˚ ܲîy!ê˛Ó˚ °!∏˛ §Ó˚îñ ≤ÈÏï˛ƒÜ˛!ê˛ xy°yòy xy°yòy §Ó˚ˆÏîÓ˚ Ó#çày!î!ï˛Ü˛ ˆÎyàú˛° •ˆÏÓ– xyÓyÓ˚
Îáö §Ó˚° ˆòy°à!ï˛=!° !Ó!û˛ß¨ !e´Î˚y !òܲ ˆÌˆÏܲ !e´Î˚y ܲˆÏÓ˚ñ ï˛áö °!∏˛ §Ó˚î •ˆÏÓ ˆû˛QÓ˚ ˆÎyàú˛°–
ò%•z Óy ï˛ˆÏï˛y!ôܲ ï˛Ó˚D ˛õÓ˚flõˆÏÓ˚Ó˚ !öÓ˚ˆÏ˛õ«˛û˛yˆÏÓ ˆÜ˛yöÄ fliyˆÏö ~ܲ•z ˛õˆÏÌ x@ˇÃ§Ó˚ •ˆÏï˛ ˛õyˆÏÓ˚– Î!ò ò%•z
ï˛Ó˚D myÓ˚y §,‹T §Ó˚î ~ܲ•z !òˆÏܲ •Î˚ñ ï˛ˆÏÓ ï˛Ó˚D˛õˆÏÌ ˆÜ˛yöÄ Ü˛îyÓ˚ °!∏˛ §Ó˚î fl∫ï˛sfû˛yˆÏÓ ï˛Ó˚D=!°Ó˚ myÓ˚y
ܲîy!ê˛ˆÏï˛ ≤Ãò_ §Ó˚î=!°Ó˚ Ó#çày!î!ï˛Ü˛ ˆÎyàú˛ˆÏ°Ó˚ §üyö •ˆÏÓ–
185
ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ xˆÏöܲ í˛zòy•Ó˚î•z xy˛õöyÓ˚ çyöy xyˆÏåÈ– ˆÜ˛yöÄ xˆÏÜ≈˛‹T…yÎ˚ Îáö xˆÏöܲ ÓyòƒÎsf
~ܲˆÏe ÓyçyˆÏöy •Î˚ñ ï˛áö ˆ§=!° ˆÌˆÏܲ í˛zͲõߨ ¢∑ï˛Ó˚D=!° !ü!°ï˛û˛yˆÏÓ ~ܲ!ê˛ ¢∑ï˛Ó˚D ˜ï˛!Ó˚ ܲˆÏÓ˚–
xyüÓ˚y !ܲv ≤Ã!ï˛!ê˛ ÎˆÏsfÓ˚ ¢∑ ˛õ,Ìܲû˛yˆÏÓ ÷öˆÏï˛ ˛õy!Ó˚– Îy ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ °!∏˛ ¢∑ï˛Ó˚ˆÏDÓ˚ üˆÏôƒ
≤Ã!ï˛!ê˛ ï˛Ó˚D ï˛yˆÏòÓ˚ ˜Ó!¢‹Tƒ ÓçyÎ˚ Ó˚yˆÏá– xy˛õöyÓ˚ ˆÓ˚!í˛ÄˆÏï˛ ˆÎ ˆÓ˚!í˛Ä ï˛Ó˚D ôÓ˚y ˛õˆÏí˛¸ñ ï˛yÄ xˆÏöܲ=!°
˛õ,Ìܲ ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ° í˛zͲõߨ– !ܲv xy˛õöyÓ˚ ˆÓ˚!í˛Ä Îsf!ê˛ ~ܲ ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ Óy!•ï˛ ¢∑
˛õ,Ìܲû˛yˆÏÓ ˛˛õ%öç≈!öï˛ Ü˛Ó˚ˆÏï˛ ˛õyˆÏÓ˚– xÌ≈yÍñ ~ˆÏ«˛ˆÏe ~ܲ ~ܲ!ê˛ ˆÓ˚!í˛Ä ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ í˛z˛õ!Ó˚˛õyï˛ö §ˆÏ_¥Ä
ÓçyÎ˚ ÌyˆÏܲ– í˛z˛õ!Ó˚˛õ!ï˛ï˛ ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ x«%˛] Ìyܲy ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ~ܲ!ê˛ ü)° ôü≈–
!Ó£ÏÎ˚!ê˛ xyÓ˚Ä û˛y°û˛yˆÏÓ ˆÓyé˛yÓ˚ çöƒ ~ܲ!ê˛ ê˛yö ܲˆÏÓ˚ Ó˚yáy ò!í˛¸Ó˚ IJõÓ˚ !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ xy§y ò%•z!ê˛
ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ °«˛ƒ ܲˆÏÓ˚ ˆòáy Îyܲ– 7.1 a Ä
b !ã˛e ò%•z!ê˛ˆÏï˛ ~Ó˚*˛õ ò%•z!ê˛ ï˛Ó˚D w1 Ä w
2 ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– a !ã˛ˆÏe ï˛Ó˚D ò%•z!ê˛Ó˚ §Ó˚î ~ܲ•z !òˆÏܲñ b
!ã˛ˆÏe §Ó˚î !Ó˛õÓ˚#ï˛ !òˆÏܲ– °«˛ƒ ܲˆÏÓ˚ ˆòá%öñ ï˛Ó˚D
ò%•z!ê˛ ˛õÓ˚flõÓ˚ˆÏܲ x!ï˛e´ü ܲÓ˚yÓ˚ xyˆÏà Ä ˛õˆÏÓ˚
ï˛yˆÏòÓ˚ Ó˚*˛õ ~ܲ•z Ó˚ˆÏÎ˚ˆÏåÈ– x!ï˛e´ü ܲÓ˚yÓ˚ §üˆÏÎ˚
ò!í˛¸Ó˚ !Ó!û˛ß¨ !Ó®%Ó˚ §Ó˚î
1 w
Ä
2 w
ï˛Ó˚ˆÏDÓ˚ çöƒ
˛õ,Ìܲû˛yˆÏÓ ˆÎ §Ó˚î •ï˛ñ ï˛yÓ˚ ˆÎyàú˛°– ~åÈyí˛¸y
§yü!Î˚ܲ í˛z˛õ!Ó˚˛õyï˛ö §ˆÏ_¥Ä ï˛Ó˚D ò%•z!ê˛ ï˛yˆÏòÓ˚
!öçfl∫ Ó˚*˛õ ˆÎ ÓçyÎ˚ Ó˚yˆÏáñ ï˛y ˛õÓ˚flõÓ˚ˆÏܲ x!ï˛e´ü ܲÓ˚yÓ˚ ˛õÓ˚ ï˛yˆÏòÓ˚ §Ó˚î ˆÌˆÏܲ•z ˆÓyé˛y ÎyˆÏFåÈ–
~ÓyÓ˚ xyüÓ˚y ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ày!î!ï˛Ü˛ !òܲ!ê˛ xyˆÏ°yã˛öy ܲÓ˚Ó– ≤ÃÌü ˛õˆÏÓ≈Ó˚ !mï˛#Î˚ ~ܲܲ ˆÌˆÏܲ
xy˛õöyˆÏòÓ˚ !öÿ˛Î˚ flõ®ˆÏöÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ày!î!ï˛Ü˛ §üyôyö §¡∫ˆÏ¶˛ ôyÓ˚îy •ˆÏÎ˚ˆÏåÈ– ôÓ˚y Îyܲñ x !ò Ïܲ
§MÈ˛Ó˚î¢#° ò%!ê˛ ï˛Ó˚D ˛õ,Ìܲû˛yˆÏÓ ~ܲ!ê˛ !Ó®%ˆÏï˛ ~ܲ!ê˛ Ü˛îyÓ˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •°– t §üˆÏÎ˚ ò%•z!ê˛ ï˛Ó˚ˆÏDÓ˚
çöƒ ˛õ,Ìܲ ˛õ,Ìܲû˛yˆÏÓ Ü˛îy!ê˛Ó˚ §Ó˚î
() 1, yxt
~ÓÇ y2(x,t)– ï˛y•ˆÏ° ܲîy!ê˛Ó˚ °!∏˛ §Ó˚î •ˆÏÓ
()()() 12 ,,, yxtyxtyxt =+
..... 7.1
xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ x x«˛ ÓÓ˚yÓÓ˚ §MÈ˛Ó˚üyö ˆÜ˛yöÄ ï˛Ó˚ˆÏDÓ˚ §Ó˚î ày!î!ï˛Ü˛û˛yˆÏÓ ˆ°áy ÎyÎ˚ :
( ) ( ), siny x t a t kxω φ= ± + ..... 7.2
ˆÎáyˆÏö a = ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ñ
ω
= ˆÜ˛Ô!îܲ ܲ¡õyB˛ñ k = §MÈ˛Ó˚î ô Óܲñ Îy ï˛Ó˚D ˜òâ≈ƒ λ ~Ó˚ §ˆÏD
2k
πλ =
§¡õÜ≈˛myÓ˚y Î%_´ñ φ, x Ä t ~Ó˚ üyö Îáö ¢)öƒ ˆ§•z xÓfliyÓ˚ ≤ÃyÓ˚!Ω˛Ü˛ ò¢yˆÏܲyˆÏî– ï˛Ó˚ˆÏDÓ˚ §Ó˚î
186
~áy Ïö y x!û˛ü%á#– ò%•z!ê˛ ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ âê˛ˆÏï˛ •ˆÏ° ≤Ã̈Ïü•z ï˛yˆÏòÓ˚ §Ó˚î ò%•z!ê˛ ˆÎö ˛õÓ˚flõÓ˚ °¡∫
x!û˛ü%á# öy •Î˚ñ ˆ§!ê˛ §%!ö!ÿ˛ï˛ ܲÓ˚ˆÏï˛ •ˆÏÓ– xyüÓ˚y ~áyˆÏö ôˆÏÓ˚ ˆöÓ ˆÎñ ò%•z!ê˛ §Ó˚î•z y ÓÓ˚yÓÓ˚ xÌ≈yÍ
~ܲ•z !òˆÏܲ– ~áö xyüÓ˚y ò%•z!ê˛ ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ˆ«˛ˆÏe ܲˆÏÎ˚ܲ!ê˛ !ӈϢ£Ï xÓfliy !ÓˆÏÓã˛öy ܲÓ˚Ó– ~=!°
•° :
(i) ï˛Ó˚DmˆÏÎ˚Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ Ä §MÈ˛Ó˚î ô &Óܲ §üyö !ܲv !ÓhflÏyÓ˚ Ä ≤ÃyÓ˚!Ω˛Ü˛ ò¢yˆÏܲyî φ !û˛ß¨– í˛zû˛Î˚•z
+ x x«˛ ÓÓ˚yÓÓ˚ §MÈ˛Ó˚üyö–
(ii) ï˛Ó˚DmˆÏÎ˚Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ §üyö !ܲv !ÓhflÏyÓ˚ !û˛ß¨ ~ÓÇ §MÈ˛Ó˚ˆÏîÓ˚ !òܲ !Ó˛õÓ˚#ï˛ñ xÌ≈yÍ Î!ò ~ܲ!ê˛
ï˛Ó˚D + x !òˆÏܲ x@ˇÃ§Ó˚ •Î˚ñ ï˛ˆÏÓ xöƒ!ê˛ -x !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏÓ–
(iii) ï˛Ó˚DmˆÏÎ˚Ó˚ !ÓhflÏyÓ˚ Ä ˆÜ˛Ô!îܲ ܲ¡õyB˛ í˛zû˛Î˚•z !ûߨ ï˛ˆÏÓñ ˛õ)ˆÏÓ≈Ó˚ üï˛ í˛zû˛Î˚•z + x !òˆÏܲ §MÈ˛Ó˚üyö–
˛õÓ˚Óï≈˛# xLjϢ xyüÓ˚y ~•z !ï˛ö!ê˛ ˆ«˛ˆÏeÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ܲÓ˚Ó– í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ° öï%˛ö ˆÜ˛yöÄ ˆû˛Ôï˛
âê˛öyÓ˚ í˛zͲõ!_ •Î˚ !ܲöyñ ï˛yÄ xy˛õ!ö ~Ó˚ ú˛ˆÏ° ˆòáˆÏï˛ ˛õyˆÏÓö–
~áö xy˛õ!ö !öÿ˛Î˚ çyˆÏöö ˆÎñ ~ܲ!ê˛ ï˛Ó˚D §¡∫ˆÏ¶˛ çyöˆÏï˛ ˆàˆÏ° ï˛yÓ˚ !ÓhflÏyÓ˚ñ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ~ÓÇ
ò¢yÓ˚ §¡∫ˆÏ¶˛ K˛yö Ìyܲ á%Ó ≤ÈÏÎ˚yçö#Î˚– !Ó!û˛ß¨ xÓfliyÎ˚ ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚àï˛ âê˛ˆÏ° fliyî% ï˛Ó˚Dñ Óƒ!ï˛ã˛yÓ˚
Óy fl∫Ó˚ ܲ¡õö §,!‹T •ˆÏï˛ ˛õyˆÏÓ˚– ~áyˆÏö xyüÓ˚y ~ˆÏòÓ˚ !ܲå%È !ܲå%ÈÓ˚ !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
~Ó˚ çöƒ !öˆÏ¡¨y_´ ï˛Ó˚D Î%àˆÏ°Ó˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ò,‹Tyhsˇ ˆöÄÎ˚y Îyܲ
(i)
() 11sin yatkx ω =−
~ÓÇ ( )2 2 siny a t kxω π= − +
(ii)
() 1111 sin =− yatkx ω
~ÓÇñ ( )2 2 2 2sin= −y a t kω(iii) ( )1 siny a t kxω= − ~ÓÇñ ( )2 siny a t kxω= + ......7.3
≤ÃÌü í˛zòy•Ó˚î!ê˛ ˆöÄÎ˚y Îyܲ– ~•z ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ xy°yòyñ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ~ÓÇ §MÈ˛Ó˚î ô &Óܲ
(k) ~ܲñ !ܲv ò%!ê˛Ó˚ üˆÏôƒ ò¢yÓ˚ ˛õyÌ≈ܲƒ = π,, °!∏˛ ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î •ˆÏÓ
( ) ( ) ( )1 2, sin siny x t a t kx a t kxω ω π= − + − +
=
()() 22 sinsin atkxatkx ωω −−−
= ( ) ( )1 2 sina a t kxω− −
ܲyˆÏç•z °!∏˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ •ˆÏÓ (a1–a
2) ~ÓÇ ≤ÃÌü ï˛Ó˚ˆÏDÓ˚ §üyö ò¢y !Ó!¢‹T •ˆÏÓ S!ãe 7.2 o‹TÓƒV
!mï˛#Î˚ ˆ«˛ˆÏe ï˛Ó˚Dò%!ê˛Ó˚ ܲ¡õyB˛ ~ÓÇ !ÓhflÏyÓ˚ ô &Óܲ (k) !Ó!û˛ß¨– Îáö ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ üˆÏôƒ ~•z ˛õyÌ≈ܲƒ
ܲü •Î˚ñ ï˛áö ~ˆÏòÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° fl∫Ó˚ܲˆÏ¡õÓ˚ §,!‹T •Î˚– ~•z fl∫Ó˚ܲ¡õ ï˛Ó˚D ò%!ê˛Ó˚ ò¢yÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°
öÎ˚– ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ ˛õ!Ó˚ÓˆÏï≈˛ ~ܲ§ˆÏD xˆÏöܲ=!° fl∫“ ï˛ú˛yˆÏï˛Ó˚ ܲ¡õyB˛!Ó!¢‹T
ï˛Ó˚ˆÏDÓ˚ Îáö í˛z˛õ!Ó˚˛õyï˛ âˆÏê˛ñ ï˛áö ~ܲ!ê˛ ï˛Ó˚DÎ%Ì §,!‹T •Î˚– ~•z ï˛Ó˚DÎ)Ì
Óy ï˛Ó˚D ò° ˆÎ ˆÓˆÏà ã˛ˆÏ°ñ ï˛yˆÏܲ Ó°y •Î˚ ò°àï˛ ˆÓàò°àï˛ ˆÓàò°àï˛ ˆÓàò°àï˛ ˆÓàò°àï˛ ˆÓà (group velocity)–
~•z ò°àï˛ ˆÓà ~ÓÇ ï˛Ó˚D ˆÓà ~ܲ öÎ˚–
ï,˛ï˛#Î˚ í˛zòy•Ó˚ˆÏî xyüÓ˚y ˆòáˆÏï˛ ˛õy!FåÈ ˆÎñ ï˛Ó˚D ò%!ê˛Ó˚ k ~Ó˚ §yüˆÏöÓ˚ !ã˛•´!ê˛
!û˛ß¨– ≤ÃÌü ï˛Ó˚D!ê˛ x ~Ó˚ ôöydܲ !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏFåÈ ~ÓÇ !mï˛#Î˚ ï˛Ó˚D!ê˛ x
187
~Ó˚ îydܲ !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏFåÈñ xÌ≈yÍ ï˛yÓ˚y ˛õÓ˚flõˆÏÓ˚Ó˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ ÎyˆÏFåÈ– ~•z ôÓ˚ˆÏöÓ˚ ò%!ê˛ ï˛Ó˚ˆÏDÓ˚
í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° fliyî% ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚– xyüÓ˚y (7.3) ˛õˆÏÓ≈ ~ §¡∫ˆÏ¶˛ !Ó¢ò xyˆÏ°yã˛öy ܲÓ˚Ó–
7.3 fliyî% ï˛Ó˚Dfliyî% ï˛Ó˚Dfliyî% ï˛Ó˚Dfliyî% ï˛Ó˚Dfliyî% ï˛Ó˚D
ò%!ê˛ §üyö ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛Ó˚ ( )ω , §üyö ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ~ÓÇ §üyö !ÓhflÏyˆÏÓ˚Ó˚ ï˛Ó˚D ~ܲ §Ó˚° ˆÓ˚áy
ÓÓ˚yÓÓ˚ !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ Îáö ˛õÓ˚flõˆÏÓ˚Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö fliyî% ï˛Ó˚D ˜ï˛!Ó˚ •Î˚– §üyö ï˛Ó˚D
˜òâ≈ƒ ~ÓÇ !ÓhflÏyˆÏÓ˚Ó˚ ò%!ê˛ ï˛Ó˚D ˆ˛õˆÏï˛ ˆàˆÏ° ≤ÃÌü!ê˛ˆÏܲ xy˛õ!ï˛ï˛ ï˛Ó˚D ~ÓÇ !mï˛#Î˚!ê˛ˆÏܲ ≤ÃÌü!ê˛Ó˚ ≤Ã!ï˛ú˛!°ï˛
ï˛Ó˚D ôˆÏÓ˚ ˆöÄÎ˚y §%!Óôyçöܲ–
~•z ≤Ã!ï˛ú˛°ö ˆÜ˛yöÄ Ók˛ ≤Ã!ï˛ú˛°ˆÏܲ âê˛ˆÏï˛ ˛õyˆÏÓ˚ SˆÎüö ˆ§ï˛yÓ˚ñ !àê˛yÓ˚ñ xà≈yö öˆÏ°Ó˚ Ók˛ !òܲV,
xÌÓy ˆÜ˛yöÄ ü%_´ ≤Ã!ï˛ú˛°ˆÏÜ˛Ä âê˛ˆÏï˛ ˛õyˆÏÓ˚ñ SÎÌy xà≈yö öˆÏ°Ó˚ ü%_´ ≤ÃyhsˇV– xy˛õöyÓ˚y §•ˆÏç•z Ó%é˛ˆÏï˛
˛õyÓ˚ˆÏåÈö ˆÎñ Ók˛ fliyˆÏö §Ó˚ˆÏîÓ˚ ˆÜ˛yöÄ §ΩyÓöy ˆö•z– °!∑ §Ó˚î ¢)öƒ •ÄÎyÓ˚ ¢ï≈˛ •° ~•z ˆÎñ xy˛õ!ï˛ï˛
Ä ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚D ~ fliyˆÏö !Ó˛õÓ˚#ï˛ ò¢yÎ%_´ •ˆÏÓ– §%ï˛Ó˚yÇ Ó°y ÎyÎ˚ ˆÎñ Ók˛ ≤Ã!ï˛ú˛°ˆÏܲ π õ!Ó˚üyî ò¢yÓ˚
˛õ!Ó˚Óï≈˛ö âˆÏê˛ñ !ܲv ü%_´ ≤Ã!ï˛ú˛°ˆÏö ò¢y x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ–
≤Ã̈Ïü xà≈yö öˆÏ°Ó˚ ü%_´ ≤ÃyˆÏhsˇ ≤Ã!ï˛ú˛°ˆÏöÓ˚ !Ó£ÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y Îyܲ– ~•z ˆ«˛ˆÏe ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ °!∏˛
§Ó˚î •ˆÏÓñ
()()() ,sinsin yxtatkxatkx ωω =−++= 2 sin cosa t kxω ......... 7.4
xÌÓyñ
()() ,2cossin yxtakxt ω =
......... 7.5
7.5 §ü#ܲÓ˚î ̈Ïܲ òáy ÎyˆÏFåÈ Îñ ï˛Ó˚D!ê˛Ó˚ !ÓhflÏyÓ˚ (2a coskx =y0) ~ܲ!ê˛ ô &Óܲ öÎ˚ñ x ~Ó˚ xÓfliyˆÏöÓ˚
˛õ!Ó˚Óï≈˛ˆÏöÓ˚ §ˆÏD §ˆÏD ˛õÎ≈yÓ,_ܲyˆÏÓ˚ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– ï˛ˆÏÓ °!∏˛ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ ~ÓÇ ï˛Ó˚ˆÏDÓ˚ ˜òâ≈ƒ
x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ–
7.4 §ü#ܲÓ˚î °«˛ƒ ܲÓ˚ˆÏ° ~Ä ˆòáy ÎyˆÏÓ ˆÎñ x x«˛ ÓÓ˚yÓÓ˚ ܲîy=!° xyˆÏ®y!°ï˛ •Î˚ñ !ܲv x ~Ó˚
!Ó!û˛ß¨ üyˆÏö ï˛yˆÏòÓ˚ !ÓhflÏyÓ˚ §üyö öÎ˚ñ ï˛ˆÏÓ ï˛yˆÏòÓ˚ ˛õÎ≈yÎ˚ܲy°=!° §üyö–
7.5 öÇ §ü#ܲÓ˚î!ê˛ !ܲv ã˛°ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î öÎ˚– ˆÜ˛ööyñ ã˛°ï˛Ó˚ˆÏD x Ä t ~üöû˛yˆÏÓ Î%_´ ˆÎñ x ~Ó˚
üyö 1 Óyí˛¸yˆÏ° Ä t ’Ó˚ üyö kω Óyí˛¸yˆÏ° ò¢y (kx -ωt) ~Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö •Î˚ öy– ò¢y ˆÎö
kω
§üˆÏÎ˚
~ܲܲ ˜òâ≈ƒ x!ï˛e´ü ܲˆÏÓ˚ ˆàˆÏåÈ– !ܲv §ü#ܲÓ˚î (7.5) -~ ò¢yÓ˚ ~Ó˚ܲü ˆÜ˛yöÄ à!ï˛•z ˆö•z– xyüÓ˚y Î!òÄ
ò%!ê˛ !Ó˛õÓ˚#ï˛àyü# ã˛°ï˛Ó˚D !öˆÏÎ˚ ÷Ó˚& ܲˆÏÓ˚!åÈ°yüñ !ܲv ˆ¢ˆÏ£Ï ~üö ~ܲ!ê˛ ï˛Ó˚ˆÏD í˛z˛õö#ï˛ •°yüñ ˆÎ!ê˛ fliyö
˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ öy– ˆÎ ï˛Ó˚D ~ˆÏàyÎ˚ öyñ ï˛yˆÏܲ fliyî% ï˛Ó˚D ӈϰ– fliyö ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ öy ӈϰ ~•z ï˛Ó˚D
¢!_´ ˛õ!Ó˚Ó•î ܲˆÏÓ˚ öy–
188
7.5 öÇ §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ !ÓhflÏyÓ˚ y(x,t) §Ó≈yˆÏ˛õ«˛y ˆÓ!¢ •ˆÏÓ Îáö
2 coscos1 kxxπλ ==±
....... 7.6
xÌ≈yÍñ
2 coscos xm
ππ λ=
ˆÎáyˆÏö m = 0,1,2
xÌÓyñ2 x
mπ πλ =
xï˛~Óñ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢ !ÓhflÏyÓ˚Î%_´ !Ó®%=!°Ó˚ fliyöyB˛ •°ñ
0, , .......,2 2
mx
λ λλ=
xyÓyÓ˚ ~ܲ•z §ü#ܲÓ˚î 7.5 ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ ˆÎñ !ÓhflÏyÓ˚ y(x,t) §ÓˆÏã˛ˆÏÎ˚ ܲü •ˆÏÓ Îáö
2cos cos 0xkx
πλ= = .......7.7
xÌ≈yÍñ
() 2 coscos212
xm
ππλ=+
ˆÎáyˆÏöñ m =0, 1, 2
xÌÓyñ ( )2 2 12
xm
π πλ = +
xÌÓyñ ( )2 14
x mλ= +
ܲyˆÏç•z §ÓˆÏã˛ˆÏÎ˚ ܲü !ÓhflÏyÓ˚Î%_´ !Ó®%=!° fliyöyB˛ •°ñ ( )3, ... 2 14 4 4
x mλ λ λ= +
§ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ !ÓhflÏyÓ˚Î%_´ !Ó®%=!°ˆÏܲ Ó°y •Î˚ §%flõ® !Ó®%
~ÓÇ §ÓˆÏã˛ˆÏÎ˚ ܲüS¢)öƒV !ÓhflÏyÓ˚ü%_´ !Ó®%=!°Ó˚ öyü !öflõ®
!Ó®%– ˛õÓ˚˛õÓ˚ ò%!ê˛ §%flõ® xÌÓy !öflõ® !Ó®%Ó˚ ò)Ó˚c ï˛Ó˚D
˜òˆÏâ≈ƒÓ˚ xˆÏô≈ˆÏܲÓ˚ §üyö 2λ⎛ ⎞
⎜ ⎟⎝ ⎠ ~ÓÇ ~ܲ!ê˛ §%flõ® Ä ~ܲ!ê˛
!öflõ® !Ó®%Ó˚ üyˆÏé˛Ó˚ ò)Ó˚c ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ~ܲ ã˛ï%˛Ì≈yLjϢÓ˚ §üyö
4λ⎛⎞
⎜⎟ ⎝⎠
– (7.3 öÇ !ã˛e o‹TÓƒV–
í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öy ˆÌˆÏܲ xy˛õöyÓ˚y çyöˆÏï˛ ˛õyÓ˚ˆÏ°ö ˆÎñ ò%•z!ê˛ ~ܲ•z Ó˚ܲü ã˛°ï˛Ó˚D !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ
189
~ܲ §Ó˚°ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ~ˆÏ§ ˛õÓ˚flõˆÏÓ˚Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •ˆÏ° fliyî%ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚– ~•z ï˛Ó˚ˆÏD ~ܲ !Ó®%
ˆÌˆÏܲ xöƒ !Ó®%ˆÏï˛ !ӈϫ˛yû˛ fliyöyhsˇ!Ó˚ï˛ •Î˚ öy– ˆÎ fliyˆÏö ï˛Ó˚D ò%!ê˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ âˆÏê˛ñ ˆ§ fliyö!ê˛ Ü˛ˆÏÎ˚ܲ!ê˛
xLjϢ !Óû˛_´ •ˆÏÎ˚ ÎyÎ˚– S!ã˛e 7.3)– ≤ÈÏï˛ƒÜ˛!ê˛ xÇ¢ ~ܲ ~ܲ!ê˛ !Ó®%ˆÏï˛ ˆ¢£Ï •Î˚– ~•z !Ó®%ˆÏï˛ Ü˛îyÓ˚ !ÓhflÏyÓ˚
¢)öƒ ~ÓÇ ~•z !Ó®%ˆÏܲ•z Ó°y •Î˚ !öflõ® !Ó®%–
ò%!ê˛ !öflõ® !Ó®%Ó˚ !ë˛Ü˛ üôƒÓï≈˛# fliyˆÏö ܲîyÓ˚ !ÓhflÏyÓ˚
§Ó≈yˆÏ˛õ«˛y ˆÓ!¢– ~•z !Ó®%Ó˚ öyü §%flõ® !Ó®§%flõ® !Ó®§%flõ® !Ó®§%flõ® !Ó®§%flõ® !Ó®%– !öflõ® Ä
§%flõ® !Ó®%Ó˚ üôƒÓï≈˛# ܲîy=!° ¢)öƒ ~ÓÇ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢
!ÓhflÏyˆÏÓ˚Ó˚ üôƒÓï≈˛# e´üÓô≈üyö !ÓhflÏyÓ˚ !öˆÏÎ˚ xyˆÏ®y!°ï˛ •Î˚
S!ã˛e 7.4)– a !Ó®%ˆÏï˛ Ü˛îy!ê˛ §Ó§üˆÏÎ˚ !fliÓ˚ ÌyˆÏܲñ b !Ó®%ˆÏï˛
ܲîy!ê˛ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢ !ÓhflÏyÓ˚ !öˆÏÎ˚ xyˆÏ®y!°ï˛ •Î˚ ~ÓÇ c
!Ó®%ˆÏï˛ Ü˛îyÓ˚ !ÓhflÏyÓ˚ Ä ò%•z ~Ó˚ üôƒÓï≈˛#–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ1 : ò%•z ≤ÃyˆÏhsˇ ò,ì˛¸û˛yˆÏÓ xyÓk˛ ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚
üˆÏôƒ í˛zͲõߨ fliyî% ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ Ü˛îyÓ˚ §Ó˚ˆÏîÓ˚ §ü#ܲÓ˚î !öî≈Î˚ ܲÓ˚&ö– ï˛yˆÏÓ˚Ó˚ xyÓk˛ ≤Ãyhsˇ ò%!ê˛
§%flõ® !Ó®% •ˆÏÓ öy !öflõ® !Ó®% •ˆÏÓ⁄ ò%•z ü%á ˆáy°y öˆÏ°Ó˚ ò%•z ≤ÃyˆÏhsˇ §%flõ® Óy !öflõ® !Ó®% ˆÜ˛yö!ê˛
•ˆÏÓ⁄ fliyî% ï˛Ó˚ˆÏD ¢!_´≤ÃÓy• •Î˚ öyÈÙÙÙÈ~•z ï˛Ìƒ!ê˛ Óƒyáƒy ܲÓ˚&ö–
7.3.1 fliyî%ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yö !Ó®%ˆÏï˛ Ó› ܲîyÓ˚ à!ï˛ˆÏÓà ~ÓÇ !ÓÜ,˛!ï˛fliyî%ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yö !Ó®%ˆÏï˛ Ó› ܲîyÓ˚ à!ï˛ˆÏÓà ~ÓÇ !ÓÜ,˛!ï˛fliyî%ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yö !Ó®%ˆÏï˛ Ó› ܲîyÓ˚ à!ï˛ˆÏÓà ~ÓÇ !ÓÜ,˛!ï˛fliyî%ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yö !Ó®%ˆÏï˛ Ó› ܲîyÓ˚ à!ï˛ˆÏÓà ~ÓÇ !ÓÜ,˛!ï˛fliyî%ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yö !Ó®%ˆÏï˛ Ó› ܲîyÓ˚ à!ï˛ˆÏÓà ~ÓÇ !ÓÜ,˛!ï˛ (straim)
!öî≈Î˚!öî≈Î˚!öî≈Î˚!öî≈Î˚!öî≈Î˚
xy˛õöyÓ˚y !öÿ˛Î˚•z çyˆÏöö ˆÎñ §üˆÏÎ˚Ó˚ §ˆÏD ˆÜ˛yö ܲîyÓ˚ §Ó˚ˆÏîÓ˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ •yÓ˚ˆÏܲ•z à!ï˛ˆÏÓà ӈϰ–
fliyî% ï˛Ó˚ˆÏD ˆÜ˛yöÄ Ü˛îyÓ˚ à!ï˛ˆÏÓà !öî≈Î˚ ܲÓ˚yÓ˚ í˛z˛õyÎ˚ •ˆÏFåÈñ °!∑ §Ó˚î y(x,t) ˆÜ˛ t- ~Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö
ܲÓ˚y S~áyˆÏö x ˆÜ˛ !fliÓ˚ Ó˚yáˆÏï˛ •ˆÏÓV–
~áö xyüÓ˚y Î!ò 7.5 öÇ §ü#ܲÓ˚îˆÏܲ t -~Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲ!Ó˚ñ ï˛y•ˆÏ° ˛õyÄÎ˚y ÎyˆÏÓñ
à!ï˛ˆÏÓà =
2coscos dyakxt
dtωω =
...... 7.8
à!ï˛ˆÏÓà §Ó≈yˆÏ˛õ«˛y ˆÓ!¢ •ˆÏÓ Îáöñ cos 1kx = ±
xÌÓyñ
2 coscos xmππ λ=
, 0,1,2m =
xÌÓyñ
2m
xλ =
xÌ≈yÍñ ˆÎ §Ó !Ó®%ˆÏï˛ 0, , .....,2 2
mx
λ λλ= – xyÓyÓ˚ à!ï˛ˆÏÓà §ÓˆÏã˛ˆÏÎ˚ ܲü S¢)öƒV•ˆÏÓ
Îáöñ coskx= 0
190
xÌ≈yÍñ ( )2cos cos 2 12
x mπ πλ = +
xÌÓyñ ( )2 14
x mπ= +
xÌ≈yÍ ˆÎ §Ó !Ó®%ˆÏï˛ 3, ........(2 1)
4 4 4x
x mλ λ= +
~Ó˚ xÌ≈ •ˆÏFåÈñ §%flõ® !Ó®%ˆÏï˛ !ÓhflÏyÓ˚ ~ÓÇ à!ï˛ˆÏÓà ò%•z•z §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ ~ÓÇ !öflõ® !Ó®%ˆÏï˛ !ÓhflÏyÓ˚
Ä à!ï˛ˆÏÓà í˛zû˛Î˚•z ¢)öƒ (7.6 öÇ §ü#ܲÓ˚î ~ÓÇ ï˛Í§¡∫¶˛#Î˚ xyˆÏ°yã˛öy ˆòá%öVñ– §%flõ® ~ÓÇ !öflõ® !Ó®%Ó˚
üˆÏôƒÜ˛yÓ˚ ܲîy=!°Ó˚ !ÓhflÏyÓ˚ §%flõ® !Ó®%ˆÏï˛ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢ ˆÌˆÏܲ e´ü¢ ܲüˆÏï˛ Ü˛üˆÏï˛ !öÉflõ® !Ó®%ˆÏï˛
¢)öƒ •Î˚– 7.4 öÇ !ã˛ˆÏe ï˛#Ó˚!ã˛•´=!°Ó˚ ˜òâ≈ƒ myÓ˚y fliyî% ï˛Ó˚ˆÏDÓ˚ ܲîy=!°Ó˚ à!ï˛ˆÏÓà !öˆÏò≈¢ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
°!∏˛ §Ó˚î ( ),y x t ˆÜ˛ x ~Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲÓ˚ˆÏ° (t ˆÜ˛ !fliÓ˚ ˆÓ˚ˆÏáV fliyî% ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yöÄ
!Ó®%ˆÏï˛ !ÓÜ,˛!ï˛ !öî≈Î˚ ܲÓ˚y ÎyÎ˚– xyüÓ˚y 7.5 öÇ §ü#ܲÓ˚îˆÏܲ x ~Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲÓ˚ˆÏ° ˛õy•zñ
!ÓÜ,˛!ï˛ (strain) =
yx∂∂
= -2ak sin kx sin
t ω
...... 7.9
ˆÎˆÏ•ï%˛ sinkx- ~Ó˚ üyö !öflõ® !Ó®%ˆÏï˛ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ñ ˆ§çöƒ !öflõ® !Ó®%ˆÏï˛ !ÓÜ,˛!ï˛ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢
S~áyˆÏö !ÓhflÏyÓ˚ ~ÓÇ ˆÓà ¢)öƒV– 7.4 öÇ !ã˛ˆÏe ~!ê˛ flõ‹T ˆÓyé˛y ÎyÎ˚– !öflõ® !Ó®%Ó˚ ܲîy=!° ò%•z ˛õyˆÏ¢
!Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ ôy!Óï˛ Ü˛îy=!°Ó˚ ê˛yö xö%û˛Ó ܲÓ˚ˆÏåÈ– !ÓÜ,˛!ï˛ §Ó≈yˆÏ˛õ«˛y ܲü •Î˚ §%flõ® !Ó®%ˆÏï˛ñ ˆÎáyˆÏö
!ÓhflÏyÓ˚ ~ÓÇ ˆÓà §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢– !ã˛ˆÏe ˆòáˆÏï˛ ˛õyÄÎ˚y ÎyˆÏFåÈ ˆÎñ §%flõ®
!Ó®%ˆÏï˛ Ü˛îy ~ÓÇ ï˛yˆÏòÓ˚ ˛õyˆÏ¢Ó˚ ܲîy=!°Ó˚ à!ï˛ü%á ~ܲ•z !òˆÏܲ– ܲyˆÏç•z
§%flõ® !Ó®%Ó˚ ܲîy=!°ˆÏï˛ !ÓܲÈ,!ï˛Ó˚ §,!‹T •Î˚ öy–
!ã˛ˆÏe ˆòáy ÎyˆÏFåÈ ˆÎñ fliyö% ï˛Ó˚ˆÏDÓ˚ ܲîy=!°ˆÏܲ P,Q,R •zï˛ƒy!ò ܲï˛Ü˛=!°
áˆÏ[˛ !Óû˛_´ ܲÓ˚y ÎyÎ˚– ≤ÈÏï˛ƒÜ˛ áˆÏ[˛Ó˚ ܲîy=!° ~ܲ•z !òˆÏܲ xyˆÏ®y!°ï˛ •Î˚
~ÓÇ ò%•z!ê˛ ˛õy¢y˛õy!¢ áˆÏ[˛Ó˚ ܲîy=!°Ó˚ xyˆÏ®y°ˆÏöÓ˚ !òܲ !Ó˛õÓ˚#ï˛– ~ܲ•z
áˆÏ[˛Ó˚ §Ó ܲîy=!° ~ܲ•z §ˆÏD ≤ÈÏï˛ƒˆÏܲÓ˚ §ˆÏÓ≈yFã˛ !ÓhflÏyˆÏÓ˚ ˆ˛õÑÔåÈÎ˚ xyÓyÓ˚
~ܲ•z §ˆÏD üôƒ !Ó®% x!ï˛e´ü ܲˆÏÓ˚– !ã˛e 7.4(a) - ~ ˆ§!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
§Ó ܲîy=!°Ó˚ ˛õÎ≈yÎ˚ܲy° T ~ܲ•z !ܲv à!ï˛ˆÏÓà !û˛ß¨ •ÄÎ˚yÓ˚ ú˛ˆÏ°•z ~!ê˛
§Ω˛Ó •ˆÏÎ˚ˆÏåÈ– ˆÎ ܲîy=!°ˆÏܲ ˆÓ!¢ ò)Ó˚c x!ï˛e´ü ܲÓ˚ˆÏï˛ •ˆÏÓñ ï˛yˆÏòÓ˚
à!ï˛ˆÏÓà ˆÓ!¢– xyÓ˚ ˆÎ ܲîy=!°ˆÏܲ ܲü ò)Ó˚c x!ï˛e´ü ܲÓ˚ˆÏï˛ •ˆÏÓñ ï˛yˆÏòÓ˚
à!ï˛ˆÏÓà ܲü–
~áö xy°yòyû˛yˆÏÓ fliyî% ï˛Ó˚ˆÏDÓ˚ ~ܲ!ê˛ Ü˛îyÓ˚ ܲÌy !Óã˛yÓ˚ ܲÓ˚ˆÏ° xyüÓ˚y
ˆòáÓñ ܲáö ~Ó˚ à!ï˛ˆÏÓà §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ñ xyÓ˚ ܲáö•z Óy ¢)öƒ •Î˚–
191
7.8 öÇ §ü#ܲÓ˚î
2coscos yakxt
tωω ∂= ∂
ˆÜ˛ !û˛ß¨ Ó˚*ˆÏ˛õ !°áˆÏ°
22 4coscos yavxt
tTππ πλ
∂= ∂
=
22 4coscos axt
TTππ πλ
í˛z˛õˆÏÓ˚y_´ §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ
4T
t=
~ÓÇ
34T
-~ ܲîyÓ˚ à!ï˛ˆÏÓà ¢)öƒ ~ÓÇ
0,2T
t=
Ä T ˆï˛ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢–
xï˛~Ó ~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yÎ˚ܲyˆÏ° üyôƒˆÏüÓ˚ ܲîy=!° Îáö üôƒ!Ó®%ˆÏï˛ ˆ˛õÔÑåÈÎ˚ñ ï˛yˆÏòÓ˚ ˆÓà §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢
~ÓÇ ˛õ%ˆÏÓ˚y ¢!_´•z à!ï˛¢!_´ ~ÓÇ Îáö ã˛Ó˚ü !ÓhflÏyˆÏÓ˚ ˆ˛õÑÔåÈÎ˚ñ ï˛áö ï˛yˆÏòÓ˚ ˆÓà ¢)öƒ ~ÓÇ ˛õ%ˆÏÓ˚y ¢!_´•z
!fli!ï˛¢!_´– ˛õˆÏÓ˚Ó˚ xLjϢ xyüÓ˚y fliyî% ï˛Ó˚ˆÏD !Ó!û˛ß¨ §üˆÏü° §,!‹TÓ˚ ܲyÓ˚î=!° §¡∫ˆÏ¶˛ çyöˆÏï˛ ˛õyÓ˚Ó–
7.3.2 fliyî% ï˛Ó˚ˆÏD §üˆÏü°fliyî% ï˛Ó˚ˆÏD §üˆÏü°fliyî% ï˛Ó˚ˆÏD §üˆÏü°fliyî% ï˛Ó˚ˆÏD §üˆÏü°fliyî% ï˛Ó˚ˆÏD §üˆÏü° (harmonics)
§ühflÏ ï˛yˆÏÓ˚Ó˚ ÓyòƒÎˆÏsfÓ˚ Óyòöû˛!D fliyî% ï˛Ó˚ˆÏDÓ˚ ï˛ˆÏ_¥Ó˚ IJõÓ˚ !öû≈˛Ó˚¢#°– ò%•z ≤ÃyˆÏhsˇ ò,ì˛¸û˛yˆÏÓ xyê˛Ü˛yˆÏöy
ï˛yˆÏÓ˚ ܲï˛Ü˛=!° !ö!ò≈‹T ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ fliyî% ï˛Ó˚ˆÏDÓ˚ §,!‹T ܲÓ˚y ÎyÎ˚–
Î!ò ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒ •Î˚ L, ï˛y•ˆÏ° fliyö% ï˛Ó˚ˆÏD §Ω˛yÓƒ ï˛Ó˚D˜Ïòâ≈ƒ =!° •° 2 22 , , , ........3 2L L L
L Ln
SˆÎáyˆÏö
n ~ܲ!ê˛ ˛õ)î≈ §ÇáƒyV !ã˛e 7.5 o‹TÓƒ–
xyüÓ˚y °«˛ƒ ܲÓ˚Ó ˆÎñ ï˛yˆÏÓ˚Ó˚ ~Ó˚*˛õ ܲ¡õˆÏö §%flõ® !Ó®%Ó˚ §Çáƒy ~ܲy!ôܲ •ˆÏï˛ ˛õyˆÏÓ˚– ï˛yˆÏÓ˚Ó˚ ò%•z
≤Ãyhsˇ xyÓk˛ ÌyܲyÎ˚ ≤Ãyhsˇ ò%!ê˛ˆÏï˛ !öflõ® !Ó®% xÓ!fliï˛– §%flõ®
!Ó®%Ó˚ §Çáƒy ~ܲ •ˆÏ° ï˛yˆÏÓ˚Ó˚ ܲ¡õö 7.5 !ã˛ˆÏeÓ˚ ≤ÃÌü !ã˛e!ê˛Ó˚
üï˛ •ˆÏÓ– ~áyˆÏö ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒ ˛õÓ˚˛õÓ˚ ò%•z !öflõ® !Ó®%Ó˚ üˆÏôƒÓ˚
ò)Ó˚ˆÏcÓ˚ §üyö xÌ≈yÍ, L =
2λ
~•z ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ §y•yˆÏ΃ xyüÓ˚y
!ö¡¨!°!áï˛ §ü#ܲÓ˚î ˆÌˆÏܲ flõ®ˆÏöÓ˚ ܲ¡õyˆÏB˛Ó˚ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛
˛õyÓ˚Ó
vv λ=
~áy Ïö v ï˛yˆÏÓ˚ !ï˛Î≈ܲ ï˛Ó˚ˆÏDÓ˚ ˆÓà =
Tm
ˆÎáyˆÏö
T = ï˛yˆÏÓ˚Ó˚ ê˛yö ~ÓÇ m = ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒ ï˛yˆÏÓ˚Ó˚ û˛Ó˚–
v = 3v0
L = λ
192
§ÓˆÏã˛ˆÏÎ˚ ܲü ܲ¡õyˆÏB˛Ó˚ flõ®öˆÏܲ Ó°y •Î˚ ü)°§%Ó˚– ~Ó˚ ܲ¡õyˆÏB˛Ó˚ üyöñ
02vlT
vLm λ ==
........ 7.10
xöƒ ܲ¡õyB˛=!°ˆÏܲ Ó°y •Î˚ í˛z˛õ§%Ó˚– ~=!°Ó˚ ܲ¡õyB˛ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ (v0)=!îï˛Ü˛–
ü)°§%Ó˚ˆÏܲ ≤ÃÌü §üˆÏü° ÓˆÏ°Ä x!û˛!•ï˛ ܲÓ˚y •Î˚– v = 2v0 ܲ¡õyB˛ Î%_´ ≤ÃÌü í˛z˛õ§%Ó˚!ê˛ˆÏܲ Ó°y •Î˚
!mï˛#Î˚ §üˆÏü°– !mï˛#Î˚ í˛z˛õ§%Ó˚ˆÏܲ (v=3v0) Ó°y •Î˚ ï,˛ï˛#Î˚ §üˆÏü°–
fliyî% ï˛Ó˚ˆÏDÓ˚ ï˛ˆÏ_¥Ó˚ IJõÓ˚ ˜ï˛!Ó˚ Îsf=!°Ó˚ üˆÏôƒ ~ܲ!ê˛ •° ÓÑy!¢–
≤ÃyÌ!üܲû˛yˆÏÓ ˆÎ !Ó£ÏÎ˚à%!°Ó˚ IJõÓ˚ ÓÑy!¢Ó˚ §%ˆÏÓ˚Ó˚ ܲ¡õyB˛ ~ÓÇ ¢ˆÏ∑Ó˚
§ü,!k˛ (a) !öû≈˛Ó˚ ܲˆÏÓ˚ ï˛y •° (1) ¢ˆÏ∑Ó˚ í˛zͧ (2) öˆÏ°Ó˚ xyÜ,˛!ï˛ ~ÓÇ
xyÎ˚ï˛öñ (3) xyD%° !òˆÏÎ˚ ã˛y˛õyÓ˚ !åȈÏoÓ˚ xyܲyÓ˚ Ä xÓfliyö– §yôyÓ˚î ÓÑy!¢ˆÏܲ
~ܲ!ê˛ ü%_´ ö° Ó°y ÎyÎ˚ [!ã˛e 7.6(a)]– ~Ó˚ !åÈo=!° Ó¶˛ ÌyܲˆÏ° öˆÏ°Ó˚
§ü@ˇÃ ˜òˆÏâ≈ƒ•z ÓyÎ˚%hflÏΩ˛ ܲ!¡õï˛ •Î˚ñ ú˛ˆÏ° ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ §ÓˆÏã˛ˆÏÎ˚ ܲü •Î˚–
ˆÜ˛yöÄ !åÈo á%°ˆÏ°•z ï˛y öˆÏ°Ó˚ ü%_´ ≤Ãyhsˇ Ó˚*ˆÏ˛õ xyã˛Ó˚î ܲˆÏÓ˚ñ ú˛ˆÏ° ÓyÎ˚%hflψÏΩ˛Ó˚
˜òâ≈ƒ ܲˆÏü Ä ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ ˆÓˆÏí˛¸ ÎyÎ˚– Îáö ˆÜ˛yöÄ öˆÏ°Ó˚ ~ܲ!òˆÏܲ
Ó¶˛ ÌyˆÏܲ ï˛yˆÏܲ Ó°y ÎyÎ˚ Ók˛ ö°– Ók˛ öˆÏ°Ó˚ Ók˛ ≤ÃyˆÏhsˇ §Ó§üˆÏÎ˚•z ~ܲ!ê˛
!öflõ® !Ó®% ~ÓÇ ü%_´ ˛≤ÃyˆÏhsˇ ~ܲ!ê˛ §%flõ® !Ó®%Ó˚ §,!‹T •Î˚ [!ã˛e 7.6(b)]–
~ܲ!òܲ Ó¶˛ öˆÏ°Ó˚ ü)°§%ˆÏÓ˚Ó˚ ï˛Ó˚D ˜òâ≈ƒ 4= Lλ – ܲyˆÏç•z ü)°§%ˆÏÓ˚Ó˚
ܲ¡õyB˛ =04
vv
L– ~•z ôÓ˚ˆÏöÓ˚ öˆÏ° ˆçyí˛¸ §ÇáƒyÓ˚ §üˆÏü°=!° xö%˛õ!fliï˛–
ò%•z ü%á ˆáy°y öˆÏ°Ó˚ ü)°§%ˆÏÓ˚Ó˚ ï˛Ó˚D ˜òâ≈ƒ λ = 2L– ܲyˆÏç•z ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛
vvL 02
=
– ü%_´ öˆÏ°
ü)°§%Ó˚ åÈyí˛¸y !mï˛#Î˚ñ ï,˛ï˛#Î˚ §Ó §üˆÏü°•z í˛zͲõߨ •ˆÏï˛ ˛õyÓ˚ˆÏÓ– °«˛ƒ ܲÓ˚ˆÏ° ˆòáy ÎyˆÏÓ ˆÎñ Ók˛ öˆÏ° í˛zͲõߨ
ü)°§)ˆÏÓ˚Ó˚ ܲ¡õyB˛ §ü ˜òˆÏâ≈ƒÓ˚ ü%_´ öˆÏ° í˛zͲõߨ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ xˆÏô≈ܲ–
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ2 : (a) ò%•z ≤Ãyhsˇ xyê˛Ü˛yˆÏöy Im °¡∫y ~ܲ!ê˛ ˛!˛õÎ˚yˆÏöyÓ˚ ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚
0.015kgm–1– ~•z !˛õÎ˚yˆÏöyÓ˚ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ Î!ò v = 200Hz •Î˚ñ ï˛ˆÏÓ ï˛yˆÏÓ˚ ܲï˛ê˛y ê˛yö ≤ÈÏÎ˚yà ܲÓ˚ˆÏï˛
•ˆÏÓ !öî≈Î˚ ܲÓ˚&ö–
(b) 1.0m °¡∫y ~ܲ!ê˛ ~ܲ ü%á Ó¶˛ xày≈ö öˆÏ°Ó˚ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ !•§yÓ Ü˛Ó˚&ö– S¢ˆÏ∑Ó˚ ˆÓà v=
350m/s) – ö°!ê˛ˆÏܲ á%Ó ˆçyˆÏÓ˚ ÓyçyˆÏ° ܲ¡õyB˛ ܲ# •ˆÏÓ⁄
7.3.3 fliyî% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒfliyî% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒfliyî% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒfliyî% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒfliyî% ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ
fliyî% ï˛Ó˚D §¡∫ˆÏ¶˛ xyˆÏ°yã˛öyÓ˚ ˛õÓ˚ xyüÓ˚y ~•z ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ•=!°ˆÏܲ !°!˛õÓk˛ ܲÓ˚yÓ˚ ˆã˛‹Ty ܲÓ˚Ó ÉÈÙÈ
(a) fliyî% ï˛Ó˚D üyôƒˆÏüÓ˚ §#!üï˛ xLjϢ §,‹T •Î˚– ~•z ˆ«˛ˆÏe ~ܲ ܲîyÓ˚ ܲ¡õˆÏöÓ˚ xÓfliy ˛õÓ˚Óï≈˛# ܲîyÎ˚
ã˛ˆÏ° ÎyÎ˚ öyñ ú˛ˆÏ° ï˛Ó˚ˆÏDÓ˚ xyÜ,˛!ï˛ ~ˆÏàyÎ˚ öy–
193
(b) fliyî% ï˛Ó˚ˆÏD üyôƒˆÏüÓ˚ ≤Ã!ï˛!ê˛ Ó› ܲîy ~ܲ•z ܲ¡õyˆÏB˛ !Ó!û˛ß¨ !ÓhflÏyÓ˚ !öˆÏÎ˚ ܲ!¡õï˛ •Î˚– §%flõ®
!Ó®%=!°ˆÏï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ §Ó≈yˆÏ˛õ«˛y ˆÓ!¢– !öflõ® !Ó®%=!°ˆÏï˛ ¢)öƒ– §%flõ® !Ó®% ˆÌˆÏܲ ÷Ó˚& ܲˆÏÓ˚
!ÓhflÏyÓ˚ ܲüˆÏï˛ Ü˛üˆÏï˛ !öflõ® !Ó®%ˆÏï˛ ¢)öƒ •Î˚–
(c) ò%!ê˛ §%flõ® Óy ò%!ê˛ !öflõ® üˆÏôƒÜ˛yÓ˚ ò)Ó˚c fliyî% ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ xˆÏô≈ܲ– üyôƒü!ê˛ Ü˛ï˛Ü˛=!°
áˆÏ[˛ !Óû˛_´ •Î˚– ≤ÈÏï˛ƒÜ˛!ê˛ áˆÏ[˛Ó˚ ˜òâ≈ƒ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ xˆÏô≈ܲ–
(d) ˛õÓ˚˛õÓ˚ ò%!ê˛ !öflõ® !Ó®%Ó˚ üôƒÓï≈˛# xMÈ˛ˆÏ° §Ó ܲîyÓ˚ ò¢y §üyö ÌyˆÏܲ ~ÓÇ ˆÜ˛yöÄ !öflõ® !Ó®%Ó˚
ò%•z !òˆÏܲ xÓ!fliï˛ ~•zÓ˚ܲü ò%!ê˛ §!ߨ!•ï˛ xMÈ˛ˆÏ° ܲ¡õˆÏöÓ˚ ò¢y !Ó˛õÓ˚#ï˛ •Î˚–
(e) !öflõ® !Ó®%ˆÏï˛ Ü˛îyÓ˚ à!ï˛ˆÏÓà ¢)öƒ ~ÓÇ §%flõ® !Ó®%ˆÏï˛ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢– üôƒÓï≈˛# !Ó®%=!°ˆÏï˛
ܲîyˆÏòÓ˚ à!ï˛ˆÏÓà §%flõ® !Ó®% ˆÌˆÏܲ e´üyß∫ˆÏÎ˚ ܲˆÏü !öflõ® !Ó®%ˆÏï˛ ¢)öƒ •Î˚–
ˆÎ üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚ !ÓFå%ÈÓ˚î âˆÏê˛ öyñ xÌ≈yÍ ï˛Ó˚DòˆÏ°Ó˚ xhsˇû)≈˛_´ §Ó ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ï˛Ó˚D•z ~ܲ•z ˆÓˆÏà
x@ˇÃ§Ó˚ •Î˚ñ ˆ§áyˆÏö ï˛Ó˚DòˆÏ°Ó˚ xyÜ,˛!ï˛ x˛õ!Ó˚Ó!ï≈˛ï˛ ˆÓ˚ˆÏá•z ˆ§!ê˛ ã˛°ˆÏï˛ ÌyˆÏܲ– ~ˆÏ«˛ˆÏe ï˛Ó˚DòˆÏ°Ó˚ ˆÓà
ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚ §üyö– !ܲv !ÓFå%ÈÓ˚î¢#° üyôƒˆÏü SˆÎáyˆÏö ï˛Ó˚DˆÏÓà ܲ¡õyˆÏB˛Ó˚ IJõÓ˚
!öû≈˛Ó˚¢#°V ï˛Ó˚DòˆÏ°Ó˚ ~ܲ ~ܲ!ê˛ ï˛Ó˚D ~ܲ ~ܲ ˆÓˆÏà ã˛ˆÏ°ñ ú˛ˆÏ° ï˛Ó˚DòˆÏ°Ó˚ xyÜ,˛!ï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏï˛
˛õyˆÏÓ˚ ~ÓÇ §ˆÏÓù≈yFã˛ !Ó®%Ó˚ à!ï˛ˆÏÓà §Ó˚° ˆòy°ï˛Ó˚Dà%!°Ó˚ à!ï˛ˆÏÓà ˆÌˆÏܲ !û˛ß¨ •Î˚– ~•z ˆÓàˆÏܲ•z Ó°y
•Î˚ ò°#Î˚ ˆÓà ï˛Ó˚ˆÏDÓ˚ ¢!_´ ~•z ˆÓˆÏà•z ï˛Ó˚DòˆÏ°Ó˚ §ˆÏD Óy!•ï˛ •Î˚–
ò°#Î˚ ˆÓà– !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ xyüÓ˚y ~ܲ•z !ÓhflÏyˆÏÓ˚Ó˚ !ܲv 1ω Ä
2 ω
~•z ò%•z §yüyöƒ ˛õ,Ìܲ ˆÜ˛Ô!îܲ
ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ ï˛Ó˚D !ö°yü– ~•z ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ §MÈ˛Ó˚î ô Óܲ k1 Ä k
2 ~ÓÇ ~=!°Ó˚ üyöÄ Ü˛yåÈyܲy!åÈ
ÌyܲˆÏÓ– ôÓ˚y Îyܲñ ï˛Ó˚D ò%•z!ê˛ ~ܲ•z !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏÎ˚ ~ˆÏܲ x˛õˆÏÓ˚Ó˚ í˛z˛õÓ˚ ˛õ!ï˛ï˛ •°–
~áö ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ °!∏˛ §Ó˚î •ˆÏÓñ
()()() 1122 ,sinsin Yxtatkxatkx ωω =−+−
= ( ) ( ) ( ) ( )1 2 1 2 1 2 1 22 sin cos
2 2t k k x t k k x
aω ω ω ω⎡ ⎤ ⎡ ⎤+ − + − − −
⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
..... 7.11
1 ω
Ä
2 ω
~ÓÇ
1 k
Ä k2 - Ó˚ üˆÏôƒ ˛õyÌ≈ܲƒ Î!ò á%Ó §yüyöƒ ÌyˆÏܲñ ï˛y•ˆÏ° ˆòáy ÎyÎ˚ñ
12 ωωω −=Δ
~ÓÇ
12 kkk −=Δ
~åÈyí˛¸y Î!ò àí˛¸ !•§yˆÏÓ !°!á
194
12
2+ =ωω ω
~ÓÇ, 1 2
2k k
k+=
xï˛~Óñ 7.11 §ü#ܲÓ˚îˆÏܲ ˆ°áy ÎyÎ˚
( ) ( ), 2 sin cos2 2
kY x t a t kx t x
∆ ∆⎛ ⎞= − −⎝ ⎠ωω ......... 7.12
~!ê˛ ~ܲ!ê˛ öï%˛ö ï˛Ó˚DÓ˚*ˆÏ˛õÓ˚ §ü#ܲÓ˚î– ≤ÃÌüï˛ñ àë˛öܲyÓ˚# (component) ò%•z ï˛Ó˚ˆÏDÓ˚ ˆÎ ˆÜ˛yöÄ!ê˛Ó˚
!ÓhflÏyˆÏÓ˚Ó˚ ï%˛°öyÎ˚ ~•z ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ !mà%î–
!mï˛#Î˚ï˛ñ ~•z ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚ˆÏî ò%!ê˛ û˛yà xyˆÏåÈ–
§y•zö xˆÏ˛õ«˛Ü˛ xÇ¢!ê˛ o&ï˛ ˛õ!Ó˚Óï≈˛ö¢#° ~ÓÇ
~Ó˚ ܲ¡õyB˛ àë˛öܲyÓ˚# ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyˆÏB˛Ó˚
àí˛¸– ˆÜ˛y§y•zö xÇ¢!ê˛ ô#ˆÏÓ˚ ˛õ!Ó˚Óï≈˛ö¢#°– ~!ê˛Ó˚
ܲ¡õyB˛ àë˛öܲyÓ˚# ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyˆÏB˛Ó˚
!ÓˆÏÎ˚yàú˛ˆÏ°Ó˚ xˆÏô≈ܲ– í˛z˛õ!Ó˚˛õy!ï˛ï˛ ï˛Ó˚ˆÏD ô#ˆÏÓ˚
˛õ!Ó˚Óï≈˛ö¢#° xÇ¢!ê˛ o&ï˛ ˛õ!Ó˚Óï≈˛¢#° xÇ¢!ê˛
o&ï˛ ˛õ!Ó˚Óï≈˛ö¢#° xLjϢÓ˚ í˛z˛õˆÏÓ˚Ó˚ ˆüyí˛¸ˆÏܲÓ˚ (envelope) ܲyç ܲˆÏÓ˚ S!ã˛e 7.7 ˆòá%ö)–
7.9 §ü#ܲÓ˚ˆÏîÓ˚ §y•zö xˆÏ˛õ«˛Ü˛!ê˛ °!∏˛ ï˛Ó˚ˆÏDÓ˚ ò¢y ˆÓà !öô≈yÓ˚î ܲˆÏÓ˚– ~•z §ü#ܲÓ˚î ˆÌˆÏܲ ò¢y ˆÓà
˛õyÄÎ˚y ÎyÎ˚–
pvkω=
xÌ≈yÍñ °!∏˛ ï˛Ó˚ˆÏDÓ˚ ò¢y ˆÓà ò%•z!ê˛ ï˛Ó˚ˆÏDÓ˚ àí˛¸ ܲ¡õyB˛ Ä àí˛¸ §MÈ˛Ó˚î ô &ÓˆÏܲÓ˚ xö%˛õyï˛– ˆÜ˛y§y•zö
xˆÏ˛õ«˛Ü˛!ê˛Ó˚ ï˛yͲõÎ≈ •° ~•z ˆÎñ !ÓhflÏyÓ˚ ˆüyí˛¸Ü˛!ê˛Ä ï˛Ó˚Dà!ï˛ˆÏï˛ x@ˇÃ§Ó˚ •Î˚– ≤Ã̈Ïü ˆòá%öñ x = 0,
t = 0 ˆï˛ ˆÜ˛y§y•zˆÏöÓ˚ üyö §Ó≈y!ôܲ ~ÓÇ !ÓhflÏyÓ˚Ä §Ó≈y!ôܲ– xyÓyÓ˚ñ
tt =Δ
§üˆÏÎ˚ Î!ò ˆÜ˛y§y•zˆÏöÓ˚ üyö
x Δ
fliyöyˆÏB˛ §Ó≈y!ôܲ •Î˚ ï˛ˆÏÓñ
022
ktk
ωΔΔ Δ=Δ=
xÌÓyñ
xtk
ω ΔΔ = ΔΔ
xÌ≈yÍñ !ÓhflÏyˆÏÓ˚Ó˚ §ˆÏÓù≈yFã˛ üyö
t Δ
§üˆÏÎ˚
0 x=
ˆÌˆÏܲ˛ x=
x Δ
!Ó®%ˆÏï˛
xt
ΔΔ
Óy
kωΔΔ
ˆÓˆÏà §ˆÏÓ˚ ˆàˆÏåÈ–
ˆÎˆÏ•ï%˛ ¢!_´Ó˚ âöc !ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õy!ï˛Ü˛ xyüÓ˚y Ó°ˆÏï˛ ˛õy!Ó˚ ˆÎñ ¢!_´ ï˛Ó˚ˆÏDÓ˚ ˆÎ xMÈ˛ˆÏ° ˛õ%O#û)˛ï˛
195
ÌyˆÏܲñ ˆ§!ê˛
kωΔΔ
à!ï˛ˆÏÓà !öˆÏÎ˚ x@ˇÃ§Ó˚ •Î˚– ~•z ˆÓàˆÏܲ Ó°y •Î˚ =FåȈÏÓà Óy ò°#Î˚ˆÏÓà (group
velocity)vR
– ~áö Î!ò í˛z˛õ!Ó˚˛õy!ï˛ï˛ ï˛Ó˚D ò%•z!ê˛Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ e´ü¢ xyÓ˚Ä Ü˛yåÈyܲy!åÈ xyˆÏ§ñ ï˛ˆÏÓ
0 ωΔ→
,
0 k Δ→
ôˆÏÓ˚ ˆöÄÎ˚y ÎyˆÏÓ– ~áö xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ ò°#Î˚ ˆÓà
g
dv
dkω =
xy˛õ!ö çyˆÏöö ˆÎñ §yòy xyˆÏ°y Îáö ܲyˆÏã˛Ó˚ !≤ÃçˆÏüÓ˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ܲyˆÏã˛Ó˚ üˆÏôƒ ~ܲ ~ܲ
ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ xyˆÏ°y ~ܲ ~ܲ ˆÓˆÏà ã˛ˆÏ°– ~Ó˚ ú˛ˆÏ°•z !≤ÃçˆÏü xyˆÏ°yÓ˚ !ÓFå%ÈÓ˚î âˆÏê˛– ˆÎ üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚
ò¢yˆÏÓà ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#°ñ ï˛yˆÏܲ xyüÓ˚y Ó!° !ÓFå%ÈÓ˚ܲ üyôƒü (dispersive medim)–
!ÓFå%ÈÓ˚îܲyÓ˚# üyôƒˆÏü ò°àï˛ ˆÓàÄ §yôyÓ˚îû˛yˆÏÓ ï˛Ó˚ˆÏDÓ˚ ˜òˆÏâ≈ƒÓ˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– í˛z˛õˆÏÓ˚ ò%!ê˛ ï˛Ó˚ˆÏDÓ˚
!ü!°ï˛ ú˛ˆÏ°Ó˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ï˛yˆÏòÓ˚ ï˛Ó˚D ˜òâ≈ƒ !Ó!ãåÈߨ ôˆÏÓ˚ !öˆÏÎ˚ ˆ¢£Ï ˛õÎ≈hsˇ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚
≤ÈÏû˛ˆÏòÓ˚ §#üyhsˇ üyö (limiting value) ¢)öƒ üˆÏö ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– !ܲv ÓyhflψÏÓ ï˛Ó˚D =ˆÏFåÈÓ˚ üˆÏôƒ ï˛Ó˚D ˜òâ≈ƒ
~ܲ!ê˛ §hsˇï˛ xˆÏ˛õ«˛Ü˛ ~ÓÇ ï˛Ó˚ÏDˆÏòÓ˚ !ü!°ï˛ ú˛° ~ܲ!ê˛ §üyܲ° !òˆÏÎ˚ !öˆÏò≈¢ ܲÓ˚ˆÏï˛ •Î˚– ˆ§•z §üyܲˆÏ°Ó˚
üyö ˆÎáyˆÏö í˛zFã˛ï˛ü (maximum), ˆ§•z !Ó®%•z •° =ˆÏFåÈÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ xÓfliyö ~ÓÇ ˙ !Ó®%Ó˚ à!ï˛ˆÏÓà •ˆÏÓ
ï˛Ó˚DòˆÏ°Ó˚ à!ï˛ˆÏÓà– xÌ≈yÍñ ï˛Ó˚D=ˆÏFåÈÓ˚ §ü#ܲÓ˚î •ˆÏÓñ
y= ( )t kxfkf e
ω −∫~áyˆÏö
kf→ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚– àîöyÓ˚ §%!ÓôyÓ˚ çöƒ ÓyhflÏÓ §y•zö/ ˆÜ˛y§y•zö xˆÏ˛õ«˛ˆÏܲÓ˚ ÓòˆÏ° ç!ê˛°
§)ã˛Ü˛ (complex exponential) xˆÏ˛õ«˛Ü˛ ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
ò¢y ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ •yÓ˚ ¢)öƒ •ÓyÓ˚ ¢ï≈˛ :
()0 tkxk
ω ∂−= ∂
xÌ≈yÍñ
0 txkω∂−= ∂
~•z ¢ï≈˛!ê˛ ˆÎáyˆÏö ˛õy!°ï˛ •ˆÏÓñ ˆ§áyˆÏö §üyܲˆÏ°Ó˚ í˛zFã˛ï˛ü üyö ˛õyÄÎ˚y ÎyˆÏÓ– §%ï˛Ó˚yÇñ ˙ í˛zFã˛ï˛ü üyˆÏöÓ˚
!Ó®%Ó˚ à!ï˛ˆÏÓàñ ˆÎ!ê˛ ò° xÌÓy =ˆÏFåÈÓ˚ à!ï˛ˆÏÓàñ ï˛yÓ˚ üyö
g
xwv
tk ω∂ ==
.......... 7.14
ò¢y ˆÓà ~ÓÇ ò°àï˛ SxÌÓy =FåÈV ˆÓˆÏàÓ˚ üˆÏôƒ §¡õÜ≈˛ !ö¡¨!°!áï˛ û˛yˆÏÓ !öî≈Î˚ ܲÓ˚y ÎyÎ˚ :
ò°àï˛ ˆÓà ( )∂ ∂= =∂ ∂g pv kvk kω xÌÓyñ p
g p
vv v k
k
∂= + ∂ ......... 7.15
196
Î!ò ˆ°áy ÎyÎ˚ñ
2k
πλ =
ï˛y•ˆÏ°ñ 22
kk
π λ∂ = − ∂
7.15 öÇ §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y ˛õy•zñ
2
22∂∂
=+=−∂ ⎛⎞ −∂ ⎜⎟ ⎝⎠
ppp
vvvgvpv
πλ λλ πλλ
......... 7.16
ò°àï˛ ˆÓà ~ÓÇ ò¢yàï˛ ˆÓˆÏàÓ˚ ˛õyÌ≈ܲƒ !öî≈ˆÏÎ˚Ó˚ çöƒ xyüÓ˚y ~ܲ!ê˛ í˛zòy•Ó˚ˆÏîÓ˚ §y•y΃ ˆöÓ– ˆ§!ê˛
•°ñ àû˛#Ó˚ çˆÏ°Ó˚ í˛z˛õ!Ó˚ï˛ˆÏ°Ó˚ ï˛Ó˚Dñ ÎyˆÏܲ üyôƒyܲ£Ï≈î ç!öï˛ ï˛Ó˚D (gravity waves) Ó°y •Î˚– ~ çï˛#Î˚
ï˛Ó˚ˆÏD ˆÜ˛Ó°üye üyôƒyܲ£Ï≈î•z ç°ï˛ˆÏ°Ó˚ í˛z˛õÓ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° §,!‹T ܲˆÏÓ˚ ~ÓÇ ~•z ï˛Ó˚ˆÏDÓ˚ x!ôܲ !ÓFå%ÈÓ˚î
°«˛ƒ ܲÓ˚y ÎyÎ˚– ~•z ï˛Ó˚D=!°Ó˚ ò¢y ˆÓà ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ Óà≈ü)ˆÏ°Ó˚ §üyö%˛õy!ï˛Ü˛ xÌ≈yÍ
12
pv cλ= ˆÎáyˆÏö c ~ܲ!ê˛ !fliÓ˚ §Çáƒy–
xÌÓy12
1pv c k−
= SˆÎˆÏ•ï%˛ 2
kπλ= )
~áyˆÏö öï%˛ö !fliÓ˚ §Çáƒy!ê˛ 1 2c π=
xyüÓ˚y çy!öñ vpkω=
12
1ck ω ∴=ω ˆÜ˛ k ~Ó˚˛ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲÓ˚ˆÏ°
12
11122 gp
dvckv
dkω−
===
xï˛~Óñ üyôƒyܲ£Ï≈îç!öï˛ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ò°#Î˚ ˆÓà ò¢y ˆÓˆÏàÓ˚ xˆÏô≈ܲ– xÌ≈yÍñ ~•z ôÓ˚ˆÏöÓ˚ ï˛Ó˚ˆÏD
àë˛öܲyÓ˚# (component) ï˛Ó˚D ¢#£Ï≈ ï˛Ó˚DòˆÏ°Ó˚ ˆã˛ˆÏÎ˚ o&ï˛ï˛Ó˚ ˆÓˆÏà ÎyÎ˚–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 3 : (ܲV ˆÜ˛yöÄ üyôƒˆÏü ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ ò¢yˆÏÓà 1 2v c c λ= + ˆÎáyˆÏö c1 ~ÓÇ c2 !fliÓ˚
§Çáƒy– ï˛Ó˚ˆÏDÓ˚ ò°#Î˚ ˆÓà ܲï˛⁄
SáV ˆÜ˛yöÄ ~ܲ!ê˛ üyôƒˆÏü xyˆÏ°yܲï˛Ó˚ˆÏDÓ˚ ˆÓà
, cv
n=
ˆÎáyˆÏö n = üyôƒˆÏüÓ˚ ≤Ã!ï˛§Ó˚yÇܲñ c = ¢)ˆÏöƒ
xyˆÏ°yˆÏܲÓ˚ à!ï˛ˆÏÓà– ˆòáyö ˆÎñ ò°#Î˚ ˆÓà
1 g
dnvv
ndλ
λ⎛⎞ =+ ⎜⎟ ⎝⎠
–
7.4 fl∫Ó˚ܲ¡õfl∫Ó˚ܲ¡õfl∫Ó˚ܲ¡õfl∫Ó˚ܲ¡õfl∫Ó˚ܲ¡õ
xyüÓ˚y ˆòá°yü ˆÎñ ò%!ê˛ §yüyöƒ ˛õyÌ≈ˆÏܲƒÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ Î%_´ (
1 ω
~ÓÇ
2 ω
) ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚
ú˛ˆÏ° ï˛Ó˚DòˆÏ°Ó˚ §,!‹T •Î˚–
197
7.7 !ã˛ˆÏe xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ x x«˛ ÓÓ˚yÓÓ˚ ò)Ó˚c ~ÓÇ y x«˛ ÓÓ˚yÓÓ˚ °!∏˛ §Ó˚îˆÏܲ
xÑyܲy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö §üÎ˚ t ˆÜ˛ x˛õ!Ó˚Ó!ï≈˛ï˛ Ó˚yáy •ˆÏÎ˚ˆÏåÈ– ~ˆÏܲ Ó°y ˆÎˆÏï˛ ˛õyˆÏÓ˚ ò)Ó˚c §yˆÏ˛õˆÏ«˛
í˛z˛õ!Ó˚˛õyï˛– xyüÓ˚y xöƒ ~ܲ ≤ÃܲyÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ܲÌy Ó°ˆÏï˛ ˛õy!Ó˚ ˆÎáyˆÏö xyüÓ˚y ~ܲ!ê˛ !ö!ò≈‹T !Ó®%ˆÏï˛
§Ó˚î Y(x,t) ˆÜ˛ t-~Ó˚ §yˆÏ˛õˆÏ«˛ xÑyÜ˛Ó ~ÓÇ ~ˆÏܲ Ó°y ˆÎˆÏï˛ ˛õyˆÏÓ˚ §üÎ˚ §yˆÏ˛õˆÏ«˛ í˛z˛õ!Ó˚˛õyï˛– ~áyˆÏö
x x˛õ!Ó˚Óï≈˛ö#Î˚ §Çáƒy–
¢∑ï˛Ó˚ˆÏDÓ˚ §üÎ˚ §yˆÏ˛õˆÏ«˛ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° ˆÎ xyܲ£Ï≈î#Î˚ âê˛öy!ê˛ âˆÏê˛ñ ï˛yÓ˚ öyü fl∫Ó˚ܲ¡õ– §yüyöƒ
ï˛ú˛yˆÏï˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ò%!ê˛ ¢ˆÏ∑Ó˚ í˛zͧˆÏܲ Î!ò ~ܲ§ˆÏD ôù!öï˛ Ü˛Ó˚y •Î˚ñ ï˛ˆÏÓ ~Ó˚ ú˛ˆÏ° ˆÎ ¢∑ í˛zͲõߨ •ˆÏÓ
ï˛yÓ˚ ï˛#Ó ï˛y ˛õÎ˚≈yÎ˚e´ˆÏü ~ܲÓyÓ˚ Óyí˛¸ˆÏÓ ~ÓÇ ~ܲÓyÓ˚ ܲüˆÏÓ– ¢ˆÏ∑ ï˛#Ó ï˛yÓ˚ ~•zÓ˚ܲü ˛õÎ≈yÎ˚e´ˆÏü •…y§Ó,!mÓ˚
âê˛öyˆÏܲ fl∫Ó˚ܲ¡õ ӈϰ–
~ܲ!ê˛ í˛zòy•Ó˚î ˆòÄÎ˚y Îyܲ– ò%•z!ê˛ §%Ó˚¢°yܲy (A, B) ˆöÄÎ˚y •° ÎyˆÏòÓ˚ ܲ¡õyB˛ 5 ~ÓÇ 6Hz– ~ˆÏòÓ˚
~ܲ§ˆÏD ôù!öï˛ Ü˛Ó˚ˆÏ° ˆÎ ¢∑ï˛Ó˚D=!° §,!‹T •ˆÏÓñ ˆ§ ò%!ê˛ ˛õÓ˚flõˆÏÓ˚Ó˚ IJõÓ˚ xy˛õ!ï˛ï˛ •ˆÏÎ˚ ~ܲ!ê˛ °!∏˛
ï˛Ó˚D (C) §,!‹T ܲÓ˚ˆÏÓ– 7.8 !ã˛ˆÏe ~!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˆÎ xÓfliyÎ˚ ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ ~ܲ•z ò¢y •ˆÏÓñ ï˛áö•z
ˆ◊yï˛y ˆçyÓ˚ ¢∑ ÷öˆÏÓö– ~áyˆÏö °!∏˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ (E !Ó®%V– 1 ˆ§ˆÏܲ[˛ ˛õˆÏÓ˚ !mï˛#Î˚
§%Ó˚ ¢°yܲy!ê˛Ó˚ ï˛Ó˚D ≤ÃÌü!ê˛Ó˚ xˆÏ˛õ«˛y ~ܲ!ê˛ ˛õ)î≈ ï˛Ó˚ˆÏD x@ˇÃ§Ó˚ •ˆÏÓ ~ÓÇ ~ˆÏòÓ˚ ò¢y ˛õ%öÓ˚yÎ˚ §üyö •ˆÏÓ–
ï˛áö xyÓyÓ˚ ˆçyÓ˚ ¢∑ ˆ¢yöy ÎyˆÏÓ– ~!ê˛ G !Ó®% !òˆÏÎ˚ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– !ܲv
12
ˆ§ˆÏܲ[˛ ˛õˆÏÓ˚ !mï˛#Î˚
198
§%Ó˚¢°yܲy!ê˛ ≤ÃÌü!ê˛ xˆÏ˛õ«˛y xô≈ ï˛Ó˚D ~!àˆÏÎ˚ ÎyˆÏÓ– ï˛áö ~ˆÏòÓ˚ ò¢y §¡õ)î≈ !Ó˛õÓ˚#ï˛ •ˆÏÓ– ~•z fliyˆÏö
(F !Ó®%V ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ §ÓˆÏã˛ˆÏÎ˚ ܲü– ~•zû˛yˆÏÓ Îï˛ §üÎ˚ ÎyˆÏÓ ï˛ï˛ ¢ˆÏ∑Ó˚ ≤ÃyӈϰƒÓ˚ ˛õÎ≈ye´ˆÏü •…y§Ó,!k˛
âê˛ˆÏÓ ~ÓÇ ˆ◊yï˛y ~ܲ!ê˛ Ü˛!¡õï˛ ¢∑ ÷öˆÏÓö–
fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy ~ܲ ˆ§ˆÏܲˆÏ[˛ ˆÎ ܲÎ˚ÓyÓ˚ ≤ÃÓ° xÌÓy ü,ò% ¢∑ ˆ¢yöy ÎyÎ˚ñ ˆ§•z §ÇáƒyˆÏܲ fl∫Ó˚ܲˆÏ¡õÓ˚
§Çáƒy Ó°y •Î˚– í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öy ˆÌˆÏܲ ˆòáy ˆà° ˆÎñ í˛zͧ ò%!ê˛Ó˚ ܲ¡õyB˛ 5 ~ÓÇ 6 SxÌ≈yÍ Ü˛¡õyˆÏB˛Ó˚
˛õyÌ≈ܲƒ =1) •ˆÏ° ï˛yÓ˚y ≤Ã!ï˛ ˆ§ˆÏܲˆÏ[˛ ~ܲ!ê˛ ≤ÃÓ° Óy ~ܲ!ê˛ ü,ò% ¢∑ í˛zͲõߨ ܲÓ˚ˆÏÓ–
xï˛~Óñ xyüÓ˚y Ó°ˆÏï˛ ˛õy!Ó˚ ˆÎñ
fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy = í˛zͧmˆÏÎ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒí˛zͧmˆÏÎ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒí˛zͧmˆÏÎ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒí˛zͧmˆÏÎ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒí˛zͧmˆÏÎ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒ
xÓ¢ƒ ~•z í˛zòy•Ó˚ˆÏî 5 Óy 6Hz ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ܲÌy Ó°y •ˆÏ°Ä ÓyhflψÏÓ ~ï˛ Ü˛ü ܲ¡õyˆÏB˛Ó˚ ¢∑
ˆ¢yöy ÎyÎ˚ öy–
fl∫Ó˚ܲˆÏ¡õÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïîfl∫Ó˚ܲˆÏ¡õÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïîfl∫Ó˚ܲˆÏ¡õÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïîfl∫Ó˚ܲˆÏ¡õÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïîfl∫Ó˚ܲˆÏ¡õÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî :
ôÓ˚y Îyܲ §ü !ÓhflÏyÓ˚ !ܲv §yüyöƒ ܲ¡õyB˛ ˛õyÌ≈ˆÏܲƒÓ˚ ò%!ê˛ §Ó˚° ˆòy°à!ï˛ ï˛Ó˚D ~ܲ•z !òˆÏܲ x@ˇÃ§Ó˚ •ˆÏFåÈ–
~ˆÏòÓ˚ !ö¡¨!°!áï˛ §ü#ܲÓ˚î !òˆÏÎ˚ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚ñ
11 sin yat ω =
~ÓÇ
22 sin yat ω =
ˆÎáyˆÏö a = ï˛Ó˚DmˆÏÎ˚Ó˚ !ÓhflÏyÓ˚ n1,n
2 = ܲ¡õyB˛
=
1 sin2 ant π
y1 ~ÓÇ y
2 =
2 sin2 ant π
n1>n
2 ~ÓÇ n
1 Ä n
2 -Ó˚ ˛õyÌ≈ܲƒ á%Ó ˆÓ!¢ öÎ˚–
ôÓ˚y Îyܲñ ò%!ê˛ ï˛Ó˚D §ü ò¢yÎ˚ ˆÌˆÏܲ Îyey ÷Ó˚& ܲÓ˚°– ~ˆÏòÓ˚ í˛z˛õ!Ó˚˛õyˆÏï˛Ó˚ ú˛ˆÏ° ˆÎ °!∏˛ §Ó˚î •ˆÏÓ
ï˛yˆÏܲñ y ôÓ˚ˆÏ°ñ
1212 sin2sin2t yyyantan =+=+ ππ
=
1212 2cos2sin222
t
nnnnat
−+ ⎛⎞⎛⎞⎜⎟⎜⎟ ⎝⎠⎝⎠ ππ
....... 7.17
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î!ê˛ ~ܲ!ê˛ §Ó˚° ˆòy°à!ï˛Ó˚ !ܲv ~Ó˚ !ÓhflÏyÓ˚ 1 22 cos22
n nA a t
−⎛ ⎞= ⎜ ⎟⎝ ⎠π §üˆÏÎ˚Ó˚ §ˆÏD
˛õ!Ó˚Óï≈˛ö¢#°– xï˛~Óñ Îï˛ §üÎ˚ x!ï˛Óy!•ï˛ •ˆÏÓñ °!∏˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ ï˛ï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÓ– ܲáˆÏöy ܲáˆÏöy
~•z !ÓhflÏyÓ˚ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ •ˆÏÓñ xyÓyÓ˚ ܲáˆÏöy §ÓˆÏã˛ˆÏÎ˚ ܲü •ˆÏÓ– ú˛ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃyӈϰƒÓ˚ •…y§Ó,!k˛ •ˆÏÓ
~ÓÇ fl∫Ó˚ܲ¡õ §,!‹T •ˆÏÓ–
°!∏˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ (A) §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ •ˆÏÓ Îáö
( )1 2cos2 12
n n−= ±π
xÌÓyñ ( )1 22
0, ,2 ......2
n nm
−=
π π π π [ m =0, 1, 2......•zï˛ƒy!ò]
xÌ≈yÍñ
121212
12 0,,,.....mt
nnnnnn=−−−
§üˆÏÎ˚ §ÓˆÏã˛ˆÏÎ˚ ˆçyˆÏÓ˚ ¢∑ ˆ¢yöy ÎyˆÏÓ–
199
xï˛~Óñ ò%!ê˛ ˆçyˆÏÓ˚ ¢∑ ˆ¢yöyÓ˚ xhsˇÓ≈ï˛# xÓܲy¢ =
12
1nn −
ˆ§ˆÏܲ[˛
∴
1 ˆ§ˆÏܲˆÏu˛ (n1–n
2) - ÓyÓ˚ ≤ÃÓ° ¢∑ ˆ¢yöy ÎyˆÏÓ–
xyÓyÓ˚ °!∏˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ (A) §ÓˆÏã˛ˆÏÎ˚ ܲü •ˆÏÓ Îáöñ
() 12 cos20
2nnt −
=π
xÌÓyñ ( ) ( )1 22 3, ..... 2 1
2 2 2 2n n
m−
= +π π π π [ m =0, 1, 2......]
xÌ≈yÍñ
()()()
() 121212
21 13 ,,......222
mt
nnnnnn
+=
−−−
§üˆÏÎ˚ !ÓhflÏyÓ˚ §ÓˆÏã˛ˆÏÎ˚ ܲü •ˆÏÎ˚ ¢)öƒ •ˆÏÎ˚ ÎyˆÏÓ–
xï˛~Óñ ˛õÓ˚˛õÓ˚ ò%!ê˛ !öÉ¢ˆÏ∑Ó˚ xhsˇÓ≈ï˛# xÓܲy¢ = 1 2
1n n− ˆ§ Ïܲ[˛–
∴ 1 ˆ§ˆÏܲˆÏu˛ (n1 -n
2) ÓyÓ˚ ˆÜ˛yö ¢∑ ˆ¢yöy ÎyˆÏÓ öy–
ܲyˆÏç•z xyüÓ˚y í˛z˛õˆÏÓ˚Ó˚ xyˆÏ°yã˛öy ˆÌˆÏܲ ˆòá°yü ˆÎ fl∫Ó˚ܲˆÏ¡õÓ˚ ܲ¡õyB˛ =
() 12 nn −
= í˛zͧmˆÏÎ˚Ó˚
ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ܲƒ– ÓyòƒÎsfy!òÓ˚ §%Ó˚ !ü°yˆÏï˛ fl∫Ó˚ܲˆÏ¡õÓ˚ §y•y΃ ˆöÄÎ˚y •Î˚– ò%!ê˛ ÓyòƒÎsf ~ܲ §%ˆÏÓ˚ xyöˆÏï˛
ò%!ê˛ˆÏܲ ~ܲ§ˆÏD Óy!çˆÏÎ˚ fl∫Ó˚ܲˆÏ¡õÓ˚ í˛z˛õ!fli!ï˛ °«˛ƒ ܲÓ˚y •Î˚– §%Ó˚ !ü°ˆÏ° xyÓ˚ fl∫Ó˚ܲ¡õ ÷öˆÏï˛ ˛õyÄÎ˚y ÎyÎ˚
öy–
7.5 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
(1) ~ܲ•z üyôƒˆÏü §MÈ˛Ó˚üyö ò%!ê˛ ï˛Ó˚D Îáö ~ˆÏܲ x˛õˆÏÓ˚Ó˚ í˛z˛õ!Ó˚˛õy!ï˛ï˛ •Î˚ñ ï˛áö ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛
°!∏˛ !ÓhflÏyÓ˚ ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ xy°yòy !ÓhflÏyˆÏÓ˚Ó˚ Ó#çày!î!ï˛Ü˛ ˆÎyàú˛ˆÏ°Ó˚ §üyö–
(2) §üyö !ÓhflÏyÓ˚ñ ܲ¡õyB˛ ~ÓÇ ï˛Ó˚D ˜òâ≈ƒÎ%_´ ò%!ê˛ ï˛Ó˚D !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ x@ˇÃ§Ó˚ •ˆÏÎ˚ ~ˆÏܲ x˛õˆÏÓ˚Ó˚
í˛z˛õÓ˚ ˛õí˛¸ˆÏ° fliyî% ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚– ~•z ï˛Ó˚D ò%!ê˛ !Ó®%Ó˚ üˆÏôƒ §#üyÓk˛ ÌyˆÏܲ–
(3) fliyî% ï˛Ó˚ˆÏDÓ˚ üˆÏôƒ §%flõ® !Ó®%ˆÏï˛ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ !ÓhflÏyÓ˚ ~ÓÇ !öflõ® !Ó®%ˆÏï˛ !ÓhflÏyÓ˚ ¢)öƒ– ò%!ê˛
˛õy¢y˛õy!¢ !öflõ® xÌÓy §%flõ® !Ó®%Ó˚ ò)Ó˚c fliyî% ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ xˆÏô≈ܲ–
(4) §yüyöƒ ˛õyÌ≈ˆÏܲƒÓ˚ ܲ¡õyB˛Î%_´ ò%!ê˛ ï˛Ó˚D ~ܲ•z !òˆÏܲ ºüî ܲÓ˚ˆÏ° ï˛Ó˚D ò° ~ÓÇ fl∫Ó˚ܲ¡õ §,!‹T ܲˆÏÓ˚–
(5) ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ fl∫Ó˚ܲˆÏ¡õÓ˚ §Çáƒy ò%!ê˛ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyˆÏB˛Ó˚ ˛õyÌ≈ˆÏܲƒÓ˚ §üyö–
(6) ~ܲ!ê˛ ï˛Ó˚Dò° ˆÎ ˆÓˆÏà ã˛ˆÏ°ñ ï˛yˆÏܲ Ó°y •Î˚ ò°àï˛ ˆÓà– ò%!ê˛ àë˛öܲyÓ˚# ï˛Ó˚ˆÏDÓ˚ ˆÓà §üyö •ˆÏ°
ò°àï˛ ˆÓàñ ï˛Ó˚D ˆÓˆÏàÓ˚ §üyö •Î˚– xöƒ ˆ«˛ˆÏeñ ò°àï˛ˆÏÓà ï˛Ó˚DˆÏÓˆÏàÓ˚ §üyö •Î˚ öy–
200
(7) xÇ¢@ˇÃ•îܲyÓ˚# ï˛Ó˚Dà%!°Ó˚ üˆÏôƒ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ˛õyÌ≈ܲƒ Îï˛ Ü˛ü •ˆÏÓñ ï˛Ó˚D òˆÏ°Ó˚ ˜òâ≈ƒ ï˛ï˛ Ó,!k˛
˛õyˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -4 : 560Hz ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ §%Ó˚¢°yܲy xöƒ ~ܲ!ê˛ §%ˆÏÓ˚Ó˚ §ˆÏD ~ܲˆÏe ôù!öï˛ •ˆÏ° ≤Ã!ï˛
ˆ§ˆÏܲˆÏ[˛ 8!ê˛ fl∫Ó˚ܲ¡õ ˆ¢yöy ÎyÎ˚– xy˛õ!ö !ܲ §%Ó˚!ê˛Ó˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö⁄
7.6 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ~ܲ!ê˛ üyôƒˆÏü 1 0x = ~ÓÇ x2 =lm !Ó®%ˆÏï˛ ï˛Ó˚ˆÏDÓ˚ §Ó˚îˆÏܲ !ö¡¨!°!áï˛ û˛yˆÏÓ ˆ°áy ÎyÎ˚ñ
10.2sin3 yt =π
y2 = 0.2 sin
() 38 t+π π
(a) ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ ܲï˛⁄ S•y͈ϧ≈ ≤Ãܲy¢ ܲÓ˚&öV–
(b) ï˛Ó˚D˜Ïòâ≈ƒ ܲï˛⁄
(c) ï˛Ó˚D ˆÓà ܲï˛⁄
2. e´üÓô≈üyö ܲ¡õyB˛ xö%ÎyÎ˚# 50!ê˛ §%Ó˚¢°yܲy §yçyˆÏöy •°– ~ˆÏòÓ˚ ˛õÓ˚˛õÓ˚ ò%!ê˛ˆÏܲ ~ܲ§ˆÏD ÓyçyˆÏ°
ˆ§ˆÏܲˆÏu˛ 5!ê˛ fl∫Ó˚ܲˆÏ¡õÓ˚ §,!‹T •Î˚– §ÓˆÏ¢ˆÏ£ÏÓ˚ §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ Î!ò ≤ÃÌü §%Ó˚¢°yܲyÓ˚ ܲ¡õyˆÏB˛Ó˚ ~ܲ
§Æܲ ˆÓ!¢ •Î˚ñ ï˛y•ˆÏ° ≤ÃÌü!ê˛Ó˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚&ö– SˆÜ˛yöÄ §%Ó˚ˆÏܲ !mï˛#Î˚ ~ܲ!ê˛ §%ˆÏÓ˚Ó˚ §Æܲ ˆÓ!¢
Ó°y •Î˚ ï˛áö•zñ Îáö ≤ÃÌü §%Ó˚!ê˛Ó˚ ܲ¡õyB˛ !mï˛#Î˚ §%Ó˚!ê˛Ó˚ !m=î •Î˚V–
3. 25 cm °¡∫y x!:ˆÏçö û˛Ó˚y ~ܲ!ê˛ Ók˛ ö° ~ܲ!ê˛ §%Ó˚¢°yܲyÓ˚ §ˆÏD xö%öyò §,!‹T ܲˆÏÓ˚– •y•zˆÏí»˛yˆÏçö
û˛Ó˚y Ü˛ï˛ ˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ Ók˛ ö° ˙ ~ܲ•z §%Ó˚¢°yܲyÓ˚ §ˆÏD xö%öyò §,!‹T ܲÓ˚ˆÏÓ⁄ Sx!:ˆÏçˆÏöÓ˚ Óyï˛yˆÏ§Ó˚
ˆÓà =320m/s ~ÓÇ •y•zˆÏí»˛yˆÏçˆÏö Óyï˛yˆÏ§Ó˚ ˆÓà = 1280m/s)4. ~ܲ!ê˛ ˆÜ˛°yˆÏ§Ó˚ üˆÏôƒÓ˚ xyî!Óܲ ò)Ó˚c d– ~•z ˆÜ˛°yˆÏ§Ó˚ üˆÏôƒ §,‹T !ï˛Î≈ܲ ï˛Ó˚ˆÏDÓ˚ ò¢y ˆÓà (v)
= 'sin 2
,
2
kdc
kd
⎛ ⎞⎝ ⎠
⎛ ⎞⎝ ⎠
~áyˆÏö c' ~ܲ!ê˛ !fliÓ˚ §Çáƒy
ˆòáyö ˆÎñ ï˛Ó˚ˆÏDÓ˚ ò°àï˛ ˆÓà •ˆÏÓ 'cos2kd
c ⎛ ⎞⎝ ⎠
7.7 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 1 : ˆÜ˛yöÄ ò,ì˛¸ ≤Ã!ï˛ú˛°ˆÏܲ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ú˛ˆÏ° π ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö •Î˚– ܲyˆÏç•z ≤Ã!ï˛ú˛!°ï˛
ï˛Ó˚D!ê˛Ó˚ §ü#ܲÓ˚î •ˆÏÓ :
201
() 2sin yatkx ω =−+
xï˛~Óñ °!∏˛ §Ó˚î y(x,t) •ˆÏÓñ
( ) ( ) ( ), sin sinY x t a t kx a t kxω ω= − − +
= 2 sin cosa kx tω−
=
cos At ω
ˆÎáyˆÏö
2sin, Aakx =−
ò,ì˛¸ ≤Ã!ï˛ú˛°ˆÏܲ ~ܲ!ê˛ !öflõ® !Ó®% §,!‹T •ˆÏÓñ ˆÜ˛ööy ˆÎáyˆÏö §Ó˚î ¢)öƒ– ôöydܲ x !òܲ ˆÌˆÏܲ xy˛õ!ï˛ï˛
~ÓÇ îydܲ x !òܲ ˆÌˆÏܲ ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚D ò%!ê˛ ~ܲ!ê˛ fliyî% ï˛Ó˚D §,!‹T ܲÓ˚ˆÏÓ– xy˛õ!ï˛ï˛ ~ÓÇ ≤Ã!ï˛ú˛!°ï˛
ò%!ê˛ ï˛Ó˚D•z §üyö ˛õ!Ó˚üyî ¢!_´ !Ó˛õÓ˚#ï˛ !òܲ ˆÌˆÏܲ Ó•ö ܲÓ˚ˆÏÓ– ú˛°ï˛ ˆÜ˛yöÄ ¢!_´ ≤ÃÓy!•ï˛ •ˆÏÓ öy–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# -2 : (a) ü)°§)ˆÏÓ˚Ó˚ ï˛Ó˚D ˜òâ≈ƒ
222 Lxlmm λ===
ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà
220 vHz =
× 2m = 440m/s
xyüÓ˚y çy!öñ
Tv
m=
xÌÓyñ 2T v m=
=
()2440/0.015/ mskgm ×
= 2.9 × 103 N
(b) ü)°§%ˆÏÓ˚Ó˚ ï˛Ó˚D ˜òâ≈ƒ
4 4 4L lm mλ = = × =
ܲ¡õyB˛
350/87.5884
vmsvHzHz
m λ ===≈
ö°!ê˛ˆÏï˛ ˆçyˆÏÓ˚ úÑ˛% !òˆÏ° §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y 3 =î Ó,!k˛ ˆ˛õˆÏÎ˚ ˛õˆÏÓ˚Ó˚ §üˆÏüˆÏ° ˆ˛õÑÔåȈÏÓ ~ÓÇ ï˛yÓ˚ ܲ¡õyB˛
• ÏÓ 3 88 264v hz= × =
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ3 : (ܲV xyüÓ˚y çy!ö ˆÎñ
g
dvvv
dλλ =−
~áyˆÏöñ 2dv
cdλ = ܲyˆÏç•z í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚ˆÏî v ~ÓÇ
dvdλ
~Ó˚ üyö Ó§yˆÏ°ñ
122 g vcccc λλ =+−=
SáV ~ˆÏ«˛ˆÏe
2,1.1. g
cdcccdncdndnvv
ndnndnndnd n⎛⎞⎛⎞⎛⎞ =−=+=+=+ ⎝⎠⎝⎠⎝⎠
λλ λλ λλλλ
xö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈxö%¢#°ö# ÈÙÈ4 : Î!ò fl∫Ó˚!ê˛Ó˚ ܲ¡õyB˛ v ôÓ˚y •Î˚ñ ï˛y•ˆÏ° 6 = [560-v] xÌÓy 6 = [v-560]
∴
554 vHz =
xÌÓy 566Hz
~áyˆÏö fl∫Ó˚!ê˛Ó˚ ܲ¡õyB˛ !fliÓ˚û˛yˆÏÓ !öî≈Î˚ ܲÓ˚y ÎyˆÏÓ öy– í˛z˛õˆÏÓ˚Ó˚ ˆÎ ˆÜ˛yöÄ!ê˛•z ~Ó˚ ܲ¡õyB˛ •ˆÏï˛ ˛õyˆÏÓ˚–
202
7.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. (a) ܲ¡õyB˛
3 1.5231.52
⎡⎤ ===∴== ⎢⎥ ⎣⎦∵ vHzvvHz ωππ
(b) ã˛°ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚D ˆû˛QÓ˚ πλ= 2
k
∴ 28
π πλ = Óyñ 16mλ =
(c)
31624/sec.2
vvm λ ==×=
2. ôÓ˚y Îyܲ ≤ÃÌü fl∫Ó˚!ê˛Ó˚ ܲ¡õyB˛ n!mï˛#Î˚ §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ •ˆÏÓ = n + 5 = n + (2 –1)5ï,˛ï˛#Î˚ §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ •ˆÏÓ = n + 10 = n + (3-1)550 ï˛ü §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ •ˆÏÓ = n + (50 -1)5
= n + 245ˆÎˆÏ•ï%˛ 50 ï˛ü §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ 2n∴ n + 245 = 2n xï˛~Ó n = 245Hz
3. ≤ÃÌü ö°!ê˛Ó˚ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ υ= 0
114
vl , v0 = x!:ˆÏçˆÏö ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà l = ≤ÃÌü öˆÏ°Ó˚ ˜òâ≈ƒ–
!mï˛#Î˚ ö°!ê˛Ó˚ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛
υ =21 4h
vl
, vh = •y•zˆÏí»˛yˆÏçˆÏö ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓàñ l
2 =!mï˛#Î˚ öˆÏ°Ó˚ ˜òâ≈ƒ
ˆÎˆÏ•ï%˛ ò%•z!ê˛ ö°•z ~ܲ•z §%Ó˚¢°yܲyÓ˚ §ˆÏD xö%öyò §,!‹T ܲˆÏÓ˚ñ ܲyˆÏç•z 12 vv = xÌÓy
0
12 44h vv
ll=
∴ 2 10
hvl l
v= ×
0 ,h vv
~ÓÇ l1 ~Ó˚ üyö Ó!§ˆÏÎ˚
21280/25100
320/ms
lcmcmms
=×=
4. ò°ˆÏÓà g
dv
dkω= ~ÓÇñ kvω =
xÌÓy
()()'sin2
2
kd kc
kd= ω
= ( )2 sin 2c kdd
′
xï˛~Óñ2 ' cos .
2 2g
dw c kd dv
dk d⎛ ⎞= = ⎝ ⎠
=
'cos2kd
c⎛⎞⎝⎠
203
BLOCK—2
ˆòy°à!ï˛ñ ï˛Ó˚D Ä fl∫î!Óòƒyˆòy°à!ï˛ñ ï˛Ó˚D Ä fl∫î!Óòƒyˆòy°à!ï˛ñ ï˛Ó˚D Ä fl∫î!Óòƒyˆòy°à!ï˛ñ ï˛Ó˚D Ä fl∫î!Óòƒyˆòy°à!ï˛ñ ï˛Ó˚D Ä fl∫î!Óòƒy
205
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 8 : !Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚D §MÈ˛yÓ˚ !Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚D §MÈ˛yÓ˚ !Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚D §MÈ˛yÓ˚ !Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚D §MÈ˛yÓ˚ !Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚D §MÈ˛yÓ˚ (Propagation of Waves in
different Media)
àë˛ö
8.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
8.2 ~ܲüy!eܲ ã˛°ï˛Ó˚D
ê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚D
≤ÃÓy•#ˆÏï˛ ï˛Ó˚D
~ܲ!ê˛ §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D
8.3 ï˛Ó˚D ≤ÃÓy• Ä ≤Ã!ï˛ˆÏÓ˚yô
ï˛yÓ˚ myÓ˚y §,‹T ≤Ã!ï˛ˆÏÓ˚yô ≠ xö%˜Ïòâ≈ƒ ï˛Ó˚D
àƒy§ myÓ˚y §,‹T ≤Ã!ï˛ˆÏÓ˚yô ≠ ¢∑ ï˛Ó˚D
8.4 !müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D
8.5 ≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ (transmitted) ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛
xö%≤ÃhflÏ ~ÓÇ xö%˜Ïòâ≈ƒ ï˛Ó˚D
8.6 ˛≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛
8.7 §yÓ˚yÇ¢
8.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
8.9 í˛z_Ó˚üy°y
8.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
•z!ï˛üˆÏôƒ xy˛õ!ö ï˛Ó˚ˆÏDÓ˚ §,!‹T ~ÓÇ ï˛Ó˚Dà!ï˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ˆÏåÈö–
~•z ≤çˆÏD xy˛õ!ö °«˛ƒ ܲˆÏÓ˚ ÌyܲˆÏÓö ˆÎñ ï˛Ó˚D §MÈ˛Ó˚ˆÏîÓ˚ üyôƒü ~ÓÇ ï˛Ó˚Dy!Î˚ï˛ Ó˚y!¢ öyöy ôÓ˚ˆÏöÓ˚
•ˆÏï˛ ˛õyˆÏÓ˚– ÓyÎ˚% Óy ˆÎ ˆÜ˛yöÄ !fli!ï˛fliy˛õܲ üyôƒˆÏü ˆÎ xö%˜Ïòâ≈ƒ ã˛y˛õï˛Ó˚D ≤ÃÓy!•ï˛ •Î˚ ï˛y xyüyˆÏòÓ˚ ¢ˆÏ∑Ó˚
xö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚– ~çöƒ ~•z ï˛Ó˚D=!°ˆÏܲ xyüÓ˚y ¢∑ï˛Ó˚D Ó!°– ê˛yö ܲˆÏÓ˚ Ó˚yáy ï˛yˆÏÓ˚Ó˚ !Ó!û˛ß¨ !Ó®%ˆÏï˛
xö%≤Ãfli §Ó˚î âê˛ˆÏï˛ ˛õyˆÏÓ˚– ~•z §Ó˚î ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ï˛Ó˚ˆÏDÓ˚ Ó˚*ˆÏ˛õ §MÈ˛y!Ó˚ï˛ •Î˚– xyÓyÓ˚ ܲ!ë˛ö ôyï˛Ó
òˆÏ[˛Ó˚ üˆÏôƒÄ üyôƒˆÏüÓ˚ !fli!ï˛fliy˛õܲï˛yÓ˚ ú˛ˆÏ° ˆ§!ê˛Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ §ˆÏB˛yã˛ö Ä ≤çyÓ˚ˆÏîÓ˚ ï˛Ó˚D ã˛°ˆÏï˛ ˛õyˆÏÓ˚–
~•z !ӈϢ£Ï ï˛Ó˚D=!°Ó˚ §¡∫ˆÏ¶˛ !ܲå%Èê˛y xyˆÏ°yã˛öy ~•z ~ܲˆÏܲ fliyö ˛õyˆÏÓ– xÓ¢ƒ ~=!° åÈyí˛¸yÄ xyÓ˚Ä xˆÏöܲ
ôÓ˚ˆÏöÓ˚ ï˛Ó˚ˆÏDÓ˚ §ˆÏD xyüyˆÏòÓ˚ ˛õ!Ó˚ã˛Î˚ xyˆÏåÈñ ˆÎüö ï˛!í˛¸Í ã%˛¡∫ܲ#Î˚ ï˛Ó˚Dñ û)˛˛õ,¤˛ Óy ˛õ,!ÌÓ#Ó˚ xû˛ƒhsˇˆÏÓ˚ ã˛°üyö
û)˛Ü˛¡õÈÙÈï˛Ó˚D Ä ç°ï˛ˆÏ°Ó˚ ï˛Ó˚D– ~Ó˚ ˆÜ˛yö!ê˛ §¡∫ˆÏ¶˛ xy˛õ!ö ˛õˆÏÓ˚ xyÓ˚Ä çyöyÓ˚ §%ˆÏÎyà ˛õyˆÏÓöñ xyÓyÓ˚
ˆÜ˛yö!ê˛ ç!ê˛°ï˛yÓ˚ çöƒ xyüÓ˚y ~•z ˛õyë˛e´ˆÏüÓ˚ xhsˇû%≈˛_´ ܲÓ˚Ó öy–
206
ï˛Ó˚D §MÈ˛Ó˚ˆÏîÓ˚ §ˆÏD üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô ~ÓÇ ï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î x!ï˛ â!ö¤˛û˛yˆÏÓ ç!í˛¸ï˛–
~=!° ÷ô% ï˛y!_¥Ü˛ ܲyÓ˚ˆÏî•z =Ó˚&c˛õ)î≈ öÎ˚ñ ï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚ˆÏîÓ˚ xçflÀ í˛zòy•Ó˚î xyüÓ˚y ˜òö!®ö
ç#ÓˆÏö ˆòáˆÏï˛ ˛õy•z– ~•z ~ܲˆÏܲ xy˛õ!ö ï˛Ó˚ˆÏDÓ˚ ~§Ó ôü≈=!°Ó˚ §¡∫ˆÏ¶˛ öï%˛öû˛yˆÏÓ !ܲå%Èê˛y çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
í˛zˆÏj¢ƒ ≠ í˛zˆÏj¢ƒ ≠ í˛zˆÏj¢ƒ ≠ í˛zˆÏj¢ƒ ≠ í˛zˆÏj¢ƒ ≠ ~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° x˛õ!ö
~ܲüy!eܲñ !müy!eܲ Ä !eüy!eܲ ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î !°áˆÏï˛ ˛õyÓ˚ˆÏÓö–
!Ó!û˛ß¨ üyôƒˆÏü ï˛Ó˚ˆÏDÓ˚ ˆÓà !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô (characteristic impedance) ~ÓÇ ¢ˆÏ∑ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ (transmitted) !ÓhflÏyÓ˚ =îyB˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ §«˛ü •ˆÏÓö–
8.2 ~ܲüy!eܲ ã˛°ï˛Ó˚D~ܲüy!eܲ ã˛°ï˛Ó˚D~ܲüy!eܲ ã˛°ï˛Ó˚D~ܲüy!eܲ ã˛°ï˛Ó˚D~ܲüy!eܲ ã˛°ï˛Ó˚D
§D#ï˛ xyüÓ˚y §Ü˛ˆÏ°•z û˛y°Óy!§– !ܲv xy˛õ!ö !ܲ ˆû˛ˆÏÓ ˆòˆÏáˆÏåÈöñ !ë˛Ü˛ ˆÜ˛yö ≤Ã!e´Î˚yÎ˚ §D#ˆÏï˛Ó˚
¢∑ï˛Ó˚D xy˛õöyÓ˚ ܲyˆÏö ~ˆÏ§ ˆ˛õÔÑåÈÎ˚⁄ ï˛Ó˚ˆÏDÓ˚ §MÈ˛Ó˚ˆÏîÓ˚ çöƒ !ܲ ˆÜ˛yö üyôƒˆÏüÓ˚ ≤ÈÏÎ˚yçö xyˆÏåÈ⁄ ~ÓÇ
Îáö ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •Î˚ ï˛áö ï˛Ó˚ˆÏDÓ˚ ˆÓà ˆÜ˛yö‰ ˆÜ˛yö‰ !ӣψÏÎ˚Ó˚ í˛z˛õÓ˚ !öû˛≈Ó˚ ܲˆÏÓ˚⁄ Îáö ò)ˆÏÓ˚ ârê˛y ÓyˆÏç
ï˛áö ârê˛yôù!öÓ˚ üˆÏôƒ ˆÎ !Ó!û˛ß¨ ܲ¡õyˆÏDÓ˚ ¢∑ï˛Ó˚D ÌyˆÏܲ ˆ§=!° ~ܲ§ˆÏD xyüyˆÏòÓ˚ ܲyˆÏö ˆ˛õÔÑåÈÎ˚– ~
ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ ÓyÎ˚%ˆÏï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓà ¢ˆÏ∑Ó˚ ܲ¡õyB˛ Óy ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– ܲyˆÏç•z
~!ê˛ flõ‹T•z ˆÓyé˛y ÎyÎ˚ ˆÎñ ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓà üyôƒˆÏüÓ˚ ˆû˛Ôï˛ ôˆÏü≈≈Ó˚ í˛z˛õˆÏÓ˚•z !öû≈˛Ó˚¢#°– ÷ô% ÓyÎ˚%ˆÏï˛
¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe•z öÎ˚ñ xöƒ üyôƒˆÏü Ä xöƒ ôÓ˚ˆÏöÓ˚ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÄ ~•z ~ܲ•z ܲÌy áyˆÏê˛– ~•z ï˛ˆÏ̃Ó˚
§ï˛ƒï˛y Îyã˛y•z ܲÓ˚yÓ˚ çöƒ xyüÓ˚y ê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚D §,!‹TÓ˚ ≤Ã!e´Î˚yÓ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
8.2.1 ê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚Dê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚Dê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚Dê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚Dê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚D
˛≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒ m û˛Ó˚ Î%_´ ~ܲ!ê˛
§%£Ïü ï˛yÓ˚ˆÏܲ T Ó° myÓ˚y xö%û)˛!üܲ !òˆÏܲ
ˆê˛ˆÏö Ó˚yáy •°– ï˛yÓ˚!ê˛Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ x-
x«˛ ˆöÄÎ˚y •°– ôÓ˚y Îyܲñ ï˛yÓ˚!ê˛ˆÏܲ
ˆÜ˛yö çyÎ˚àyÎ˚ y x«˛ ÓÓ˚yÓÓ˚ ê˛yöy •°–
~ÓyÓ˚ ï˛yÓ˚!ê˛ˆÏܲ ˆåȈÏí˛¸ !òˆÏ° ï˛yˆÏÓ˚Ó˚ üˆÏôƒ
xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ §,!‹T •ˆÏÓ– ~•z ï˛Ó˚ˆÏDÓ˚
ˆÓà çyöˆÏï˛ ˆàˆÏ° üyôƒˆÏüÓ˚ çí˛¸c ~ÓÇ
!fli!ï˛fliy˛õܲï˛y §¡∫ˆÏ¶˛ ôyÓ˚îy Ìyܲy òÓ˚ܲyÓ˚–
!ã˛e!ã˛e!ã˛e!ã˛e!ã˛e 8.1 ê˛yöy ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ «%˛o xLjϢÓ˚ í˛z˛õÓ˚ ≤ÃÎ%_´ Ó°–
207
ê˛yöy ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ï˛yˆÏÓ˚Ó˚ ê˛yˆÏöÓ˚ ˆÌˆÏܲ !fli!ï˛fliy˛õܲï˛yÓ˚ ~ÓÇ m ˆÌˆÏܲ çí˛¸ˆÏcÓ˚ üy˛õ ˛õyÄÎ˚y ÎyÎ˚– xyüÓ˚y
~áyˆÏö ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ ˆåÈyR xÇ¢ !ö°yü– ôˆÏÓ˚ !öö ˆÎñ ï˛yÓ˚!ê˛ x!ï˛ §yüyöƒ !ÓÜ,˛!ï˛ âê˛° ÎyˆÏï˛ ï˛yˆÏÓ˚Ó˚ IJõÓ˚
ê˛yˆÏöÓ˚ üyˆÏöÓ˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö öy âˆÏê˛– 8.1 !ã˛ˆÏe Ó!ô≈ï˛ xyܲyˆÏÓ˚ !ÓÜ,˛ï˛ ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ ˆåÈyê˛ xÇ¢ˆÏܲ ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ–
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ~•z ˆåÈyê˛ xÇ¢ ÓÓ˚yÓÓ˚ ӈϰÓ˚ !òܲ ˛õ!Ó˚Óï≈˛ö •ˆÏFåÈ– ~Ó˚ ܲyÓ˚î •ˆÏFåÈ
ˆÎñ ï˛yÓ˚!ê˛ ~áö ˆ§yçy öÎ˚– ~Ó˚ ú˛ˆÏ° ˆåÈyê˛ xÇ¢!ê˛Ó˚ ò%•z ≤ÃyˆÏhsˇ ~ܲ•z üyˆÏöÓ˚ ê˛yö Ó° T, T ≤ÃÎ%_´ •ˆÏ°Ä
Ó° ò%!ê˛ ˛õÓ˚flõÓ˚ˆÏܲ !ö!‹;˛Î˚ ܲÓ˚ˆÏåÈ öy– x ~ÓÇ y ~Ó˚ !òˆÏܲ ≤ÃÎ%_´ °!∏˛Ó° F !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ T Ó°ˆÏܲ
ò%!ê˛ í˛z˛õyLjϢ û˛yà ܲÓ˚ˆÏï˛ •ˆÏÓ– «%˛o á[˛!ê˛Ó˚ í˛yö ~ÓÇ Óyü !òˆÏܲ ê˛yˆÏöÓ˚ x Äy í˛z˛õyLjϢÓ˚ ˛õyÌ≈ܲƒ=!° •°≠
Fx = T cosθ 2 – T cos
θ
1
~ÓÇ Fx = T sin
θ
2 – T sin
θ
1
θ
1 ~ÓÇ
θ
2 «%˛o xÇ¢!ê˛Ó˚ ò%•z ≤ÃyˆÏhsˇ x!B˛ï˛ flõ¢≈ܲ ~ÓÇ x xˆÏ«˛Ó˚ üôƒÓï˛#≈ ˆÜ˛yî–
«%˛o ˆòy°ˆÏöÓ˚ çöƒ
θ
1 Ä
θ
2 x!ï˛ «%˛o ˆÜ˛yî– ~•z xÓfliyÎ˚ cos
θ
1 cos
θ
2 1
xÌ≈yÍñ x x«˛ ÓÓ˚yÓÓ˚ ˆÜ˛yö °!∏˛ Ó° ܲyç ܲÓ˚ˆÏÓ öyñ Fx = 0 ~ÓÇ ï˛yÓ˚!ê˛ ≤ÃyÎ˚ xö%û)˛!üܲ xÓfliyÎ˚ ÌyˆÏܲ–
~•z xÌ≈ ~•z ˆÎñ ˆÜ˛yî=!°Ó˚ §y•zˆÏöÓ˚ üyö ~ÓÇ ê˛ƒyöˆÏçˆÏrê˛Ó˚ üyö ≤ÃyÎ˚ ~ܲñ xÌ≈yÍ sinθ 1≈ tanθ 1– sinθ 2
≈ tan
θ
2
!ܲv ˆÜ˛yî ò%!ê˛Ó˚ ê˛ƒyöˆÏçrê˛ •ˆÏFåÈ «%˛o xÇ¢!ê˛Ó˚ ò%•z ≤ÃyˆÏhsˇÓ xÓܲ°
dydx
~Ó˚ §üyö– xï˛~Ó «%˛o xÇ¢!ê˛Ó˚
í˛z˛õÓ˚ ≤ÃÎ%_´ ӈϰÓ˚ y í˛z˛õyÇ¢ Fy= T (tan
θ
2 – tan
θ
1)
= T
()() ,,|| xxx
dyxtdyxtdxdx +Δ
⎛⎞− ⎜⎟ ⎜⎟ ⎝⎠
......8.1
Ó¶˛ö#Ó˚ üˆÏôƒÓ˚ §Çáƒy=!° ~ܲ ≤Ãyhsˇ ˆÌˆÏܲ x˛õÓ˚ ≤Ãyhsˇ ˛õÎ≈hsˇ xÓܲˆÏ°Ó˚ ˛õ!Ó˚Óï≈˛ö !öˆÏò≈¢ ܲÏÓ˚ˆÏåÈ– ~•z
˛õ!Ó˚Óï≈˛öˆÏܲ
x Δ
!òˆÏÎ˚ û˛yà ܲÓ˚ˆÏ°ñ
0 x Δ→
§#üyÓ˚ üˆÏôƒ ~!ê˛ ≤ÃÌü xÓܲˆÏ°Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ •yÓ˚ !öˆÏò≈¢ ܲÓ˚ˆÏÓ–
!ܲv xyüÓ˚y çy!ö ˆÎñ ï˛yˆÏÓ˚Ó˚ §Ó˚î xÓfliyö ~ÓÇ §üÎ˚ í˛zû˛ˆÏÎ˚Ó˚ í˛z˛õˆÏÓ˚•z !öû≈˛Ó˚ ܲˆÏÓ˚– ~•z ò%!ê˛Ó˚ ˆÜ˛yöÄ
~ܲ!ê˛Ó˚ ˛õ!Ó˚Óï≈˛ö âê˛ˆÏ°•z §Ó˚ˆÏîÓ˚ ˛õ!Ó˚Óï≈˛ö •Î˚– ܲyˆÏç•z §ü#ܲÓ˚î 8.1 ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ !ӈϢ£Ï §üˆÏÎ˚Ó˚
çöƒ §ï˛ƒ– xï˛~Óñ xyüyˆÏòÓ˚ §üÎ˚ˆÏܲ !fliÓ˚ ôˆÏÓ˚ !öˆÏÎ˚ ~•z §ü#ܲÓ˚ˆÏî xÓܲ° !öî≈Î˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– xy˛õ!ö
çyˆÏööñ xöƒ ã˛°Ó˚y!¢ˆÏòÓ˚ !fliÓ˚ ˆÓ˚ˆÏá ˆÜ˛Ó° ~ܲ!ê˛ ã˛°Ó˚y!¢Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲÓ˚yˆÏܲ xyÇ!¢Ü˛ xÓܲ°ö
208
ӈϰ– ~ÓÇ xyÇ!¢Ü˛ xÓܲ°ˆÏöÓ˚ ˆ«˛ˆÏe d !ã˛ˆÏ•´Ó˚ ÓòˆÏ°
∂
!ã˛•´ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ï˛y•ˆÏ° 8.1 §ü#ܲÓ˚îˆÏܲ
!ö¡¨Ó˚*ˆÏ˛õ ˆ°áy ÎyÎ˚≠
Fy = T
()2
2
, ⎛⎞ ∂ ∂∂ ∆=∆ ⎜⎟ ∂∂∂ ⎝⎠t
yxty xTxxxx
~áyˆÏö xyüÓ˚y ( ),
+∆
⎛ ⎞∂⎜ ⎟∂⎝ ⎠x x
y x tx ˆÜ˛ x !Ó®%Ó˚ §yˆÏ˛õˆÏ«˛ §¡±§yÓ˚î ܲˆÏÓ˚!åÈ–
( ) ( ) ( )2
2
, , , )
+∆
⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂ ∂= + ∆⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠x x x x
y x t y x t y x tx
x x x
~•z §ü#ܲÓ˚î!ê˛ xΔ ~Ó˚ í˛z˛õÓ˚ ≤Ãò_ Ó° ˛≤Ãܲy¢ ܲÏÓ˚ˆÏåÈÈ– !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚ §)e xö%ÎyÎ˚# xyüÓ˚y û˛Ó˚ ~ÓÇ
cÓ˚ˆÏîÓ˚ =îú˛°ˆÏܲ ~•z ӈϰÓ˚ §üyö !•§yˆÏÓ !°áˆÏï˛ ˛õy!Ó˚–
x Δ
xLjϢÓ˚ û˛Ó˚ m
x Δ
– xï˛~Óñ
m
x Δ()() 22
22,, ∂∂
=∆∂∂yxtyxt
Txtx
xÌÓyñ()() 22
22,, ∂∂
=∂∂yxtyxt T
m tx.......8.2
~•z §ü#ܲÓ˚î!ê˛ ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ «%˛o xLjϢÓ˚ IJõÓ˚ !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚ §)e ≤ÈÏÎ˚yà ܲˆÏÓ˚ ˛õyÄÎ˚y ˆàˆÏåÈ– ï˛ˆÏÓ
§ü#ܲÓ˚ˆÏî
x Δ
xö%˛õ!fliï˛ ÌyܲyÎ˚ §ü#ܲÓ˚î!ê˛ §¡õ)î≈ ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe•z ≤ÈÏÎyçƒ •ˆÏÓ– xyüyˆÏòÓ˚ í˛zˆÏj¢ƒ ï˛Ó˚ˆÏDÓ˚
ˆÓà !öî≈Î˚ ܲÓ˚y– ˆ§çöƒ !öˆÏã˛ Ó!î≈ï˛ ~ܲ!ê˛ §Ó˚° ˆòy° ï˛Ó˚D ˆöÄÎ˚y Îyܲ ≠
y(x, t) = asin (
ω
0t – kx)
ˆÎ!ê˛ ï˛yˆÏÓ˚Ó˚ IJõÓ˚ §,!‹T •ˆÏÎ˚ˆÏåÈ– ~•z §ü#ܲÓ˚î Î!ò !öí˛zê˛ˆÏöÓ˚ §)e xÌ≈yÍ 8.2 §ü#ܲÓ˚î ˆüˆÏö ã˛ˆÏ°ñ ï˛y•ˆÏ°
xyüÓ˚y !ö!ÿ˛ï˛Ó˚*ˆÏ˛õ Ó°ˆÏï˛ ˛õy!Ó˚ ˆÎñ ï˛yˆÏÓ˚Ó˚ üˆÏôƒ ~•z ôÓ˚ˆÏöÓ˚ ã˛°üyö ï˛Ó˚D §,!‹T •ˆÏï˛ ˛õyˆÏÓ˚– ~!ê˛ •Î˚
!ܲöy ˆòáyÓ˚ çöƒ xyüÓ˚y ܲîyÓ˚ §Ó˚ˆÏîÓ˚ !mï˛#Î˚ xyÇ!¢Ü˛ xÓܲ° !öî≈Î˚ ܲÓ˚Ó–
2
2yx
∂∂
= – k2asin(
ω
0t – kx) ~ÓÇ
2
2yt
∂∂
= –
ω
02asin(
ω
0t – kx)
8.2 §ü#ܲÓ˚ˆÏî ~•z üyö Ó!§ˆÏÎ˚
– k2a sin (
ω
0t – kx) =
mT
−ω
02 asin(
ω
0t – kx)
209
xÌÓyñ k2 =
mT
ω
02
xÌÓyñ
20Tkmω⎛⎞= ⎜⎟ ⎝⎠
xÌÓy ï˛Ó˚ˆÏDÓ˚ ˆÓà v = 0 Tk m
ω= ...8.3
~•z §¡õÜ≈˛ ˆÌˆÏܲ xyüÓ˚y Ó°ˆÏï˛˛˛õy!Ó˚ñ ê˛yöy ï˛yˆÏÓ˚ í˛zͲõߨ §Ó˚° ˆòy°ï˛Ó˚ˆÏDÓ˚ ï˛Ó˚DˆÏÓà •ˆÏÓ Tm
–
8.3 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ ï˛yˆÏÓ˚ §MÈ˛Ó˚üyö xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆÓàñ ê˛yö ~ÓÇ ï˛yˆÏÓ˚Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚
û˛ˆÏÓ˚Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ñ ï˛Ó˚D ˜òâ≈ƒ xÌÓy ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy–
8.3 §ü#ܲÓ˚î ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y 8.2 ï˛Ó˚D §ü#ܲÓ˚î!ê˛ˆÏܲ !°áˆÏï˛˛ ˛õy!Ó˚
22
2221 yy
xt υ∂∂ =∂∂
....8.4
~•z §ü#ܲÓ˚î!ê˛ ˆÜ˛Ó°üye x !òܲ ÓÓ˚yÓÓ˚ §MÈ˛Ó˚üyö ï˛Ó˚D !öˆÏò≈¢ ܲˆÏÓ˚ xÌ≈yÍñ ~!ê˛ ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚
§ü#ܲÓ˚î– ~•z §ü#ܲÓ˚î!ê˛ «%˛o !ÓhflÏyˆÏÓ˚Ó˚ ï˛Ó˚ˆÏDÓ˚ ˛õˆÏ«˛•z ≤ÈÏÎy烖 !ÓhflÏyÓ˚ ˆÓ!¢ •ˆÏ°
θ
1Ä
θ
2 ˆÜ˛yî=!°
Óí˛¸ •Î˚ ~ÓÇ xyˆÏàÓ˚ àîöyÓ˚ xˆÏöܲ xD#ܲyÓ˚ xyÓ˚ áyˆÏê˛ öy– ï˛Ó˚ˆÏDÓ˚ ˆÓà ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#°
•ˆÏÎ˚ ˛õˆÏí˛¸– ~•z xÓfliyÎ˚ ~áö xy˛õ!ö §•ˆÏç•z ö#ˆÏã˛Ó˚ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1
1 gm û˛Ó˚ §¡õߨ ~ܲ!ê˛ 1m °¡∫y ï˛yÓ˚ˆÏܲ 10N Ó° !òˆÏÎ˚ ˆê˛ˆÏö Ó˚yáy •°– ~•z ï˛yˆÏÓ˚ ˆÎ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚
§,!‹T •ˆÏÓ ï˛yÓ˚ ˆÓà !öî≈Î˚ ܲÓ˚&ö–
ï˛Ó˚ˆÏDÓ˚ ˆÓà üyôƒˆÏüÓ˚ çí˛¸c ~ÓÇ !fli!ï˛fliy˛õܲï˛yÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– !fli!ï˛fliy˛õܲï˛y ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚
(restoring force) §,!‹T ܲˆÏÓ˚ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° üyôƒˆÏüÓ˚ ܲ# ≤Ã!ï˛!e´Î˚y •ˆÏÓ ï˛y !fliÓ˚ ܲˆÏÓ˚ çí˛¸c– ˆÜ˛yö üyôƒˆÏü
xö%≤Ãfli !fli!ï˛fliy˛õܲ ï˛Ó˚D §,!‹T •ˆÏï˛ üyôƒü!ê˛Ó˚ Ü,˛hsˇö ˛õ#í˛¸ö §•ƒ ܲÓ˚yÓ˚ «˛üï˛y Ìyܲy ã˛y•z !ܲv ˆÎˆÏ•ï%˛ ≤ÃÓy•#ˆÏï˛
Sï˛Ó˚° xÌÓy àƒy§V ò,ì˛¸ï˛y ˆö•zñ ˆ§•z ܲyÓ˚ˆÏî xö%≤Ãfli !fli!ï˛fliy˛õܲ ï˛Ó˚D ˆÜ˛Ó° ܲ!ë˛ö ˛õòyˆÏÌ≈•z §MÈ˛y!Ó˚ï˛
•ˆÏï˛ ˛˛õyˆÏÓ˚– !ܲv xö%˜Ïòâ≈ƒ ï˛Ó˚D àƒy§ñï˛Ó˚° ~ÓÇ Ü˛!ë˛ö ˛õòyÌ≈ §ˆÏÓÓ˚ üˆÏôƒ•z xö%û˛Óö ~ÓÇ âö#û˛ÓˆÏöÓ˚
xyܲyˆÏÓ˚ ≤ÃÓy!•ï˛ •Î˚– ~áö xyüÓ˚y ≤ÃÓy•#ˆÏï˛ ï˛Ó˚ˆÏDÓ˚ ≤ÃÓyˆÏ•Ó˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚Ó–
8.2.2 ≤ÃÓy•#ˆÏï˛ ï˛Ó˚D §,!‹T≤ÃÓy•#ˆÏï˛ ï˛Ó˚D §,!‹T≤ÃÓy•#ˆÏï˛ ï˛Ó˚D §,!‹T≤ÃÓy•#ˆÏï˛ ï˛Ó˚D §,!‹T≤ÃÓy•#ˆÏï˛ ï˛Ó˚D §,!‹T
ôˆÏÓ˚ !ööñ A ≤ÃfliˆÏFåÈò Î%_´ ~ܲ!ê˛ °¡∫y öˆÏ° ~ܲ!ê˛ !ö!ò≈‹T û˛ˆÏÓ˚Ó˚ ~ÓÇ
ρ
ôöˆÏcÓ˚ ≤ÃÓy•# ˆöÄÎ˚y •°
ÎyÓ˚ IJõÓ˚ ã˛y˛õ P0– ï˛yˆÏÓ˚Ó˚ üï˛ ~áyˆÏöÄ ≤ÃÓy•#Ó˚ ~ܲ!ê˛ «%˛o ˜òâ≈ƒ ˆöÄÎ˚y •°– ôÓ˚y Îyܲ ~•z «%˛o ˜òˆÏâ≈ƒÓ˚
≤ÃÓy•# PQRSρ xÇ¢!ê˛Ó˚ üˆÏôƒ xyÓk˛ xyˆÏåÈ– ~•z xÇ¢!ê˛ x ~ÓÇ x +
x Δ
ï˛ˆÏ°Ó˚ üˆÏôƒ xÓ!fliï˛– S!ã˛e
8.2 (a) ˆòá%öV–
210
!ã˛e 8.2 - (a) ≤ÃfliˆÏFåÈòÎ%_´ ~ܲ!ê˛ °¡∫y öˆÏ°Ó˚ üˆÏôƒ xÓ!fliï˛ PQRS hflψÏΩ˛Ó˚ ~ܲ!ê˛ ≤ÃÓy•#Ó˚
§yüƒyÓfliy– (b) ã˛yˆÏ˛õÓ˚ ˛õyÌ≈ܲƒÓ˚ çöƒ hflÏΩ˛!ê˛Ó˚ fliyöyhsˇ!Ó˚ï˛ xÓfliy–
~•z ˆåÈyê˛ hflÏΩ˛ PQRSρ !ê˛Ó˚ û˛Ó˚
ρ
A
x Δ
– ~áö ~•z ≤ÃÓy•#ˆÏï˛ Ü˛# ܲˆÏÓ˚ xö%˜ÏòˆÏâ≈ƒÓ˚ ï˛Ó˚D §,!‹T ܲÓ˚y ÎyÎ˚⁄
xçöƒ xy˛õ!ö ~ܲ!ê˛ Ü˛¡õö¢#° §%Ó˚¢°yܲyˆÏܲ ~•z ≤ÃÓy•#Ó˚ ~ܲ!òˆÏܲ ôÓ˚ˆÏï˛ ˛õyˆÏÓ˚öñ xÌÓy ~ܲ!ê˛ !˛õfiê˛ö
!òˆÏÎ˚ ≤ÃyÓ•#ˆÏܲ í˛yö!òˆÏܲ §!Ó˚ˆÏÎ˚ !òˆÏï˛ ˛õyˆÏÓ˚ö– ï˛Ó˚D Îáö ~•z ˆåÈyê˛ hflÏΩ˛!ê˛Ó˚ üˆÏôƒ !òˆÏÎ˚ ÎyˆÏÓ ï˛áö hflÏΩ˛!ê˛Ó˚
ã˛y˛õñ âöc ~ÓÇ xyÎ˚ï˛ö ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÓ– üˆÏö ܲÓ˚y Îyܲñ t §üÎ˚ ˛õˆÏÓ˚ PQ ~ÓÇ SR ï˛° §ˆÏÓ˚ P'Q' ~ÓÇ
S'R' ÈÙÈ~ ˆàÏ° S!ã˛e 8.2 (b)) – Î!ò PQ,
Ψ
(x) ò)Ó˚c §ˆÏÓ˚ ~ÓÇ SR
Ψ
(x +
x Δ
) ò)Ó˚c §ˆÏÓ˚ ï˛y•ˆÏ° hflψÏΩ˛Ó˚
ˆÓˆÏôÓ˚ Ó,!k˛ •ˆÏÓ
Δ
l =
Ψ
(x +
x Δ
) –
Ψ
(x) = –
Ψ
(x) +
xψ∂∂
x Δ
–
Ψ
(x) Sˆê˛°ˆÏÓ˚Ó˚ ˆ◊î#Ó˚ ≤çyÓ˚ˆÏîÓ˚ §y•yÏ΃
=
xψ∂∂
x Δ
!öˆÏÎ˚ ≤ÃÌü ò%•z!ê˛ ˛õò ˆöÄÎ˚y •°V
xï˛~Óñ xyÎ˚ï˛ˆÏöÓ˚ Ó,!k˛
Δ
v ˆÜ˛ ˆ°áy ÎyÎ˚ ≠
211
Δ
v = A
Δ
l = A
Δ
x
xψ∂∂
xyÎ˚ï˛ˆÏöÓ˚ !ÓÜ,˛!ï˛ (volume strain) xÌ≈yÍñ ≤Ã!ï˛ ~ܲܲ xyÎ˚ï˛ˆÏö xyÎ˚ï˛ˆÏöÓ˚ Ó,!k˛
AxxAxx
ψνψν
∂ Δ Δ∂ ∂ ==Δ∂
....8.5
8.5 §ü#ܲÓ˚î ˆÌˆÏܲ ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ hflÏΩ˛!ê˛Ó˚ xyÎ˚ï˛ˆÏöÓ˚ ˛õ!Ó˚Óï≈˛ö âˆÏê˛ˆÏåÈ– ~Ó˚ ܲyÓ˚î •°ñ hflÏΩ˛!ê˛Ó˚ üˆÏôƒ
ã˛y˛õ P0 ÈÙÈ ~Ó˚ §üyö öÎ˚– ôÓ˚y Îyܲ ã˛yˆÏ˛õÓ˚ üyö P0 ˆÌˆÏܲ ˆÓˆÏí˛¸
ρ
(x) •ˆÏÎ˚ˆÏåÈ– hflÏΩ˛!ê˛Ó˚ í˛z˛õÓ˚ x !òˆÏܲ ≤ÃÎ%_´
Ó° •ˆÏÓ A[–
ρ
(x +
x Δ
) +
ρ
(x)]
ˆê˛°ˆÏÓ˚Ó˚ ˆ◊î#Ó˚ §y•y΃ !öˆÏÎ˚
x Δ
~Ó˚ ˆÜ˛Ó°üye ≤ÃÌü âyˆÏï˛Ó˚ ˛õò!ê˛ ˆÓ˚ˆÏá xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
F = – A
() xx
xρ∂
Δ∂
= – A
()() 0PxAx
xxρρ ∂+Δ∂Δ
Δ=−Δ∂∂
ˆÎáyˆÏö P0 §yüƒyÓfliyÎ˚ ã˛y˛õ ~ÓÇ
Δρ
ï˛Ó˚Dç!öï˛ x!ï˛!Ó˚_´ ã˛y˛õ–
[ §yüƒyÓfliyÓ˚ ã˛y˛õ ô &Óܲ ӈϰ 0Px
∂∂
= 0 ]
xï˛~Óñ !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚ §)e xö%ÎyÎ˚# hflÏΩ˛!ê˛Ó˚ à!ï˛ §ü#ܲÓ˚î •ˆÏÓ É
ρx Δ
A
2
2tψ∂
∂
= – A
x Δ()x
ρ ∂Δ∂
....8.6
xy˛õöyÓ˚y !öÿ˛Î˚•z xyÎ˚ï˛ö =îyB˛ (bulk modulus of elasticity) E ~ÓÇ
Δρ
ÈÙÈÓ˚ üˆÏôƒ ö#ˆÏã˛Ó˚ §¡õÜ≈˛
§¡∫ˆÏ¶˛ xÓàï˛ xyˆÏåÈö≠
E = =
ρν
ν
Δ −Δ
ã˛y˛õ Óyí˛¸ˆÏ° xyåÈï˛ö ܲüˆÏÓ ~•z âê˛öy!ê˛ˆÏܲ•z îydܲ !ã˛•´ !òˆÏÎ˚ ˆÓyé˛yö •ˆÏÎ˚ˆÏåÈ–
ܲyˆÏç•z xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ ≠
Δ
ρ
= – E
ννΔ⎛⎞
⎜⎟ ⎝⎠
8.5 §ü#ܲÓ˚î ˆÌˆÏܲ
ννΔ
~Ó˚ üyö Ó!§ˆÏÎ˚ ˛õy•zñ
xyÎ˚ï˛ö õ#í˛¸ö
xyÎ˚ï˛ö !ÓÜ,˛!ï˛
212
Δρ
= – E
xψ∂∂
xyÓyÓ˚
Δρ
üyö 8.6 §ü#ܲÓ˚ˆÏî Ó!§ˆÏÎ˚ ˛õy•zñ
22
22 EExx tx
ψψψ ρ∂∂∂∂ ⎛⎞ == ⎜⎟ ∂∂ ∂∂ ⎝⎠
xÌÓyñ
2 2
2 2E
t xψ ψ
ρ∂ ∂=∂ ∂
...8.7
xyüÓ˚y Î!ò
E νρ
=
...8.7(a)
ˆÜ˛ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆÓà Ó!° ï˛y•ˆÏ°ñ 8.7 §ü#ܲÓ˚îˆÏܲ ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î Ó°y ÎyÎ˚–
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ï˛Ó˚ˆÏDÓ˚ ˆÓà ˆÜ˛ÏÓ°üye E ~ÓÇ ρ myÓ˚y ≤Ãû˛y!Óï˛ •Î˚– ˆÎ üyôƒˆÏü
ï˛Ó˚D ≤ÃÓy!•ï˛ •ˆÏFåÈñ ~•z E ~ÓÇ
ρ
ˆ§•z üyôƒˆÏüÓ˚ ôü≈–
~ÓyÓ˚ xyüÓ˚y àƒyˆÏ§Ó˚ üôƒ !òˆÏÎ˚ ¢∑ï˛Ó˚ˆÏDÓ˚ ≤ÃÓyˆÏ•Ó˚ܲÌy xyˆÏ°yã˛öy ܲ!Ó˚–
(a) àƒyˆÏ§Ó˚ üˆÏôƒ ¢∑ï˛Ó˚D ≤ÃÓy•àƒyˆÏ§Ó˚ üˆÏôƒ ¢∑ï˛Ó˚D ≤ÃÓy•àƒyˆÏ§Ó˚ üˆÏôƒ ¢∑ï˛Ó˚D ≤ÃÓy•àƒyˆÏ§Ó˚ üˆÏôƒ ¢∑ï˛Ó˚D ≤ÃÓy•àƒyˆÏ§Ó˚ üˆÏôƒ ¢∑ï˛Ó˚D ≤ÃÓy•
~ܲ!ê˛ àƒy§#Î˚ üyôƒˆÏüÓ˚ SôÓ˚y Îyܲ Óyï˛y§V üˆÏôƒ Îáö xö%˜Ïòâ≈ƒ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •Î˚ñ ï˛áö üyôƒˆÏüÓ˚ xyÎ˚ï˛ö
=îyB˛ üyôƒˆÏüÓ˚ ï˛y˛õ àï˛#Î˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° •ˆÏÎ˚ ˛õˆÏí˛¸– ~•z ˛õ!Ó˚Óï≈˛ö §ˆÏüy£è Óy Ó˚&k˛ï˛y˛õñ
ˆÎ ˆÜ˛yöÄ!ê˛ •ˆÏï˛ ˛õyˆÏÓ˚– S~ܲ!ê˛ ≤Ã!e´Î˚yˆÏܲ §ˆÏüy£è ï˛áö•z Ó°y •Î˚ñ Îáö ≤Ã!e´Î˚y ã˛°yܲy°#ö í˛z£èï˛yÓ˚
ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö •Î˚ öy– Ó˚&k˛ï˛y˛õ ≤Ã!e´Î˚yÎ˚ ï˛ˆÏsfÓ˚ ˆüyê˛ ¢!_´Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö •Î˚ öy–V ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe
!öí˛zê˛ö ôˆÏÓ˚ !öˆÏÎ˚!åȈϰö ˆÎñ üyôƒˆÏüÓ˚ ˛õ!Ó˚Óï≈˛ö §ˆÏüy£è ≤Ã!e´Î˚yÎ˚ •ˆÏÓ– §ˆÏüy£è ≤Ã!e´Î˚yÎ˚ ÓˆÏÎ˚ˆÏ°Ó˚ §)e
xö%ÎyÎ˚#
ρν
= ô &Óܲ–
ρ
~ÓÇ
ν
ÈÙÈ~Ó˚ §yüyöƒ ˛õ!Ó˚Óï≈˛ö •ˆÏ° xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚–
νρρν Δ+Δ
= 0
xÌÓyñ E =
ρρ νν
Δ −=Δ
xÌ≈yÍñ xyÎ˚ï˛ö =îyB˛ ã˛yˆÏ˛õÓ˚ §üyö–
§ü#ܲÓ˚î 8.7a ˆï˛ E - ~Ó˚ üyö Ó!§ˆÏÎ˚ xy˛õ!ö ˛õyˆÏÓö ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓà ρνρ ...8.8
~!ê˛ˆÏܲ !öí˛zê˛ˆÏöÓ˚ ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ §)e Ó°y •Î˚–
Óyï˛yˆÏ§Ó˚ ˆ«˛ˆÏe ≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ
ρ
= 1·29kgm–3 ~ÓÇ
ρ
= 1·01 × 105 Nm–2
!öí˛zê˛ˆÏöÓ˚ §)ˆÏeÓ˚ §y•yˆÏ΃ !öˆÏÎ˚ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓà •ˆÏÓ
52
31.0110Nm
1.29kgm
−
−× = ν
= 280ms–1
213
!ܲv ˛õÓ˚#«˛y ܲˆÏÓ˚ ˆòáy ˆàˆÏåÈ ˆÎñ ≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓà = 332ms–1
í˛z˛õˆÏÓ˚Ó˚ ò%!ê˛ ú˛° ˆÌˆÏܲ fl∫yû˛yÓï˛•z ≤ß¿ çyˆÏà ˆÎñ !öí˛zê˛ö ܲ# ܲˆÏÓ˚ §!ë˛Ü˛ í˛z_ˆÏÓ˚Ó˚ ܲyåÈyܲy!åÈ ~ˆÏ§Ä §!ë˛Ü˛
üyö ˆÌˆÏܲ ≤ÃyÎ˚ 15% ܲü üyö ˆ˛õˆÏ°ö⁄ ~•z ˛õyÌ≈ܲƒ Óyï˛yˆÏ§Ó˚ ã˛y˛õ Óy âöˆÏcÓ˚ ˛õ!Ó˚üy˛õàï˛ §%-ï˛yÓ˚
xû˛yˆÏÓÓ˚ çöƒ •ˆÏï˛ ˛õyˆÏÓ˚ öy– fl∫yû˛yÓï˛•z xy˛õöyˆÏòÓ˚ üˆÏö •ˆÏÓ ˆÎñ !öí˛zê˛ˆÏöÓ˚ §)ˆÏe !öÿ˛Î˚•z ˆÜ˛yöÄ ï˛_¥àï˛
e&!ê˛ xyˆÏåÈ– ú˛Ó˚y§# à!îï˛!Óò °y@’y§ !öí˛zê˛ˆÏöÓ˚ §)ˆÏeÓ˚ §ÇˆÏ¢yôˆÏöÓ˚ ≤ÃfliyÓ Ü˛ˆÏÓ˚ö– !ï˛!ö ~•z Î%!_´ ˆòö ˆÎñ
Îáö ¢∑ï˛Ó˚D ~ܲ!ê˛ üyôƒˆÏüÓ˚ üˆÏôƒ !òˆÏÎ˚ ÎyÎ˚ ï˛áö üyôƒˆÏüÓ˚ ܲîy=!° á%Ó o&ï˛ xyˆÏ®y!°ï˛ •Î˚– üyôƒˆÏüÓ˚
âö#û˛Óö Ä ï˛ö%û˛Óöó í˛zû˛ˆÏÎ˚•z âˆÏê˛ Ó˚&k˛ï˛y˛õ ≤Ã!e´Î˚yÎ˚– ú˛ˆÏ° âö#û˛ÓˆÏöÓ˚ fliyö=!° àÓ˚ü •ˆÏÎ˚ ĈÏë˛ ~ÓÇ
ï˛ö%û˛ÓˆÏöÓ˚ fliyö=!° ë˛y[˛y •Î˚– ~Ó˚ xÌ≈ ¢∑ï˛Ó˚D Óyï˛yˆÏ§Ó˚ ˆÎ xÇ¢ !òˆÏÎ˚ ≤ÃÓy!•ï˛ •Î˚ ˆ§•z fliyˆÏö í˛z£èï˛yÓ˚
˛õ!Ó˚Óï≈˛ö âˆÏê˛–
Ó˚&k˛ï˛y˛õ ≤Ã!e´Î˚yÓ˚ çöƒ xyò¢≈ àƒy§ ÓˆÏÎ˚ˆÏ°Ó˚ §)e öy ˆüˆÏö P γν = ô &Óܲñ ~•z §)e!ê˛ ˆüˆÏö ã˛ˆÏ°– ~áyˆÏö
γ
!ö!ò≈‹T àyˆÏ§Ó˚ ˆ«˛ˆÏe ~ܲ!ê˛ ô &Óܲ ~ÓÇ ~Ó˚ üyö !fliÓ˚ ã˛y˛õ Ä !fliÓ˚ xyÎ˚ï˛ˆÏö àƒyˆÏ§Ó˚ ò%!ê˛ xyˆÏ˛õ!«˛Ü˛ ï˛yˆÏ˛õÓ˚
xö%˛õyˆÏï˛Ó˚ §üyö–
P
γ ν
= ô &Óܲ §)e ˆÌˆÏܲ §%Ó˚& ܲˆÏÓ˚
ρ
~ÓÇ
ν
~Ó˚ §yüyöƒ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ çöƒ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
10 γγ νρργνν − Δ+Δ=xÌÓyñ
E ργρννΔ −==Δ
xï˛~Óñ 8.7a §ü#ܲÓ˚î ˆÌˆÏܲ ¢ˆÏ∑Ó˚ ˆÓà •ˆÏÓ γρνρ
= ...8.9
~ܲ @ˇÃyü xî%àƒyˆÏ§Ó˚ ˆ«˛ˆÏe ρV = RT, R = àƒy§ ô &Óܲñ T = ˛õÓ˚ü í˛z£èï˛yñ V = xyî!Óܲ xyÎ˚ï˛öñ
M Î!ò xyî!Óܲ û˛Ó˚ •Î˚ ï˛y•ˆÏ° V =
== MRTM
θρρ
xÌ≈yÍñ V = RTM
γ....8.9a
Óyï˛yˆÏ§Ó˚ çöƒ γ = 1·4– ܲyˆÏç ܲyˆÏç•z 8.9 ˆÌˆÏܲ ˛õyˆÏÓö ≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓà
331ms–1– ˛õÓ˚#«˛y°∏˛ üyˆÏöÓ˚ §ˆÏD ~•z üyˆÏöÓ˚ !ü° §%flõ‹T ~ÓÇ ~Ó˚ myÓ˚y °y≤’yˆÏ§Ó˚ §ÇˆÏ¢yôˆÏöÓ˚ ÎÌyÌ≈ï˛y
≤Ãüy!îï˛ •Î˚–
8.9 Ä 8.9a §)e xyüyˆÏòÓ˚ ܲyˆÏåÈ á%Ó•z =Ó˚&c˛õ)î≈– ~•z ò%•z §)e ˆÌˆÏܲ xyüÓ˚y ¢ˆÏ∑Ó˚ ˆÓà ã˛y˛õñ í˛z£èï˛y
≤Ãû,˛!ï˛Ó˚ í˛z˛õÓ˚ ܲ#û˛yˆÏÓ !öû≈˛Ó˚ ܲˆÏÓ˚ ï˛y ÓyÓ˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚–
ã˛yˆÏ˛õÓ˚ ≤Ãû˛yÓ ≠ ã˛yˆÏ˛õÓ˚ ≤Ãû˛yÓ ≠ ã˛yˆÏ˛õÓ˚ ≤Ãû˛yÓ ≠ ã˛yˆÏ˛õÓ˚ ≤Ãû˛yÓ ≠ ã˛yˆÏ˛õÓ˚ ≤Ãû˛yÓ ≠ ˆÎÈÙÈˆÜ˛yö í˛z£èï˛yÎ˚ ~ܲ!ê˛ àƒyˆÏ§Ó˚ ˆ«˛ˆÏe
ρρ
= ô &Óܲ– xï˛~Óñ 8.9 §ü#ܲÓ˚î ˆÌˆÏܲ
xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆÓà ã˛yˆÏ˛õÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° öÎ˚–
214
í˛z£èï˛yÓ˚ ˛≤Ãû˛yÓ ≠ í˛z£èï˛yÓ˚ ˛≤Ãû˛yÓ ≠ í˛z£èï˛yÓ˚ ˛≤Ãû˛yÓ ≠ í˛z£èï˛yÓ˚ ˛≤Ãû˛yÓ ≠ í˛z£èï˛yÓ˚ ˛≤Ãû˛yÓ ≠ ˆÜ˛yöÄ àƒyˆÏ§Ó˚ í˛z£èï˛y ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏ° ã˛y˛õ §üyö ˆÌˆÏÜ˛Ä âöc ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚–
í˛z£èï˛y Óyí˛¸ˆÏ° xyÎ˚ï˛ö ÓyˆÏí˛¸ñ §%ï˛Ó˚yÇ âöc ܲˆÏü– í˛z£èï˛y ܲüˆÏ° âöc ÓyˆÏí˛¸– 8.9a §)e xö%ÎyÎ˚# !ö!ò≈‹T
àƒyˆÏ§ ¢ˆÏ∑Ó˚ ˆÓà ˛õÓ˚ü í˛z£èï˛yÓ˚ Óà≈ü)ˆÏ°Ó˚ §üyö%˛õyï˛#– xÌ≈yÍ
T ν∝
¢ˆÏ∑Ó˚ ˆÓà ˛õ!Ó˚üyˆÏ˛õÓ˚ ˛õk˛!ï˛ xyüyˆÏòÓ˚ ≤Ã!ï˛Ó˚«˛y Óy!•ö#ˆÏï˛ Ó‡û˛yˆÏÓ ÓƒÓ•*ï˛ •ˆÏÎ˚ˆÏåÈ– ~Ó˚ §y•yˆÏ΃ ≤ÃÌü
ü•yÎ%ˆÏk˛Ó˚ §üÎ˚ ¢e&˛õˆÏ«˛Ó˚ ܲyüyˆÏöÓ˚ xÓfliyö !öî≈Î˚ §Ω˛Ó •ˆÏÎ˚!åÈ°– ~•z ˛õk˛!ï˛Ó˚ öyü ¢ˆÏ∑Ó˚ í˛zͧ §¶˛yö
(Sound ranging)
(b) ï˛Ó˚ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃÓy• ÈÙÙÙÈ ï˛Ó˚ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃÓy• ÈÙÙÙÈ ï˛Ó˚ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃÓy• ÈÙÙÙÈ ï˛Ó˚ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃÓy• ÈÙÙÙÈ ï˛Ó˚ˆÏ° ¢ˆÏ∑Ó˚ ≤ÃÓy• ÈÙÙÙÈ ï˛Ó˚ˆÏ°Ó˚ xyÎ˚ï˛ö =îyB˛ §yôyÓ˚îï˛ ~ï˛ ˆÓ!¢ •Î˚ ˆÎñ ï˛Ó˚°ˆÏܲ ≤ÃyÎ˚ x§Çöüƒ
(incompressible) ӈϰ ôÓ˚y •Î˚– çˆÏ°Ó˚ çöƒ E = 2·22 × 109Nm–2 ~ÓÇ
ρ
= 103kgm–3– ~•z ò%•z
üyö ˆÌˆÏܲ 8,7a §)ˆÏeÓ˚ §y•yˆÏ΃ ï˛Ó˚ˆÏDÓ˚ ˆÓà !öî≈Î˚ ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎ˚
ν
= 15000ms–1– ~•z üyˆÏöÓ˚
§ˆÏD ≤Ãüyî ã˛y˛õ Ä í˛z£èï˛yÎ˚ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ üyˆÏöÓ˚ ï%˛°öy ܲÓ˚&ö– ˆòáy ÎyˆÏÓ ˆÎñ Î!òÄ Óyï˛y§ ç°
xˆÏ˛õ«˛y ≤ÃyÎ˚ §•flÀ =î ܲü âöñ ï˛Ìy!˛õ ¢∑ Óyï˛yˆÏ§Ó˚ ˆÌˆÏܲ çˆÏ° ≤ÃyÎ˚ 4 =î o&ï˛ ÎyÎ˚– ~Ó˚ ܲyÓ˚ˆÏî Óyï˛yˆÏ§Ó˚
ï%˛°öyÎ˚ çˆÏ°Ó˚ x§Çöüƒï˛y– çˆÏ°Ó˚ üˆÏôƒ ¢∑ï˛Ó˚ˆÏDÓ˚ x!ôܲ à!ï˛ˆÏÓà ~ÓÇ xöƒ ܲˆÏÎ˚ܲ!ê˛ Ü˛yÓ˚ˆÏîñ ˆÎüö
çˆÏ° ¢ˆÏ∑Ó˚ ≤Ã!ï˛Ó˚§îñ x“ ˆ¢y£Ïî Ä !ӈϫ˛˛õîñ Óyï˛yˆÏ§Ó˚ ï%˛°öyÎ˚ §ü%ˆÏoÓ˚ çˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ¢∑ ˆ≤ÃÓ˚î
xˆÏöܲ §%!Óôyçöܲ– ~ çˆÏöƒ•z §ü%ˆÏo ~ܲ çy•yç ˆÌˆÏܲ xöƒ çy•yˆÏç Óyï≈˛y ˛õyë˛yˆÏï˛ ~ÓÇ ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚
§¶˛yö ܲÓ˚ˆÏï˛ ˆ§yöyÓ˚ (SONAR = Sound Navigation and Ranging) ÓƒÓfliy ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ~•z
ÓƒÓfliyÎ˚ í˛zFã˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ §,!‹T ܲˆÏÓ˚ §ü%ˆÏoÓ˚ çˆÏ°Ó˚ üˆÏôƒ ˆ≤ÃÓ˚î ܲÓ˚y •Î˚ ~ÓÇ ≤Ã!ï˛ú˛!°ï˛ ¢ˆÏ∑Ó˚ !òܲ
Ä !Ó°ˆÏ¡∫Ó˚ ˛õ!Ó˚üy˛õ ܲˆÏÓ˚ ï˛yÓ˚ ˆÌˆÏܲ §ü%ˆÏoÓ˚ àû˛#Ó˚ï˛yñ üyˆÏåÈÓ˚ é˛yÑܲñ í%˛ˆÏÓyçy•yç ~üö !ܲ ¢e&˛õˆÏ«˛Ó˚ ˆåÈyÑí˛¸y
ê˛ˆÏ˛õ≈ˆÏí˛yÓ˚ xÓfliyö Ä ò)Ó˚c !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
≤ÃÓy•#Ó˚ ˆ«˛ˆÏe xyüÓ˚y 8.7a §ü#ܲÓ˚î!ê˛ˆÏï˛ ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ ܲ!ë˛ö ˆÎ Ó˚y!¢üy°y!ê˛ ˆ˛õˆÏÎ˚!åÈ ˆ§!ê˛ xyüÓ˚y
~ÓyÓ˚ ~ܲ!ê˛ òˆÏ[˛Ó˚ ˆ«˛ˆÏe ≤ÈÏÎ˚yà ܲÓ˚Ó–
8.2.3 ~ܲ!ê˛ Ü˛!ë˛ö §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D~ܲ!ê˛ Ü˛!ë˛ö §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D~ܲ!ê˛ Ü˛!ë˛ö §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D~ܲ!ê˛ Ü˛!ë˛ö §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D~ܲ!ê˛ Ü˛!ë˛ö §%£Ïü òˆÏ[˛Ó˚ üˆÏôƒ ï˛Ó˚D
~ܲ!ê˛ Ü˛!ë˛ö !fli!ï˛fliy˛õܲ òˆÏ[˛Ó˚ ˆ«˛ˆÏe ˛õ!Ó˚Óï≈˛ö ˆÜ˛Ó°üye ˜òˆÏâ≈ƒ•z •Î˚– ˜òˆÏâ≈ƒÓ˚ ≤çyÓ˚î Óy §ˆÏB˛yã˛ö
âê˛ˆÏ° ≤ÈÏfli ÎÌye´ˆÏü §ˆÏB˛yã˛ö Ä ≤çyÓ˚î âˆÏê˛ñ ú˛ˆÏ° xyÎ˚ï˛ö ≤ÃyÎ˚ !fliÓ˚ ÌyˆÏܲ– ܲyˆÏç•z ≤ÃÓy•#Ó˚ ˆ«˛ˆÏe
ÓƒÓ•*ï˛ xyÎ˚Èï˛ö =îyˆÏB˛Ó˚ fliyˆÏö ~áyˆÏö xyüÓ˚y ˆöÓ •zÎ˚ÇÈÙÈ=îyB˛ xÌ≈yÍ Y = ï˛y•ˆÏ°
8.7 §ü#ܲÓ˚ˆÏîÓ˚ ˛õ!Ó˚Ó!ï≈˛ï˛ Ó˚*˛õ •ˆÏÓ
22
22∂∂ =∂∂
Ytxψψ
ρ
......8.10
xö% Ïòâ≈ƒ õ#í ¸ö
xö)% Ïòâ≈ƒ !ÓÜ, !ï˛
215
~!ê˛•z òˆÏ[˛Ó˚ üˆÏôƒ ≤ÃÓy!•ï˛ ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î– ~Ó˚ ˆÌˆÏܲ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà
Yνρ
= ....8.10a
ˆòáy ÎyˆÏFåÈ ˆÎñ ν ~Ó˚ üyö òˆÏ[˛Ó˚ ≤ÃfliˆÏFåȈÏòÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° öÎ˚–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2
~ܲ!ê˛ •zflõyˆÏï˛Ó˚ òˆÏ[˛Ó˚ ˆ«˛ˆÏe Y = 1·95 × 1011Nm–2 ~ÓÇ
ρ
= 7800kgm–3 òˆÏ[˛Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚
ˆÓà !öî≈Î˚ ܲÓ˚&ö–
í˛z˛õˆÏÓ˚Ó˚ xö%¢#°ö#!ê˛ Ü˛Ó˚ˆÏ° xy˛õ!ö ˆòáˆÏï˛ ˛õyˆÏÓö ˆÎñ xö%˜Ïòâ≈ƒ ï˛Ó˚D ï˛Ó˚° !ܲÇÓy àƒy§ xˆÏ˛õ«˛y ܲ!ë˛ö
˛õòyˆÏÌ≈Ó˚ o&ï˛ ≤çy!Ó˚ï˛ •Î˚– ~•z ܲyÓ˚ˆÏî °¡∫y ˆ°y•yÓ˚ öˆÏ°Ó˚ ~ܲ≤ÃyˆÏhsˇ xyâyï˛ Ü˛Ó˚ˆÏ° ï˛yÓ˚ ¢∑ ÓyÎ˚%Óy!•ï˛
•ˆÏÎ˚ xy§yÓ˚ xyˆÏà•z öˆÏ°Ó˚ ôyï%˛Ó˚ üôƒ !òˆÏÎ˚ Óy!•ï˛ •ˆÏÎ˚ xyˆÏ§ ~ÓÇ ◊&!ï˛ˆÏàyã˛Ó˚ •Î˚È–
8.3 ï˛Ó˚D≤ÃÓy• ~ÓÇ ≤Ã!ï˛ˆÏÓ˚yôï˛Ó˚D≤ÃÓy• ~ÓÇ ≤Ã!ï˛ˆÏÓ˚yôï˛Ó˚D≤ÃÓy• ~ÓÇ ≤Ã!ï˛ˆÏÓ˚yôï˛Ó˚D≤ÃÓy• ~ÓÇ ≤Ã!ï˛ˆÏÓ˚yôï˛Ó˚D≤ÃÓy• ~ÓÇ ≤Ã!ï˛ˆÏÓ˚yô (Impedance)
ˆÜ˛yö üyôƒˆÏü Îáö ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚ñ ï˛áö üyôƒˆÏüÓ˚ ӛܲîyÓ˚ ï˛Ó˚Dà!ï˛Ó˚ ú˛ˆÏ° í˛zͲõߨ ˆÓà ˙ ï˛Ó˚D
í˛zͲõyòˆÏöÓ˚ çöƒ ≤ÃÎ%_´ ӈϰÓ˚ §üyö%˛õyï˛# •Î˚– xy˛õ!ö •Î˚ï˛ çyˆÏöö ˆÎñ ˜Óò%ƒ!ï˛Ü˛ Óï≈˛ö#ˆÏï˛ ˛õ!Ó˚Óï˛#≈
ï˛!í˛¸Fã˛y°Ü˛ Ó° ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° Óï≈˛ö#Ó˚ üˆÏôƒ ˛õÓ˚Óï˛#≈ ≤ÃÓyˆÏ•Ó˚ §,!‹T •Î˚– ~ÓÇ ≤ÃÎ%_´ ï˛!í˛¸Fã˛y°Ü˛ Ó°Ä ≤ÃÓyˆÏ•Ó˚
xö%˛õyï˛ˆÏܲ Óï≈˛ö#Ó˚ ≤Ã!ï˛ˆÏÓ˚yô (Impedance) Ó°y •Î˚– ˜Óò%ƒ!ï˛Ü˛ Óï≈˛ö#Ó˚ ≤Ã!ï˛ˆÏÓ˚yô Óï≈˛ö#Ó˚ ï˛!í˛¸Í ≤ÃÓyˆÏ•
ܲï˛ê˛y Óyôy ˆòÎ˚ ï˛yÓ˚•z ˛õ!Ó˚üy˛õ– xö%Ó˚*˛õû˛yˆÏÓñ ï˛Ó˚D í˛zͲõyòöܲyÓ˚# Ó°Ä Ó›Ü˛îyÓ˚ ˆÓˆÏàÓ˚ !ÓhflÏyˆÏÓ˚Ó˚
xö%˛õyï˛ˆÏܲ ï˛Ó˚D ≤Ã!ï˛ˆÏÓ˚yô (wave impedance) Ó°y •Î˚ ~ÓÇ ~•z ï˛Ó˚D ≤Ã!ï˛ˆÏÓ˚yô üyôƒü!ê˛ ï˛Ó˚D §MÈ˛yÓ˚î
ܲï˛ê˛y ÓyôyÓ˚ §,!‹T ܲˆÏÓ˚ ï˛yÓ˚•z ˛õ!Ó˚üy˛õ– ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ~«˛ˆÏe ~•z ≤Ã!ï˛ˆÏÓ˚yôˆÏܲ xyüÓ˚y ï˛yˆÏÓ˚Ó˚
!Ó!¢‹T (characteristic) ≤Ã!ï˛ˆÏÓ˚yô Ó!°– xö%≤Ãfli ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ~!ê˛ •Î˚ üyôƒˆÏüÓ˚ ¢∑ (acoustic)
≤Ã!ï˛ˆÏÓ˚yô–
≤ß¿ í˛zë˛ˆÏï˛ ˛õyˆÏÓ˚ñ ~•z ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ §,!‹T •Î˚ ˆÜ˛ö⁄ xy§ˆÏ° üyôƒˆÏüÓ˚ ≤Ã!ï˛!ê˛ Ó›Ü˛îyÓ˚ à!ï˛ ï˛yÓ˚ ˛õÓ˚Óï˛#≈
ӛܲîyˆÏܲ ≤Ãû˛y!Óï˛ Ü˛ˆÏÓ˚– ˆÎ ܲîy!ê˛ xˆÏ®y!°ï˛ •ˆÏFåÈ ˆ§!ê˛ ˛õÓ˚Óï˛#≈ ܲîy!ê˛ˆÏܲÈÙÈà!ï˛¢#° ܲÓ˚yÓ˚ çöƒ ˆ§!ê˛ˆÏï˛
¢!_´ •hflÏyhsˇÓ˚ ܲˆÏÓ˚ñ xyÓyÓ˚ ˆÎ ܲîy !fliÓ˚ xyˆÏåÈ ˆ§!ê˛ ˛õÓ˚Óï˛#≈ ܲîyÓ˚ à!ï˛ ü!®ï˛ ܲˆÏÓ˚– ~•zû˛yˆÏÓ ≤Ã!ï˛!ê˛ Ü˛îy
~ܲ!ê˛ ˛õÿ˛yÍܲ£Ï≈î (drag) xö%û˛Ó ܲˆÏÓ˚– ~•z Ó°•z ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ ܲyÓ˚î ~ÓÇ ~!ê˛ Óy•zˆÏÓ˚ ˆÌˆÏܲ ≤ÃÎ%_´ ӈϰÓ˚
§üüyö–
xyüÓ˚y ~áö ܲï˛Ü˛=!° !ӈϢ£Ï í˛zòy•Ó˚î xyˆÏ°yã˛öy ܲÓ˚Ó–
216
8.3.1 xö%≤Ãfli ï˛Ó˚ˆÏD ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ˆÏÓ˚yôxö%≤Ãfli ï˛Ó˚ˆÏD ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ˆÏÓ˚yôxö%≤Ãfli ï˛Ó˚ˆÏD ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ˆÏÓ˚yôxö%≤Ãfli ï˛Ó˚ˆÏD ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ˆÏÓ˚yôxö%≤Ãfli ï˛Ó˚ˆÏD ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ˆÏÓ˚yô
!ã˛e 8.3-ˆòy°à!ï˛ Ó° F = F0cos
ω
0t ≤Ãû˛yˆÏÓ xyˆÏ®y!°ï˛ ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚ !ã˛e
ôÓ˚&öñ x x«˛ ÓÓ˚yÓÓ˚ ~ܲ!ê˛ ê˛yöy ï˛yˆÏÓ˚Ó˚ x = 0 !Ó®%ˆÏï˛ ~ܲ!ê˛ ˆòy°à!ï˛ Ó° F = F0 cos
ω0t ≤ÈÏÎ˚yà
ܲˆÏÓ˚ ˙ ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚D §,!‹T ܲÓ˚y •° (8.3 !ã˛e) – ˆÎ ˆÜ˛yöÄ §üÎ˚ t ˆï˛ x xÓfliyˆÏö ï˛yˆÏÓ˚Ó˚ ܲîyÓ˚
§Ó˚ˆÏîÓ˚ §ü#ܲÓ˚î •ˆÏÓ–
y(x, t) = a sin(
ω
0t – kx) ...8.11
~•z §ü#ܲÓ˚î!ê˛Ó˚ §ˆÏD !öÿ˛Î˚•z xy˛õöyÓ˚ ˛õ!Ó˚ã˛Î˚ xyˆÏåÈ–
ôÓ˚y Îyܲñ ï˛yˆÏÓ˚Ó˚ ê˛yö T– xï˛~Óñ F0 cos
ω
0t = – Tsin
θ
ˆÎˆÏ•ï%˛ x = 0 !Ó®%ˆÏï˛ ï˛yˆÏÓ˚Ó˚ ӛܲîyÓ˚ í˛z˛õÓ˚ °!∏˛ Ó° ¢)öƒ •ˆÏÓ
xï˛~Óñ F0 cos
ω
0t + T sin
θ
x = 0= 0
θ
~Ó˚ üyö «%˛o •ˆÏ° ôÓ˚y ÎyÎ˚ sin
θ
≈ tan
θ
ˆ§ˆÏ«˛ˆÏe xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ F0cos
ω
0t = – Ttan
θ
x = 0 ...8.12
~•z ö!ï˛Ó˚ (tan
θ
) üyö!ê˛ x = 0 ˆï˛ ≤ÈÏÎy烖
8.11 §ü#ܲÓ˚ˆÏîÓ˚ §y•yˆÏ΃ xyüÓ˚y
dydx
~ÓÇ
dydt
~Ó˚ üˆÏôƒ §¡∫¶˛ ˛õy•z≠
dydx
= – ka cos(
ω
0t – kx) ~ÓÇ
dydt
=
ω
0a cos(
ω
0t – kx)
217
xÌ≈yÍñ
0=− dykdy
dxdt ω
~•z üyö 8.12 ~ Ó!§ˆÏÎ˚ xyüÓ˚y ˛õy•z
F0cosω0t =
00=
⎛⎞⎜⎟ ⎝⎠x
kTdydt ω
!ܲv ˆÎˆÏ•ï%˛0x
dydt =
⎛ ⎞⎜ ⎟⎝ ⎠ = a
ω
0 cos
ω
0t.
xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ F0 cos
ω
0t =
0
Tkω
a
ω
0 cos
ω
0t =
Tν
a
ω
0cos
ω
0t, ˆÎáyˆÏö
0
kω ν=
Î!ò ˆ°áy ÎyÎ˚ a ω0 =
ν
0 ˆÎáyˆÏö
ν
0 = ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚ !ÓhflÏyÓ˚ (velocity amplitude)
ï˛y•ˆÏ°ñ F0cos
ω
0t =
00 cos Tt νω
ν
xÌÓyñ F0=
0 Tνν
!ܲÇÓyñ
0
0
FTνν
=
...8.13
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î!ê˛ ≤ÃÎ%_´ ӈϰÓ˚ !ÓhflÏyÓ˚ ~ÓÇ Ü˛îyÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ xö%˛õyï˛ˆÏܲ ï˛yˆÏÓ˚Ó˚
ê˛yö ~ÓÇ Ü˛îyÓ˚ ˆÓˆÏàÓ˚ xö%˛õyï˛ Ó˚*ˆÏ˛õ ≤Ãܲy¢ ܲÓ˚ˆÏåÈ– ~•z §ü#ܲÓ˚î ˆÌˆÏܲ ï˛yˆÏÓ˚Ó˚ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô ˛õyÄÎ˚y
ÎyÎ˚– Z ~Ó˚ §ÇK˛y xö%ÎyÎ˚#
Z =
8.13 §ü#ܲÓ˚î ˆÌˆÏܲ üyö Ó!§ˆÏÎ˚
Z =
0
0
FTνν
=
...8.14
8.7a §)e ˆÌˆÏܲñ
Tm
ν=
ˆÎáyˆÏöñ m = ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚
§%ï˛Ó˚yÇñ 8.14 §ü#ܲÓ˚îˆÏܲ xy˛õ!ö !°áˆÏï˛ ˛õyˆÏÓ˚ö ≠
Z = T Tmν
= ...8.15a
≤ÃÎ%_´ ӈϰÓ˚ !ÓhflÏyÓ˚ (F0)ôÓ˚ˆÏD !ï˛Î≈à ˆÓˆÏàÓ˚ !ÓhflÏyÓ˚ (V0)
218
xÌÓy T ˆÜ˛ x˛õ§yÓ˚î ܲÓ˚ˆÏï˛ ã˛y•zˆÏ°
Z = 2m mν ν
ν= ...8.15b
8.15a §)e ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô ï˛yˆÏÓ˚Ó˚ ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚ ~ÓÇ
ï˛yˆÏÓ˚Ó˚ ê˛yˆÏöÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ~Ó˚˛xÌ≈ ~•z ˆÎñ ~ܲ!ê˛ ˆ§yˆÏöy!üê˛yˆÏÓ˚Ó˚ (Sonemeter) ï˛yˆÏÓ˚Ó˚ ≤ÃyˆÏhsˇ !Ó!û˛ß¨
û˛Ó˚ ã˛y˛õyˆÏ° ï˛yÓ˚ ≤Ã!ï˛ˆÏÓ˚yôÄ !Ó!û˛ß¨ •Ó– 8.15b §ü#ܲÓ˚î ˆÌˆÏܲ xyÓ˚ çyöˆÏï˛ ˛õyÓ˚!åÈ ˆÎñ Z ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚
§ˆÏD §¡õÜ≈˛Î%_´ ӈϰñ Z üyôƒˆÏüÓ˚ çí˛¸c ~ÓÇ !fli!ï˛fliy˛õܲï˛yÓ˚ IJõÓ˚Ä !öû≈˛Ó˚¢#°–
°«˛ƒ ܲÓ˚&ö ≤Ã!ï˛ˆÏÓ˚yô Z ~Ó˚ ~ܲܲ Nm–1s xÌÓy kgs–1–
xö%¢#°ö# ÈÙxö%¢#°ö# ÈÙxö%¢#°ö# ÈÙxö%¢#°ö# ÈÙxö%¢#°ö# ÈÙ 3
~ܲ!ê˛ ï˛yÓ˚ˆÏܲ 80N ӈϰÓ˚ §y•yˆÏ΃ ˆê˛ˆÏö Ó˚yáy xyˆÏåÈ– ï˛yÓ˚!ê˛Ó˚ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô !öî≈Î˚ ܲÓ˚&ö– ï˛yˆÏÓ˚Ó˚
≤Ã!ï˛ !üê˛yˆÏÓ˚Ó˚ û˛Ó˚ 2g– ~•z ï˛yˆÏÓ˚ 25cms–1 !ÓhflÏyˆÏÓ˚Ó˚ xö%≤Ãfli ˆÓà §,!‹T ܲÓ˚ˆÏï˛ Ü˛ï˛ !ÓhflÏyˆÏÓ˚Ó˚ Ó° °yàˆÏÓ⁄
8.3.2 ¢∑ï˛Ó˚ˆÏD àƒy§#Î˚ üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô¢∑ï˛Ó˚ˆÏD àƒy§#Î˚ üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô¢∑ï˛Ó˚ˆÏD àƒy§#Î˚ üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô¢∑ï˛Ó˚ˆÏD àƒy§#Î˚ üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô¢∑ï˛Ó˚ˆÏD àƒy§#Î˚ üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô
àyˆÏ§Ó˚ !û˛ï˛Ó˚ !òˆÏÎ˚ Îáö ¢∑ï˛Ó˚D ≤ÃÓy!•ï˛ •Î˚ñ ï˛áö ï˛Ó˚ˆÏDÓ˚ x!ï˛!Ó˚_´ ã˛y˛õ•z xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe
≤ÃÎ%_´ ӈϰÓ˚ û)˛!üܲy @ˇÃ•î ܲˆÏÓ˚–
xï˛~Óñ ¢∑≤Ã!ï˛ˆÏÓ˚yôˆÏܲ Ó°y ÎyÎ˚
Z = = t
ρψΔ
∂∂
...8.16
Îáö ~ܲ!ê˛ üyôƒü !òˆÏÎ˚ xö%˜Ïòâ≈ƒ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •Î˚ñ ï˛áö üyôƒˆÏü xö%û%˛ï˛ x!ï˛!Ó˚_´ ã˛y˛õ
Exψ ρ∂ Δ=−∂
~áyˆÏö E = üyôƒˆÏüÓ˚ xyÎ˚ï˛ö =îyB˛– ~•z §¡õÜ≈˛!ê˛ xyüÓ˚y 8.2.2. xLjϢ xyˆÏà•z ≤Ã!ï˛!¤˛ï˛ ܲˆÏÓ˚!åÈ–
ܲyˆÏç•z Z ~Ó˚ üyö !öî≈Î˚ ܲÓ˚ˆÏï˛ •ˆÏ° xyüyˆÏòÓ˚
tψ∂∂
~ÓÇ Ç
xψ∂∂
í˛zû˛Î˚•z !öî≈Î˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– ï˛y ܲÓ˚ˆÏï˛ •ˆÏ°
xyüyˆÏòÓ˚ üˆÏö Ó˚yáˆÏï˛ •ˆÏÓ ˆÎñ + x !òˆÏܲ §MÈ˛y!Ó˚ï˛ xö%˜Ïôâ≈ƒÈÙÈï˛Ó˚ˆÏDÓ˚ ܲîyÓ˚ §Ó˚î
Ψ
(x, t) = asin
() 2tx πνλ
⎡⎤ − ⎢⎥ ⎣⎦
x ~ÓÇ t ~Ó˚ §yˆÏ˛õˆÏ«˛ xÓܲ°ö ܲÓ˚ˆÏ°
() 22 cos atxxψππν
λλ∂⎡⎤ =−− ⎢⎥ ∂⎣⎦
...8.17
¢∑ï˛Ó˚D ç!öï˛ x!ï˛!Ó˚_´ ã˛y˛õ
ܲöyÓ˚ à!ï˛ˆÏÓà
219
~ÓÇ
() 22 cos atxtψπνπν
λλ∂⎛⎞⎡⎤ =− ⎜⎟⎢⎥ ∂⎝⎠⎣⎦
...8.19
∆p ~Ó˚ Ó˚y!¢üy°yÓ˚ ˆÌˆÏܲ ˛õy•zñ
() 22 cos() pEavtx ππ Δλλ⎡⎤ =− ⎢⎥ ⎣⎦
...8.19
8.19 ~ÓÇ 8.18-~Ó˚ üyö 8.16 ~ Ó!§ˆÏÎ˚ ˛õyÄÎ˚y ÎyÎ˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yô
Z =
()()()
22 cos22 cos
EatxE
atx t
ππλλν ρ
ψπνπν νλλ
⎡⎤ − Δ⎣⎦ == ∂⎛⎞⎡⎤ − ∂⎜⎟⎢⎥ ⎝⎠⎣⎦
...8.20
~áy Ïö ν = ï˛Ó˚D ˆÓà–
í˛z˛õˆÏÓ˚Ó˚ §)e ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎ ¢∑ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ ~ܲܲ •° Nm–3s(8.7a) xö%§yˆÏÓ˚ xy˛õöyÓ˚y !öÿ˛Î˚
çyˆÏöö ˆÎ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆÓà
E νρ
=ˆÎáyˆÏö ρ = üyôƒˆÏüÓ˚ âöc–
xï˛~Óñ ¢∑ ≤Ã!ï˛ˆÏÓ˚yôˆÏܲ ~û˛yˆÏÓÄ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚≠
Z =
EEρρνν
==
...8.21
ˆòáy ÎyˆÏFåÈ ˆÎñ ¢∑ ≤Ã!ï˛ˆÏÓ˚yôˆÏܲ üyôƒˆÏüÓ˚ âöc ~ÓÇ ï˛Ó˚DˆÏÓˆÏàÓ˚ =îú˛°Ó˚*ˆÏ˛õ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚– ¢∑
≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ ~ܲܲ!ê˛ xy˛õ!ö !öî≈Î˚ ܲˆÏÓ˚ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö– ~•z ~ܲܲ!ê˛ •° Nm–3s– ~Ó˚ ˛õˆ ÏÓ ˚
xy˛õöyÓ˚y í˛z˛õˆÏÓ˚Ó˚ °∏˛ ú˛°=!° ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ !ÓhflÏyÓ˚ ~ÓÇ ¢!_´Ó˚ =îyB˛ !öî≈Î˚ ܲÓ˚ˆÏï˛
ÓƒÓ•yÓ˚ ܲÓ˚ˆÏÓö–
xö%¢#°ö ÈÙÈ xö%¢#°ö ÈÙÈ xö%¢#°ö ÈÙÈ xö%¢#°ö ÈÙÈ xö%¢#°ö ÈÙÈ 4
≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ ÓyÎ˚%üyôƒˆÏüÓ˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yô !öî≈Î˚ ܲÓ˚&ö– ˆòÄÎ˚y xyˆÏåÈñ ρ = 1·29kgm–3 ~ÓÇ
ν
=
332ms–1– ¢∑ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ üyö ç° xÌÓy Óyï˛y§ñ ˆÜ˛yö‰ üyôƒˆÏüÓ˚ ˆ«˛ˆÏe ˆÓ!¢ •ˆÏÓ⁄ xy˛õöyÓ˚ í˛z_ˆÏÓ˚Ó˚
ܲyÓ˚î !°á%ö–
220
8.4 !müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D!müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D!müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D!müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D!müy!eܲ ~ÓÇ !eüy!eܲ ï˛Ó˚D
xyüÓ˚y ~ ˛õÎ≈hsˇ ê˛yöy ï˛yˆÏÓ˚ §,‹T ~ܲüy!eܲ ï˛Ó˚ˆÏDÓ˚ ܲÌy xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ˆ§ ˆ«˛ˆÏe ï˛Ó˚D ï˛yÓ˚ ÓÓ˚yÓÓ˚
≤ÃÓy!•ï˛ •Î˚ ~ÓÇ Ü˛îy=!° xö%≤Ãfliû˛yˆÏÓ xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyˆÏܲ– !ܲv §Ó ÓyòƒÎsf ï˛yˆÏÓ˚Ó˚ ˜ï˛!Ó˚ •Î˚ öy–
xy˛õöyÓ˚y !öÿ˛Î˚•z ï˛Ó°y ÓyçyˆÏöy ˆòˆÏáˆÏåÈö ~ÓÇ ÷ˆÏöˆÏåÈö– Îáö ~ܲ!ê˛ ï˛Ó°y xÌÓy ~ܲ!ê˛ ì˛yˆÏܲÓ˚ ˛õò≈yÎ˚
(membrane) ˆ§•z ˛õò≈yÓ˚ ï˛ˆÏ°Ó˚ °¡∫ ÓÓ˚yÓÓ˚ ây ˆòÄÎ˚y •Î˚ñ ï˛áö ˛õò≈yÓ˚ ܲîy=!° ≤ÃÎ%_´ ӈϰÓ˚ x!û˛ü%ˆÏá
flõ!®ï˛ •ˆÏï˛ ÌyˆÏܲ– !ܲv ˛õò≈yÓ˚ ê˛yˆÏöÓ˚ çöƒ ~•z !ӈϫ˛yû˛ §yÓ˚y ï˛ˆÏ° åÈ!í˛¸ˆÏÎ˚ ÎyÎ˚– ~Ó˚ xÌ≈ñ ~ܲ!ê˛ ≤çy!Ó˚ï˛
˛õò≈yÓ˚ (stretched membrane) í˛z˛õˆÏÓ˚ ˆÎ ï˛Ó˚D §,!‹T •Î˚ ï˛y !müy!eܲ– ~•z §Ó ˆ«˛ˆÏe §Ó˚î x, y ~ÓÇ
t ~Ó˚ xˆÏ˛õ«˛Ü˛ •ˆÏÓñ xÌ≈yÍ
Ψ
=
Ψ
(x, y,t) – ~áö xy˛õöyˆÏòÓ˚ üˆÏö ≤ß¿ çyàˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ !müy!eܲ ï˛Ó˚ˆÏDÓ˚
§ü#ܲÓ˚î ܲ# Ó˚ܲü •ˆÏÓ⁄ ˆû˛Ôï˛ Ü˛yÓ˚î=!° !ÓˆÏÓã˛öy ܲˆÏÓ˚ xyüÓ˚y xöyÎ˚yˆÏ§ 8.4 §ü#ܲÓ˚îˆÏܲ ~ܲ!ê˛ !müy!eܲ
§ü#ܲÓ˚ˆÏî ˛õ!Ó˚Ó!ï≈˛ï˛ ܲÓ˚ˆÏï˛ ˛õy!Ó˚– x ~ÓÇ y x«˛ ÓÓ˚yÓÓ˚ Ó° Îáö !öÓ˚ˆÏ˛õ«˛û˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ñ ï˛áö ≤ÈÏï˛ƒˆÏܲÓ˚
çöƒ•z §ü#ܲÓ˚ˆÏî §ò,¢ Ó˚y!¢ í˛z˛õ!fliï˛ ÌyܲˆÏÓ– xï˛~Óñ 8.4 §ü#ܲÓ˚îˆÏܲ §ÇˆÏÎyçö Ä ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ !°áˆÏï˛
˛õy!Ó˚ñ
2tψ2 ∂
∂
(x, y, t) =
22
22E
xyψ
ρ⎛⎞ ∂∂ + ⎜⎟ ⎜⎟ ∂∂ ⎝⎠
(x, y, t) ...8.22
í˛z˛õˆÏÓ˚y_´ §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö •ˆÏFåÈñ
Ψ (x, y, t) = asin( ω0t – k·r), ˆÎáyˆÏö ...8.23
= ()()()() ++=+ xyxy
ˆˆˆˆ kikjk.rxiyjkxky~ÓÇ r2 = x2+y2
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ¢∑ ~ÓÇ xyˆÏ°yܲï˛Ó˚D ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ !müy!eܲ ï˛ˆÏ° Óƒy§yô≈ ÓÓ˚yÓÓ˚
(radially) åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸ öy– ~=!° xy§ˆÏ° !eüy!eܲ ï˛ˆÏ° §MÈ˛y!Ó˚ï˛ •Î˚– û)˛Ü˛¡õ ï˛Ó˚D xÌÓy ˆÜ˛yöÄ
!fli!ï˛fliy˛õܲ ܲ!ë˛ö ˛õòyˆÏÌ≈Ó˚ üˆÏôƒ §MÈ˛y!Ó˚ï˛ !eüy!eܲ ï˛Ó˚ˆÏDÓ˚ Óî≈öy ܲ# ܲˆÏÓ˚ ˆòÄÎ˚y ÎyÎ˚⁄ xyüyˆÏòÓ˚ !müy!eܲ
ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚ˆÏîÓ˚ §¡±§yÓ˚î ܲˆÏÓ˚ !eüy!eܲ §ü#ܲÓ˚ˆÏî ˆÎˆÏï˛ •Î˚– ~ ܲyç!ê˛ xy˛õöyÓ˚ çöƒ ~ܲ!ê˛
xö%¢#°ö# !•§yˆÏÓ Ó˚•z°–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 5
8.22 §ü#ܲÓ˚î!ê˛ˆÏܲ !eüy!eܲ §ü#ܲÓ˚ˆÏî Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚&ö–
~ÓyÓ˚ xyüÓ˚y ï˛Ó˚ˆÏDÓ˚ ܲˆÏÎ˚ܲ!ê˛ =Ó˚&c˛õ)î≈ ôˆÏü≈Ó˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚Ó– xyˆÏ°yܲï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛°ö Ä
≤Ã!ï˛§Ó˚ˆÏîÓ˚ âê˛öy xy˛õöyˆÏòÓ˚ ܲyˆÏåÈ !öÿ˛Î˚•z x!ï˛ ˛õ!Ó˚!ã˛ï˛– ¢∑ï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ú˛ˆÏ° í˛zͲõߨ
≤Ã!ï˛ôù!öÄ xy˛õöyˆÏòÓ˚ ܲyˆÏåÈ x˛õ!Ó˚!ã˛ï˛ öÎ˚– ܲyˆÏç•z §yôyÓ˚îû˛yˆÏÓ ï˛Ó˚ˆÏDÓ˚ ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î §¡∫ˆÏ¶˛
xy˛õ!ö !öÿ˛Î˚•z çyöˆÏï˛ ã˛y•zˆÏÓö– xy§%öñ ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚ˆÏîÓ˚ !Ó£ÏÎ˚=!° xyüÓ˚y ày!î!ï˛Ü˛û˛yˆÏÓ ˆòáyÓ˚
ˆã˛‹Ty ܲ!Ó˚–
221
8.5 ≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ (transmitted) ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛ï˛Ó˚D !ÓhflÏyˆÏÓ˚Ó˚ =îyB˛
~ܲ!ê˛ ï˛Ó˚D Îáö ò%!ê˛ !Ó!û˛ß¨ üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° xyâyï˛ Ü˛ˆÏÓ˚ñ ï˛áö ~Ó˚ !ܲå%Èê˛y xÇ¢ ≤Ã!ï˛ú˛!°ï˛ •Î˚
~ÓÇ !ܲå%È xÇ¢ ≤Ã!ï˛§,ï˛ •Î˚– !Ó!û˛ß¨ üyôƒü ï˛yˆÏòÓ˚ üˆÏôƒ ≤ÃÓy!•ï˛ ï˛Ó˚ˆÏD !Ó!û˛ß¨ ≤Ã!ï˛ˆÏÓ˚yô §,!‹T ܲˆÏÓ˚– ~áö
xyüÓ˚y ˆòáÓ ˆÎñ ò%!ê˛ üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ •ë˛yÍ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ú˛ˆÏ° ï˛Ó˚ˆÏDÓ˚ !ܲÓ˚*˛õ ˛õ!Ó˚Óï≈˛ö
•Î˚– ~çöƒ ≤Ã̈Ïü xyüÓ˚y xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚Ó–
8.5.1 xö%≤Ãfli ï˛Ó˚Dxö%≤Ãfli ï˛Ó˚Dxö%≤Ãfli ï˛Ó˚Dxö%≤Ãfli ï˛Ó˚Dxö%≤Ãfli ï˛Ó˚D
~ܲ!ê˛ §Ó˚& ï˛yÓ˚ AO ~ÓÇ xyÓ˚ ~ܲ!ê˛ ˆüyê˛y ï˛yÓ˚ OB ˆÜ˛ O !Ó®%ˆÏï˛ ç%ˆÏí˛¸ ê˛yö ܲˆÏÓ˚ Ó˚yáy •°– ò%!ê˛
ï˛yˆÏÓ˚Ó˚ ê˛yö•z (T) ~áö §üyö– ôÓ˚y Îyܲñ ï˛yÓ˚ ò%!ê˛Ó˚ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô Z1 ~ÓÇ Z2
ôÓ˚&öñ ~ܲ!ê˛ ï˛Ó˚D x x«˛ ÓÓ˚yÓÓ˚ x@ˇÃ§Ó˚ •ˆÏÎ˚ O !Ó®%ˆÏï˛ xyÇ!¢Ü˛ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ xyÇ!¢Ü˛ ≤Ã!ï˛§,ï˛
•°– xy˛õ!ï˛ï˛ñ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ §Ó˚î ~•zû˛yˆÏÓ ˆ°áy ÎyÎ˚ñ
yi(x, t) = aisin(
ω
0t – k1x) ...8.24
yi(x, t) = arsin(
ω
0t + k1x) ...8.25
~ÓÇ yt(x, t) = atsin(
ω
0t – k2x) ...8.26
222
ˆÎáyˆÏö §Ó˚î ~ÓÇ !ÓhflÏyˆÏÓ˚Ó˚ ˛õòyB˛ (Subscripts) i, r ~ÓÇ t xy˛õ!ï˛ï˛ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚D
ˆÓyé˛yÎ˚– xy˛õöyÓ˚y !öÿ˛Î˚ °«˛ƒ ܲˆÏÓ˚ ÌyܲˆÏÓö ˆÎñ ~•z ï˛Ó˚D=!°Ó˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ §üyö Ó˚ˆÏÎ˚ˆÏåÈ– ~åÈyí˛¸y
xy˛õ!ï˛ï˛ ~ÓÇ ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ §MÈ˛Ó˚î ô &Óܲ (propagation constant) ~ܲ•zñ !ܲv ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚
ˆ«˛ˆÏe ~!ê˛ xy°yòy– xy˛õ!ö !ܲ ~Ó˚ ܲyÓ˚î ˆÓyé˛yÓ˚ ˆã˛‹Ty ܲˆÏÓ˚ˆÏåÈö⁄ ~Ó˚ ܲyÓ˚î •ˆÏFåÈ ˆÎñ üyôƒˆÏüÓ˚ ˛õ!Ó˚Óï≈˛ö
•ˆÏ°•z ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚ Ä ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ˛õ!Ó˚Óï˛≈ö âˆÏê˛– xy˛õ!ö !öÿ˛Î˚ xyÓ˚Ä °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ≤Ã!ï˛ú˛!°ï˛
ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe xyüÓ˚y k1x ~Ó˚ xyˆÏà ôöydܲ !ã˛•´ !òˆÏÎ˚!åÈ– ~Ó˚ ܲyÓ˚îñ ~•z ï˛Ó˚D!ê˛ x ~Ó˚ îydܲ !òˆÏܲ
x@ˇÃ§Ó˚ •ˆÏFåÈ–
≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î =îyˆÏB˛Ó˚ ≤ÃÜ,˛ï˛ xÌ≈ Ó%é˛ˆÏï˛ •ˆÏ° xyüyˆÏòÓ˚ §#üyhsˇ ¢ï≈˛yÓ°# (boundary
conditions) ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛ •ˆÏÓ–
ò%•z üyôƒˆÏüÓ˚ §ÇˆÏÎyàfliˆÏ° ≤ÃÌü üyôƒˆÏü ܲîyÓ˚ ˆüyê˛ §Ó˚î ~ÓÇ ê˛yˆÏöÓ˚ xö%≤Ãfli í˛z˛õyLjϢÓ˚ ˆüyê˛ üyö
ÎÌye´ˆÏü !mï˛#Î˚ üyôƒˆÏüÓ˚ ܲîyÓ˚ ˆüyê˛ §Ó˚î Ä ê˛yˆÏöÓ˚ xö%≤Ãfli í˛z˛õyLjϢÓ˚ §üyö •ˆÏÓ– ˆÎˆÏ•ï%˛ xy˛õï˛ˆÏöÓ˚
!òˆÏܲÓ˚ üyôƒˆÏü xy˛õ!ï˛ï˛ Ä ≤Ã!ï˛ú˛!°ï˛ ~•z ò%•z ï˛Ó˚D ~ÓÇ xöƒ !òˆÏܲÓ˚ üyôƒˆÏü ˆÜ˛Ó°üye ≤Ã!ï˛§,ï˛ ï˛Ó˚D
xyˆÏåÈñ ~ˆÏ«˛ˆÏe §#üyhsˇ ¢ï≈˛=!° •°≠
ò%•z üyôƒˆÏüÓ˚ §ÇˆÏÎyàfliˆÏ°ñ xÌ≈yÍ x = 0 !Ó®%ˆÏï˛
1. xy˛õ!ï˛ï˛ Ä ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ §Ó˚ˆÏîÓ˚ ˆÎyàú˛° = ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ §Ó˚îñ
2. xy˛õ!ï˛ï˛ Ä ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ ê˛yˆÏöÓ˚ xö%≤Ãfli í˛z˛õyÇ¢=!°Ó˚ ˆÎyàú˛° = ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ ê˛yˆÏöÓ˚
xö%≤Ãfli í˛z˛õyÇ¢–
~•z ò%!ê˛ ¢ï≈˛ ˆÌˆÏÜ ˛õy•zñ
yi(x, t)
|
x=0+ yr(x, t)
|
x = 0 yt(x, t)
|
x = 0 ...8.27
~ÓÇ
000it r
xxxyy y |T|T|xxx ===
Τ∂∂ ∂ −−=−∂∂∂
...8.28
§ü#ܲÓ˚î 8.24, 8.25, 8.26 ÓƒÓ•yÓ˚ ܲˆÏÓ˚ 8.27 ¢ï≈˛ ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ñ
aisin
ω
0t + arsin
ω
0t = atsin
ω
0t
xÌÓy ai + ar = at ...8.29
xyÓyÓ˚ 8.28 ¢ï≈˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õyˆÏÓöñ
aik1Tcos
ω
t – ark1Tcos
ω
t =ark2Tcos
ω
t
xÌÓy k1T(ai – ar) = k2Tat ...8.30
223
xyüÓ˚y çy!ö ˆÎñ
k1 T =
11
222 == TTZ ππνπνλυ
(8.15a §ü#ܲÓ˚î xö%ÎyÎ˚#)
~áyˆÏö Z1 = ≤ÃÌü üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yô– ~ܲ•zû˛yˆÏÓ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚
k2T = 2π
ν
Z2 ˆÎáyˆÏö Z2
→
!mï˛#Î˚ üyôƒü ܲï,≈˛Ü˛ ≤Ãò_ ≤Ã!ï˛ˆÏÓ˚yô
~•z ú˛°=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y §ü#ܲÓ˚î 8.30 - ˆÜ˛ !°áˆÏï˛ ˛õy!Ó˚ñ
2
πν
Z1(ai – ar) = 2
πν
Z2at
xÌÓy Z1(ai– ar) = Z2at ...8.31
(8.29) ~ÓÇ (8.31) §ü#ܲÓ˚î ˆÌˆÏܲ xy˛õ!ö §•ˆÏç•z
r
i
aa
~ÓÇ
t
i
aa
xö%˛õyï˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– ~•z
xö%˛õyï˛=!° ˆÌˆÏܲ xyüÓ˚y xy˛õ!ï˛ï˛ !ÓhflÏyˆÏÓ˚Ó˚ ܲï˛ê˛y xÇ¢ ≤Ã!ï˛ú˛!°ï˛ •ˆÏFåÈ xyÓ˚ ܲï˛ê˛y xÇ¢ ≤Ã!ï˛§,ï˛
•ˆÏFåÈ ï˛yÓ˚ üyö çyöˆÏï˛ ˛õy!Ó˚– ~•z xö%˛õyï˛=!°ˆÏܲ Ó°y •Î˚ ≤Ã!ï˛ú˛°ö ~ÓÇ ≤Ã!ï˛§Ó˚ˆÏîÓ˚ (transmission)
!ÓhflÏyÓ˚ =îyB˛– xyüÓ˚y ~ ò%!ê˛ˆÏܲ R12 ~ÓÇ T12 myÓ˚y ≤Ãܲy¢ ܲÓ˚Ó≠
R12 =
12
12
r
t
aZZaZZ
− =+
...8.32
~ÓÇ T12 = 1
1 2
2t
i
a Za Z Z
=+ ...8.33
xyüÓ˚y ˆòáˆÏï˛ ˛õy!FåÈ ˆÎñ ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î !ÓhflÏyÓ˚ =îyB˛ üyôƒü ò%!ê˛Ó˚ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚
ܲˆÏÓ˚–
~áö 8.32 ~ÓÇ 8.33 §ü#ܲÓ˚î=!° ˆÌˆÏܲ ≤ÃyÆ ú˛°=!°Ó˚ xÌ≈ !öî≈ˆÏÎ˚Ó˚ ˆã˛‹Ty ܲÓ˚y Îyܲ–
(i) üˆÏö ܲÓ˚y Îyܲñ ï˛yÓ˚!ê˛Ó˚ ~ܲ≤Ãyhsˇ ˆòÄÎ˚yˆÏ° xyê˛Ü˛yˆÏöy xyˆÏåÈ– ~Ó˚xÌ≈ •°ñ !mï˛#Î˚ üyôƒü!ê˛ xï˛ƒhsˇ
û˛yÓ˚# xÌ≈yÍ Z2 = ∞– ~ˆÏ«˛ˆÏe R12 = – 1 ~ÓÇ T12 = 0– ú˛°fl∫Ó˚*˛õ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
ar= –ai ~ÓÇ at= 0 xÌ≈yÍñ ˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ xy˛õ!ï˛ï˛ !ÓhflÏyˆÏÓ˚Ó˚ üyö ~ܲ•z !ܲv !ã˛•´ !Ó˛õÓ˚#ï˛
~ÓÇ ≤Ã!ï˛§,ï˛ ˆÜ˛yöÄ ï˛Ó˚D ˆö•z– ~Ó˚ xÌ≈ñ xy˛õ!ï˛ï˛ ï˛Ó˚D xï˛ƒhsˇ âö üyôƒü ˆÌˆÏܲ ≤Ã!ï˛ú˛!°ï˛
•ˆÏ°
π
ò¢yÓ˚ ˛˛õ!Ó˚Óï≈˛ö •Î˚–
(ii) Îáö Z2 > Z1 xÌ≈yÍñ !mï˛#Î˚ ï˛yÓ˚ üyôƒü!ê˛ âöï˛Ó˚ñ R12 ï˛áöÄ îydܲ •ˆÏÓ– ~Ó˚ xÌ≈ñ ~ˆÏ«˛ˆÏeÄ
≤Ã!ï˛ú˛ˆÏ°öÓ˚ ˛õˆÏÓ˚
π
ò¢y ˛õ!Ó˚Óï≈˛ö •ˆÏÓ– ï˛ˆÏÓ ~áyˆÏö xy˛õ!ï˛ï˛ ï˛Ó˚D xÇ¢ï˛ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ
xÇ¢ï˛ ≤Ã!ï˛§,ï˛ •ˆÏÓ–
(iii) Îáö Z2 < Z1, R12 ôöydܲ xÌ≈yÍñ ~ˆÏ«˛ˆÏe ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õˆÏÓ˚ ò¢yÓ˚ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö •Î˚ öy–
~ˆÏ«˛ˆÏeÄ ï˛Ó˚D ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ •Î˚–
224
(iv) Îáö Z1 = Z2, R12 = 0 xÌ≈yÍ ˆÜ˛yöÄ ˛≤Ã!ï˛ú˛°ö âˆÏê˛ öy– ~ˆÏ«˛ˆÏe T12 = 1 ~ÓÇ ai = at
~Ó˚ xÌ≈ ~•z ˆÎñ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ xy˛õ!ï˛ï˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ §üyö– ï˛Ó˚D!ê˛ ~ˆÏ«˛ˆÏe ò%•z
üyôƒˆÏüÓ˚ §#üyˆÏhsˇÓ˚ x!hflÏc xö%û˛Ó ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ öy–
(i), (ii), (iii) ̈Ïܲ ~!ê˛ §%flõ‹T •ˆÏFåÈ Îñ ܲü ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ ~ܲ!ê˛ üyôƒˆÏü ≤ÃÓy•# ï˛Ó˚D í˛zFã˛ ≤Ã!ï˛ˆÏÓ˚yôÎ%_´
ˆÜ˛yöÄ üyôƒˆÏüÓ˚ SÓyï˛y§ ˆÌˆÏܲ ç°V !ÓˆÏû˛òï˛° ˆ˛õÔÑåȈϰ ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚
π
ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö âˆÏê˛–
!ܲv í˛zFã˛ ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ üyôƒˆÏü ≤ÃÓy•# ï˛Ó˚D Îáö ܲü ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ ˆÜ˛yöÄ üyôƒˆÏüÓ˚ !ÓˆÏû˛ò ï˛ˆÏ° ˆ˛õÔÑåÈyÎ˚
Sç° ˆÌˆÏܲ Óyï˛y§V ï˛áö ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ ˆÜ˛yöÄ ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö •Î˚ öy– xy˛õöyÓ˚y !öÿ˛Î˚ °«˛ƒ
ܲˆÏÓ˚ˆÏåÈö ˆÎñ T12 §ü§üÎ˚ ôöydܲ ÌyܲˆÏåÈ– xÌ≈yÍñ ˆÜ˛yö §üˆÏÎ˚•z xy˛õ!ï˛ï˛ ï˛Ó˚ˆÏ°Ó˚ ï%˛°öyÎ˚ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚
ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö •Î˚ öy– !ã˛e (8.5) ~ ~•z ú˛°=!° ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ˆÏåÈ–
!ã˛e 8.5 - ≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ Ó˚*˛õ Îáö (a) ï˛Ó˚D ܲü ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ üyôƒü ˆÌˆÏܲ
í˛zFã˛≤Ã!ï˛ˆÏÓ˚yôÎ%_´ üyôƒˆÏü ÎyˆÏFåÈ ~ÓÇ (b) Îáö !Ó˛õÓ˚#ï˛ âê˛öy âê˛ˆÏåÈ–
8.14 §ü#ܲÓ˚ˆÏî xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ ~ܲ!ê˛ !ö!ò≈‹T ˛ê˛yˆÏö ܲü ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ üyôƒˆÏü ï˛Ó˚DˆÏÓà ˆÓ!¢
•ˆÏÓ– ã˛ï%˛Ì≈ ˆ«˛ˆÏeñ Îáö Z1 = Z2 Î!ò ò%•z!ê˛ ï˛yˆÏÓ˚Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚ m1 Ä m2 •Î˚ñ ï˛ˆÏÓ
12
TTmm
=
Óy
m1 = m2 – ~ˆÏ«˛ˆÏe ï˛yÓ˚ ò%•z!ê˛ •Î˚ ~ܲ•z í˛z˛õyòyö Ä ≤ÃfliˆÏFåȈÏòÓ˚ xÌÓy xy°yòy í˛z˛õyòyˆÏöÓ˚ •ˆÏ°Ä ï˛yˆÏòÓ˚
225
âöc Ä ≤ÃfliˆÏFåȈÏòÓ˚ ~üö ˆÎ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚ §üyö– ~•z xÓfliyÎ˚ ò%!ê˛ ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ !ÓˆÏû˛òï˛°•z ÌyˆÏܲ
öyñ ú˛ˆÏ° ˛≤Ã!ï˛ú˛°öÄ •Î˚ öy–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 6
ò%!ê˛ ï˛yˆÏÓ˚Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚ m ~ÓÇ 4m– ~•z ò%!ê˛ ï˛yÓ˚ˆÏܲ ~ܲ§ˆÏD ç%ˆÏí˛¸ ï˛yˆÏòÓ˚ í˛z˛õÓ˚ T ê˛yö ≤ÈÏÎ˚yà
ܲÓ˚y •°– ˙ ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î !ÓhflÏyÓ˚ =îyB˛ !öî≈Î˚ ܲÓ˚&ö–
8.5.2 xö%˜Ïòâ≈ƒ ï˛Ó˚Dxö%˜Ïòâ≈ƒ ï˛Ó˚Dxö%˜Ïòâ≈ƒ ï˛Ó˚Dxö%˜Ïòâ≈ƒ ï˛Ó˚Dxö%˜Ïòâ≈ƒ ï˛Ó˚D
xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ≤Ã!ï˛ú˛°ö ~ÓÇ ≤Ã!ï˛§Ó˚ˆÏîÓ˚ xyˆÏ°yã˛öyÎ˚ xyüÓ˚y xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ˆÎ í˛z˛õyÎ˚
ÓƒÓ•yÓ˚ ܲˆÏÓ˚!åÈ ˆ§•z ~ܲ•z ˛õk˛!ï˛ xö%§Ó˚î ܲÓ˚Ó– ôÓ˚y Îyܲñ ò%!ê˛ üyôƒü ÎyˆÏòÓ˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yô Z1 ~ÓÇ
Z2 ñ ï˛yˆÏòÓ˚ !ÓˆÏû˛òï˛ˆÏ° (x = 0) ~ܲ!ê˛ ¢∑ï˛Ó˚D °¡∫û˛yˆÏÓ xy˛õ!ï˛ï˛ •°– xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÓ˚ üï˛
xyüÓ˚y xy˛õ!ï˛ï˛ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ ܲîyÓ˚ §Ó˚î 8.24, 8.25 ~ÓÇ 8.26 §ü#ܲÓ˚ˆÏîÓ˚ üï˛
!ï˛ö!ê˛ §ü#ܲÓ˚î !òˆÏÎ˚ ≤Ãܲy¢ ܲÓ˚ˆÏï˛ ˛õy!Ó˚≠
iψ (x, t) = aisin(
ω
0t – k1x) ...8.34
Ψ
r(x, t) = arsin(
ω
0t – k1x) ...8.35
~ÓÇ Ψt(x, t) = at sin( ω0t – k2x) ...8.36
~ˆÏ«˛ˆÏe §#üyhsˇ ¢ï≈˛yÓ°# •ˆÏÓñ
(i) ܲîyÓ˚ §Ó˚î
Ψ
1(x, t) §#üyˆÏhsˇ x!Ó!FåÈߨ (continuous) xÌ≈yÍñ ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° (x = 0)
!ë˛Ü˛ ò%•z !òˆÏܲ §Ó˚ˆÏîÓ˚ ˆüyê˛ üyö §üyö •Î˚–
(ii) ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° !ë˛Ü˛ ò%•z!òˆÏܲÓ˚ ˆüyê˛ x!ï˛!Ó˚_´ ã˛y˛õÄ §üyö •Î˚–
≤ÃÌü ¢ˆÏï≈˛Ó˚ xÌ≈ •° ≠
ai+ ar = at
8.2.2 xÇ¢ xy˛õ!ö ˆòˆÏáˆÏåÈö ˆÎñ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe
∂ ∆=−∂
Exψ ρ
, ˆÎáyˆÏö E = !fli!ï˛fliy˛õܲ
=îyB˛– xy§ˆÏ° E = γ p0 ˆÎáy Ïö
p CCν
γ=
~ÓÇ p0 = §yüƒyÓfliyÓ˚ ã˛y˛õ (Equilibrium pressure) !mï˛#Î˚
¢ï≈˛ ˆÌˆÏܲ ˛õy•z ˆÎñ
0 0 0∂ ∂∂− − = −∂ ∂ ∂
i tr
x x xψ ψψγρ γρ γρ
226
xÌ≈yÍñ ∂ ∂∂+ =∂ ∂ ∂
i tr
x x xψ ψψ
...8.38
(8.38) §ü#ܲÓ˚î ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ñ
– aik1cos
ω
0t + ark1cos
ω
0t = – atk2cos
ω
0t
xÌÓyñ k1(ai – ar) = k2at ...8.39
xyüÓ˚y çy!öñ k1 =
0
1
ων
11 ρν
!òˆÏÎ˚ =î ~ÓÇ û˛yà ܲˆÏÓ˚ k1 =
011012
0 11=Z ωρνω
γρ ρν
SˆÎˆÏ•ï%˛ Z1 = 1 1ρ ν ~ÓÇ
01
1
γρ νρ
=
)
~ܲ•zû˛yˆÏÓ xy˛õ!ö ˆòáyˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ
k2 = 0 2
0
Zωγρ
8.39 §ü#ܲÓ˚ˆÏî ~•z ú˛° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xyüÓ˚y ˛õy•zñ
() 0102
00−= irt
ZZ aaa ωωγργρ
xÌ≈yÍ Z1 (ai – ar) = Z2a1 ...8.40
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ 8.37 Ä 8.40 §ü#ܲÓ˚î ò%!ê˛ ï˛yˆÏÓ˚Ó˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe xyüÓ˚y
ˆÎ 8.29 Ä 8.31 §ü#ܲÓ˚î ò%!ê˛ ˆ˛õˆÏÎ˚!åÈ°yüñ ˆ§=!°Ó˚ xö%Ó˚*˛õ– §%ï˛Ó˚yÇñ ˆ§=!°Ó˚ §y•yˆÏ΃ xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚
ˆ«˛ˆÏe 8.32 Ä 8.33 §ü#ܲÓ˚îmˆÏÎ˚Ó˚ xö%Ó˚*˛õ §¡õÜ≈˛
R12 = 1 2
1 2
r
i
a Z Za Z Z
−=+
Ä T12 = 1
1 2
2t
i
a Za Z Z
=+
˛õyÄÎ˚y ÎyˆÏÓ– ~ˆÏ«˛ˆÏe R12Ä T12xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ≤Ã!ï˛ú˛°ö Ä ≤Ã!ï˛§Ó˚î !ÓhflÏyÓ˚ =îyB˛–
8.6 ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛
xyüÓ˚y çy!ö ˆÎñ ã˛°ï˛Ó˚D üyôƒˆÏüÓ˚ ~ܲ !Ó®% ˆÌˆÏܲ xöƒ !Ó®%ˆÏï˛ ¢!_´ fliyöyhsˇ!Ó˚ï˛ Ü˛ˆÏÓ˚– ~•z ï˛Ó˚D Îáö
!û˛ß¨ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ~•z ¢!_´Ó˚ ܲ# •Î˚ ï˛y ˆòáy•z ~áö xyüyˆÏòÓ˚
227
í z Ïj¢ƒ–
xy˛õöyÓ˚y çyˆÏöö ˆÎñ Îáö ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒ m û˛Ó˚ Î%_´ ~ܲ!ê˛ ï˛yÓ˚ a !ÓhflÏyÓ˚ ~ÓÇ ω0 ˆÜ˛Ô!îܲ ܲ¡õyˆÏB˛
flõ!®ï˛ •Î˚ ï˛áö ï˛yÓ˚ ~ܲܲ ˜òâ≈ƒ !˛õå%È ¢!_´
E =
12
ma2
ω
02 ...8.41
üˆÏö ܲÓ˚y Îyܲñ ï˛Ó˚D
ν
ˆÓˆÏà §MÈ˛y!Ó˚ï˛ •ˆÏFåÈ– ï˛y•ˆÏ° ˆÎ •yˆÏÓ˚ ï˛yˆÏÓ˚Ó˚ üˆÏôƒ !òˆÏÎ˚ ¢!_´ ≤ÃÓy!•ï˛ •ˆÏFåÈ
ï˛y ˆ˛õˆÏï˛ ˆàˆÏ° xyüyˆÏòÓ˚ ¢!_´Ó˚ Ó˚y!¢üy°yˆÏܲ (expression) ˆÓà !òˆÏÎ˚ =î ܲÓ˚ˆÏï˛ •ˆÏÓ– §%ï˛Ó˚yÇ xy˛õ!ï˛ï˛
ï˛Ó˚ˆÏDÓ˚ §ˆÏD ˆÎ •yˆÏÓ˚ ¢!_´ !ÓˆÏû˛ò ï˛ˆÏ° ~ˆÏ§ ˛õˆÏí˛¸ ï˛yˆÏܲ xyüÓ˚y !ö¡¨Ó˚*ˆÏ˛õ !°áˆÏï˛ ˛õy!Ó˚–
ρ
i=
12
m1ai2
ω
02
ν
=
12
Z1
ω
02ai
2 ...8.42
[ §)e 8.15b ˆï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ ˆÎ Z = mv ]
~ܲ•z û˛yˆÏÓñ ˆÎ •yˆÏÓ˚ ¢!_´ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ §ˆÏD !ÓˆÏû˛òï˛° ˆåȈÏí˛¸ ÎyÎ˚ñ ˆ§=!°
ÎÌye´ˆÏü
Pr =
12
Z1
ω
02ar
2 ...8.43
~ÓÇ Pt = 12
Z2 ω02at
2 ...8.44
xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈ ar = ai
12
12
ZZZZ
−+
~ÓÇ at = ai1
1 2
2ZZ Z+
~•z §¡õÜ≈˛=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˛õyÄÎ˚y
ÎyÎ˚ ≠
Pr = 12
Z1
ω
02
212
12
ZZZZ
⎛⎞ −⎜⎟ + ⎝⎠
ai2 ...8.45
~ÓÇ Pt =
12
Z2
ω
02
21
12
2ZZZ
⎛⎞⎜⎟ + ⎝⎠
ai2 ...8.46
í˛z˛õˆÏÓ˚Ó˚ ú˛°=!° ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛ RE Ä TE !öî≈ˆÏÎ˚ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚–
RE = =
212
12
r
i
PZZPZZ
⎛⎞ − =⎜⎟ + ⎝⎠
...8.47
TE= =
()12
212
4 t
i
PZZPZZ
=+
...8.48
üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛° ≤Ã!ï˛ú˛!°ï˛ ¢!_´Ó˚ •yÓ˚
üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° xy˛õ!ï˛ï˛ ¢!_´Ó˚ •yÓ˚
üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛° ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ •yÓ˚
üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° xy˛õ!ï˛ï˛ ¢!_´Ó˚ •yÓ˚
228
§ü#ܲÓ˚î 8.47 ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎ Z1 = Z2 •ˆÏ° RE = 0; ~Ó˚ xÌ≈ñ Îáö ≤Ã!ï˛ˆÏÓ˚yô
!ü°ö (impedance matching) •Î˚ñ ï˛áö ˆÜ˛yöñ ¢!_´ ≤Ã!ï˛ú˛!°ï˛ •Î˚ öy– ¢!_´ ˛õ!Ó˚Ó•ˆÏöÓ˚ ˆ«˛ˆÏe ~•z
≤Ã!ï˛ˆÏÓ˚yô !ü°ö xï˛ƒhsˇ =Ó˚&û)˛!üܲy ˛õy°ö ܲˆÏÓ˚– ¢!_´ ˛õ!Ó˚Ó•öܲyÓ˚# ò#â≈ ˆÜ˛Ó‰°‰ ÈÙÈ ~ ≤ÈÏï˛ƒÜ˛!ê˛ ˆçyˆÏí˛¸Ó˚V
çyÎ˚àyÎ˚ ~•z ≤Ã!ï˛ˆÏÓ˚yô !ü°ö xï˛ƒhsˇ çÓ˚&Ó˚#ñ öy •ˆÏ° ≤Ã!ï˛ú˛°ˆÏöÓ˚ ú˛ˆÏ° Ó‡ ˛õ!Ó˚üyî ¢!_´Ó˚ «˛Î˚ âê˛ˆÏï˛ ˛õyˆÏÓ˚–
~ܲ!ê˛ °yí˛zí˛ !flõܲyÓ˚ ˆÌˆÏܲ ˆÜ˛yöÄ âˆÏÓ˚Ó˚ üˆÏôƒ ¢∑ åÈ!í˛¸ˆÏÎ˚ ˆòÄÎ˚yÓ˚ çöƒÄ ≤Ã!ï˛ˆÏÓ˚yô !ü°ö ≤ÈÏÎ˚yçö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 7
!Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yô Z1 ~ÓÇ Z2 Î%_´ ò%•z!ê˛ üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° ~ܲ!ê˛ xö%≤Ãfli ï˛Ó˚D xy˛õ!ï˛ï˛ •ˆÏ° ˆòáyö
ˆÎ ¢!_´ §ÇÓ˚!«˛ï˛ •Î˚–
xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe ¢!_´Ó˚ •hflÏyhsˇˆÏÓ˚Ó˚ !•§yˆÏÓ §ã˛yÓ˚ã˛Ó˚ ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– àƒy§#Î˚
üyôƒˆÏü ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y≠
I = 12
ρ
a2
ω
02
ν
= 2
π
2
ν
2a2Z [ï˛Ó˚D ≤ÃÓyˆÏ•Ó˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ˆÌˆÏܲ ] ...8.49
ˆÎáyˆÏö üyôƒˆÏüÓ˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yôñ Z = ρν
(§)e 8.21)
xï˛~Óñ xy˛õ!ï˛ï˛ñ ˛≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ ≤ÃyÓ°ƒ •ˆÏÎ˚ ÎÌye´ˆÏü
Ii = 2
π
2
γ
2 ai2Z1 ...8.50
Ir = 2
π
2
γ
2 ar2Z1 ...8.51
It = 2
π
2
γ
2 ar2Z2 ...8.52
~•z !ï˛ö!ê˛ §ü#ܲÓ˚ˆÏî ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆòáyˆÏöy ÎyÎ˚ ˆÎ ≤Ã!ï˛ú˛!°ï˛ ¢!_´Ó˚ =îyB˛ ≠
RE =
212
12
r
i
IZZIZZ
⎛⎞ − =⎜⎟ + ⎝⎠
...8.53
~ÓÇ ≤Ã!ï˛§,ï˛ ¢!_´Ó˚ =îyB˛ ≠
TE =
()12
212
4 =+
t
i
IZZIZZ
...8.54
229
°«˛ƒ ܲˆÏÓ˚ ˆòá%öñ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÄ xyüÓ˚y ~ܲ•z §ü#ܲÓ˚î ˆ˛õˆÏÎ˚!åÈ°yü–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 8
ç° ~ÓÇ Ü˛yˆÏã˛Ó˚ !ÓˆÏû˛òï˛ˆÏ° ¢∑ï˛Ó˚D xy˛õ!ï˛ï˛ •ˆÏ° ˆòáyö ˆÎñ 66% ¢!_´ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÓ– ç°
~ÓÇ Ü˛yˆÏã˛Ó˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yô ÎÌye´ˆÏü 1·43 × 106 Nm–3s ~ÓÇ 13·9 × 106 Nm–3s
8.7 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~•z ~ܲˆÏܲ xyüÓ˚y ~ܲüy!eܲ ã˛°ï˛Ó˚D !•§yˆÏÓ ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚D ~ÓÇ !fli!ï˛fliy˛õܲ üyôƒˆÏü xö%˜Ïòâ≈ƒ
ï˛Ó˚ˆÏDÓ˚ ܲÌy !ÓˆÏÓã˛öy ܲˆÏÓ˚!åÈ–
~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ ~ܲüy!eܲ xö%≤Ãfli ï˛Ó˚DˆÏܲ !öˆÏã˛Ó˚ §ü#ܲÓ˚î !òˆÏÎ˚ ≤Ãܲy¢ ܲÓ˚y ÎyÎ˚ÈÙÙÙÈ
È( ) ( )2 2
22 2, ,∂ ∂
=∂ ∂
x t x tt x
ψ ψν ˆÎáyˆÏö Ψ(x, t) •ˆÏFåÈ ï˛yÓ˚!ê˛Ó˚ §Ó˚î ~ÓÇ
ν
ï˛Ó˚DˆÏÓà–
ê˛yöy ï˛yˆÏÓ˚ §,‹T ï˛Ó˚ˆÏDÓ˚ ˆÓà
=Tm
ν
ˆÎáyˆÏö T = ï˛yˆÏÓ˚Ó˚ ê˛yö ~ÓÇ m = ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚–
!fli!ï˛fliy˛õܲ üyôƒˆÏü xö%˜Ïòâ≈ƒ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe = Eνρ ˆÎáyˆÏö E = üyôƒˆÏüÓ˚ !fli!ï˛fliy˛õܲ =îyB˛ ~ÓÇ
ρ
= üyôƒˆÏüÓ˚ âöc– Óyï˛yˆÏ§ ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe üyôƒˆÏüÓ˚ ≤çyÓ˚î Ä xö%û˛Óö Ó˚&k˛ï˛y˛õ ≤Ã!e´Î˚yÎ˚ âˆÏê˛–
~•z ï˛Ó˚ˆÏDÓ˚ ˆÓàÈÙÙÙ
=p γ νρ
γ
= !fliÓ˚ ã˛y˛õ Ä !fliÓ˚ xyÎ˚ï˛ˆÏö xyˆÏ˛õ!«˛Ü˛ ï˛y˛õmˆÏÎ˚Ó˚ xö%˛õyï˛– ܲ!ë˛ö ˛õòyˆÏÌ≈Ó˚ òˆÏ[˛ ≤ÃÓy!•ï˛
¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe
=Y νρ
ˆÎáyˆÏö Y = üyôƒˆÏüÓ˚ •zÎ˚Ç =îyB˛–È
Îáö ~ܲ!ê˛ ï˛Ó˚D üyôƒˆÏüÓ˚ üˆÏôƒ x@ˇÃ§Ó˚ •Î˚ñ ï˛áö üyôƒü ï˛yˆÏܲ Óyôy ˆòÎ˚– ï˛Ó˚D˛õˆÏÌÓ˚ ~•z ÓyôyˆÏܲ
ï˛Ó˚D ≤Ã!ï˛ˆÏÓ˚yô Ó°y •Î˚– ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe üyôƒˆÏüÓ˚ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ üyö
Z =
== TTmmν ν
230
!fli!ï˛fliy˛õܲ üyôƒˆÏü ¢∑ï˛Ó˚ˆÏDÓ˚ !Ó!¢‹T ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ üyö≠
Z =
== EEρρνν
!müy!eܲ ï˛Ó˚D ≤ÃÓyˆÏ•Ó˚ §ü#ܲÓ˚î ≠
()()22
2222
,,
2⎛⎞ ∂∂∂=+ ⎜⎟ ∂∂∂ ⎝⎠
rtrt
txyψ
νψ
~•z §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≠
Ψ (r, t) = asin(
ω
0t – k.r)
~áyˆÏö k.r = kxx+ kyy ~ÓÇ k2= kx2 + ky
2
~ܲ!ê˛ üyôƒˆÏüÓ˚ üˆÏôƒ ≤ÃÓy!•ï˛ ï˛Ó˚D Îáö xöƒ ≤Ã!ï˛ˆÏÓ˚yô §¡õߨ ~ܲ!ê˛ !mï˛#Î˚ üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ°
xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ï˛yÓ˚ !ܲå%È xÇ¢ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ !ܲå%È xÇ¢ ≤Ã!ï˛§,ï˛ •Î˚– ~•z ≤Ã!ï˛ú˛!°ï˛ Ä ≤Ã!ï˛§,ï˛
!ÓhflÏyÓ˚ =îyB˛ •ˆÏFåÈ ÎÌye´ˆÏüÈÙÙÙÈ
R12= 12
12
ZZZZ
−+ ~ÓÇ T12 = 1
1 2
2ZZ Z+
!ö¡¨ ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ üyôƒü ˆÌˆÏܲ ≤ÃÓy!•ï˛ ~ܲ!ê˛ ï˛Ó˚D Îáö í˛zFã˛ ≤Ã!ï˛ˆÏÓ˚yôÎ%_´ ~ܲ!ê˛ üyôƒüï˛ˆÏ°
≤Ã!ï˛ú˛!°ï˛ •Î˚ ï˛áö ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ π ò¢y ˛õ!Ó˚Óï≈˛ö âˆÏê˛–
8.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. 500Hz ܲ¡õöÎ%_´ ~ܲ!ê˛ §%ˆÏÓ˚Ó˚ âˆÏÓ˚Ó˚ í˛z£èï˛yÎ˚ (15°c) ï˛Ó˚D˜Ïòâ≈ƒ 70m– ≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ
Óyï˛yˆÏ§Ó˚ âöc 1.29kgm–3 • Ï° γÈÙÈ~Ó˚ üyö !öî≈Î˚ ܲÓ˚&ö–
2. û)˛Ü˛ˆÏ¡õÓ˚ ú˛ˆÏ° ~ܲ!ê˛ xö%˜Ïòâ≈ƒ !ӈϫ˛yû˛ 2.5 !ü!öˆÏê˛ 103km ˆà°– Î!ò ˛õÿÏÓ˚Ó˚ (rock)
âöc 2·7 × 103kgm–3 •Î˚ñ ï˛y•ˆÏ° ˛õÿÏÓ˚Ó˚ xyÎ˚ï˛ö !ÓÜ,˛!ï˛ =îyB˛ !öî≈Î˚ ܲÓ˚&ö–
3. Óyï˛yˆÏ§ ≤ÃÓy!•ï˛ ~ܲ!ê˛ ¢∑ï˛Ó˚D °¡∫û˛yˆÏÓ ç°˛õ,ˆÏ¤˛ xyâyï˛ Ü˛Ó˚°– !mï˛#Î˚ üyôƒˆÏü ≤Ã!Ó‹T ¢∑ï˛Ó˚ˆÏD
!ÓhflÏyÓ˚ ~ÓÇ xy˛õ!ï˛ï˛ ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ xö%˛õyï˛ !öî≈Î˚ ܲÓ˚&ö– ˆòÄÎ˚y xyˆÏåÈñ Óyï˛yˆÏ§Ó˚ âöc
231
1·29kgm–3, çˆÏ°Ó˚ âöc 1000kgm–3 ~ÓÇ Óyï˛yˆÏ§ Ä çˆÏ° ¢ˆÏ∑Ó˚ ˆÓà ÎÌye´ˆÏü 350ms–1
~ÓÇ 1500ms–1–
4. ~ܲ!ê˛ ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ xÓܲ° §ü#ܲÓ˚î !öî≈Î˚ ܲÓ˚&ö ~ÓÇ ï˛yÓ˚ ˆÌˆÏܲ ˙ ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚
Ó˚y!¢üy°y!ê˛ˆÏܲ !°á%ö–
5. ˆòáyö ˆÎñ ≤ÃÓy•#ˆÏï˛ xö%˜Ïòâ≈ƒ ¢∑ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î ≠
2
22 ·2 ∂∂ =
∂∂E
txψψ
ρ
ˆÎáy Ïö Ψ = ≤ÃÓy•#Ó˚
ӛܲîyÓ˚ §Ó˚îñ E = xyÎ˚ï˛ö =îyB˛ Ä
ρ
= âöc– ~•z §ü#ܲÓ˚î ˆÌˆÏܲ ÓyÎ˚%ˆÏï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÓˆÏàÓ˚
Ó˚y!¢üy°y !öî≈Î˚ ܲÓ˚&ö–
6. í˛z˛õˆÏÓ˚Ó˚ ≤Èϟ¿Ó˚ §ü#ܲÓ˚î!ê˛ ~ܲ!ê˛ §%£Ïü ôyï˛Ó òˆÏ[˛Ó˚ ˆ«˛ˆÏe ܲ#û˛yˆÏÓ ˆ°áy ÎyˆÏÓ⁄
7. üyôƒˆÏüÓ˚ ï˛Ó˚DÈÙÈ≤Ã!ï˛ˆÏÓ˚yô ܲyˆÏܲ ӈϰ⁄ ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚D ~ÓÇ !fli!ï˛fliy˛õܲ üyôƒˆÏü
¢∑ï˛Ó˚DÈÙÙÙÈ~•z ò%•z ˆ«˛ˆÏe ï˛Ó˚D ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ Ó˚y!¢üy°y !öî≈Î˚ ܲÓ˚&ö–
8. ~ܲ!ê˛ ï˛Ó˚D Îáö ~ܲ üyôƒü ˆÌˆÏܲ xöƒ ~ܲ!ê˛ üyôƒˆÏüÓ˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ≤Ã!ï˛ú˛°ö Ä
≤Ã!ï˛§Ó˚î =îyB˛ üyö Ü˛ï˛ •Î˚ñ ò%•z üyôƒˆÏüÓ˚ ≤Ã!ï˛ˆÏÓ˚yˆÏôÓ˚ !•§yˆÏÓ !öî≈Î˚ ܲÓ˚&ö–
8.9 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#
1, xüÓ˚y çy!ö ˆÎñ ê˛yöy ï˛yˆÏÓ˚ ï˛Ó˚D ˆÓàñ
mΤ ν=
– T Ä m üyö Ó!§ˆÏÎ˚ ˛õyÄÎ˚y ÎyÎ˚ 3 110
10 kgmNν − −= = 100ms–1
2. xyüÓ˚y çy!öñ Yνρ
=
üyö Ó!§ˆÏÎ˚
11 2
11·95 10 Nm
7800kgmν
−
−×= = 5 × 103ms–1
3. m = 2·0 × 10–3kgm–1, T = 80N
232
(8.15a) §ü#ܲÓ˚î ˆÌˆÏܲñ Z = Tm
üyö Ó!§ˆÏÎ˚ Z =
() 31 802·010kgm N−− ××
= 0·4Nm–1s
F0= Z × υ0= 0·4 × 0·25 = 0·1 N
4. 8.21 §ü#ܲÓ˚î xö%§yˆÏÓ˚ñ
Z = ρ
ν
Z = (1·29kgm–3) × (332ms–1)
= (4·28 × 102)kgm–2s–1
= 428Nm–3s
çˆÏ°Ó˚ ¢∑ ≤Ã!ï˛ˆÏÓ˚yô ˆÓ!¢ •ˆÏÓ– ˆÜ˛ööyñ ≤Ãüyî í˛z£èï˛y Ä ã˛yˆÏ˛õ çˆÏ°Ó˚ âöc Óyï˛yˆÏ§Ó˚ ï%˛°öyÎ˚ ≤ÃyÎ˚
800 =î ˆÓ!¢– çˆÏ° ¢ˆÏ∑Ó˚ ˆÓàÄ Óyï˛yˆÏ§ ¢ˆÏ∑Ó˚ ˆÓˆÏàÓ˚ ≤ÃyÎ˚ 5 =î–
5. !eüy!eܲ ï˛Ó˚ˆÏD Ó° x, y ~ÓÇ z x«˛ ÓÓ˚yÓÓ˚ !öÓ˚ˆÏ˛õ«˛û˛yˆÏÓ Ü˛yç ܲÓ˚ˆÏÓ–
§ü#ܲÓ˚î!ê˛ •ˆÏÓ
() 222
2,
(,)∂
=∇∂
rtrt
tψ
νψ
ˆÎáyˆÏöñ r2 = x2 + y2 + z2
2 2 22
2 2 2x y z∂ ∂ ∂∇ = + +∂ ∂ ∂
~ÓÇñ
2E νρ
=
6. ≤Ã!ï˛ˆÏÓ˚yôñ ê˛yö ~ÓÇ ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛ˆÏÓ˚Ó˚ üˆÏôƒ §¡õÜ≈˛ !ö¡¨Ó˚*˛õ ≠
Z =
mT∴
ï˛yÓ˚ ò%!ê˛Ó˚ çöƒ §¡õÜ≈˛ •ˆÏ° Z1 =
mT
~ÓÇ Z2 =
4mT 11
22
12
ZmZm
==
233
§ü#ܲÓ˚î 8.32 ~ÓÇ 8.33 xö%§yˆÏÓ˚ñ
R12 =
12
12
r
i
aZZaZZ
− =+
~ÓÇ T12 = 1
1 2
2t
i
a Za Z Z
=−
∴R12 =
11
221 12
2
11113 1
ZZ
ZZ
−−==−
+ +
~ÓÇ T12= 1
2
12
2 231
t
i
Za Z
ZaZ
= =+
R12 îydܲ •ÄÎ˚yÓ˚ xÌ≈ •° ˆÎ ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° π ò¢yÓ˚ ˛õ!Ó˚Óï≈˛ö •ˆÏÎ˚ˆÏåÈ–
7. 8.42 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õy•zñ ˆÎ •yˆÏÓ˚ ¢!_´ ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° ˆ˛õÔÑåÈÎ˚ ï˛y •ˆÏFåÈñ
Pi =
12
Z1
ω
02 ai
2
~ÓÇ ˆÎ •yˆÏÓ˚ ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ ≤Ã!ï˛§,ï˛ ï˛Ó˚ˆÏDÓ˚ §ˆÏD ¢!_´ ò%•z üyôƒˆÏüÓ˚ !ÓˆÏû˛òï˛ˆÏ° ˆåȈÏí˛¸ ÎyÎ˚ñ ï˛y
•° ≠
Pr + Pt =
12
Z1
ω
02ar
2 +
12
Z2
ω
02at
2
ar ~ÓÇ at Ó˚ üyö ≠
ai + ar = at ~ÓÇ Z1(ai – ar) = Z2at
ˆÌˆÏܲ Ó!§ˆÏÎ˚
Pr + Pt =
12
Z1
ω
02
212
12
ZZZZ
⎛⎞ −⎜⎟ + ⎝⎠
ai +
12
Z2
ω
02
22 1
12
2i
ZaZZ
⎛⎞⎜⎟ + ⎝⎠
= ( )
22 21 2 1 2
1 0 21 2 1 2
412 i
Z Z Z ZZ aZ Z Z Z
ω⎡ ⎤⎛ ⎞−⎢ ⎥+⎜ ⎟+⎢ ⎥+⎝ ⎠⎣ ⎦
234
=
()()
12 2210
12
12i
ZZZa
ZZω
⎡⎤ +⎢⎥ + ⎢⎥ ⎣⎦
=
12
Z1
ω
02ai
2 = Pi
ˆÎˆÏ•ï%˛ñ ˆÎ •yˆÏÓ˚ ¢!_´ !ÓˆÏû˛òï˛ˆÏ° ˆ˛õÔÑåÈyˆÏFåÈ ˆ§•z •yˆÏÓ˚•z !ÓˆÏû˛òï˛° ˆåȈÏí˛¸ ÎyˆÏFåÈñ xyüÓ˚y Ó°ˆÏï˛ ˛õy!Ó˚
ˆÎñ ¢!_´ §ÇÓ˚!«˛ï˛ •ˆÏFåÈ–
8. ≤Ã!ï˛ú˛!°ï˛ ¢!_´Ó˚ =îyB˛ •ˆÏFåÈñ
R =
212
12
zzzz
⎛⎞ −⎜⎟ + ⎝⎠
=
()()
612
611·4313·910Nms1·4313·910Nms
−
−
⎡⎤ −×⎢⎥
+× ⎢⎥ ⎣⎦
=
2 12·4715·33
⎛⎞⎜⎟ ⎝⎠
= 0·66
~Ó˚ xÌ≈ñ Îáö ¢∑ï˛Ó˚D ç° Ä Ü˛yˆÏã˛Ó˚ §ÇˆÏÎyàfliˆÏ° xyâyï˛ Ü˛ˆÏÓ˚ ï˛áö ¢!_´Ó˚ 66% ≤Ã!ï˛ú˛!°ï˛ •Î˚–
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. xyüÓ˚y çy!öñ xyò¢≈ àƒyˆÏ§Ó˚ §ü#ܲÓ˚î ˆÌˆÏܲ 00
0
rr
r
PV PVTT
= ˆÎáyˆÏö ˛õyò!ã˛•´!ê˛ âˆÏÓ˚Ó˚ xÓfliyÎ˚ ~ÓÇ
o ˛õyò!ã˛•´!ê˛ ≤Ãüyî ã˛y˛õ Ä í˛z£èï˛yÎ˚ ã˛y˛õñ xyÎ˚ï˛ö Óy í˛z£èï˛y !öˆÏò≈¢ ܲÏÓ˚ˆÏåÈ– ˆÎˆÏ•ï%˛ xyÎ˚ï˛ö V Ä âöc
ρ ÓƒyhflÏyö%˛õyï˛#ñ
0
00
r
rr
P PTT ρρ
=
∴
0
00
27315 ··273
rr
r
P PTT ρρ
+ ⎛⎞ =⎜⎟ ⎝⎠
= 1·055·
0
0
Pρ
xyÓyÓ˚ ¢ˆÏ∑Ó˚ ˆÓà v = v
λ
=
γρ
r
r
P ∴γ
= (
νλ
)2 ÷
r
r
Pρ
(
νλ
)2·
0
0 1·055ρ
ρ
235
=
()2
55000·7
1·0551·01310×
××
× 1·29 (P0= 1·013 ×105 Nm–2)
= 1·48
2. û)˛Ü˛¡õö ç!öï˛ ï˛Ó˚ˆÏDÓ˚ ˆÓà
33 1010m2·560s
ν× =×
= 6·7 × 10–3ms–1
ˆÎˆÏ•ï%˛
E νρ
=
xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ E = ν 2
ρ
üyö Ó!§ˆÏÎ˚ ˛õy•z
E = (6·7 × 103ms–1)2 × 2·7 × 103kgm–3
= 1·2 × 1011Nm–2
3. ˆòÄÎ˚y xyˆÏåÈ
ρ
1 = 1·29kgm–3
ρ
2 = 1000kgm–3
ν
1 = 350ms–1
ν
2 = 1500ms–1
ˆÎˆÏ•ï%˛ ¢∑ï˛Ó˚D xö%˜ÏòÏâ≈ƒñ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ (8.33 ˆÌˆÏܲV
112
1 122
2 2
1t
i
ZaZZ
Z aZZZ
==++
ˆÎˆÏ•ï%˛ Z = ρ
ν
ñ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚
111
222
ZZ
ρνρν
=
236
= 3 1
3 11·29kgm 350ms
1000kgm 1500ms
− −
− −××
= 3·01 ×10-4
§%ï˛Ó˚yÇñ ( )( )
4
4
2 3·01 10
1 3·01 10
−
−
×=
+ ×t
i
aa = 6·02 × 10-4
°«˛ƒ ܲÓ˚&ö ˆÎñ çˆÏ°Ó˚ ≤Ã!ï˛ˆÏÓ˚yô xˆÏöܲ ˆÓ!¢ •ÄÎ˚yÎ˚ñ Óyï˛yˆÏ§Ó˚ ¢∑ï˛Ó˚ˆÏDÓ˚ x!ï˛ x“ xÇ¢•z çˆÏ°Ó˚
üˆÏôƒ ≤ÈÏÓ¢ ܲˆÏÓ˚–
4. ~Ó˚ í˛z_Ó˚ xy˛õ!ö 8.2.1 xLjϢ ˛õyˆÏÓö–
5. 8.2.2 xLjϢ ~Ó˚ í˛z_Ó˚!ê˛ ˛õyÄÎ˚y ÎyˆÏÓ–
6. 8.2.3 xÇ¢ o‹TÓƒ–
7. 8.3 xÇ¢ o‹TÓƒ–
8. 8.5 xÇ¢ o‹TÓƒ–
237
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 9 : ê˛yöy ï˛yˆÏÓ˚Ó˚ flõ®öê˛yöy ï˛yˆÏÓ˚Ó˚ flõ®öê˛yöy ï˛yˆÏÓ˚Ó˚ flõ®öê˛yöy ï˛yˆÏÓ˚Ó˚ flõ®öê˛yöy ï˛yˆÏÓ˚Ó˚ flõ®ö (Vibration of stretched strings)
9.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
9.2 ê˛yöy ï˛yÓ˚
9.2.1 xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚–
9.3 ܲ!£Ï≈ï˛ ï˛yÓ˚ (Plucked string)
9.4 xy•ï˛ ï˛yÓ˚ (Struck string)
9.5 åÈí˛¸ê˛yöy ï˛yÓ˚ (Bowed string)
9.5.1 ˆ•°‰ü‰ˆÏ•y°Í§ÈÙÈ~Ó˚ §)e
9.5.2 åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°‰üˆÏ•y°Í§‰ ~Ó˚ ï˛_¥
9.6 §yÓ˚yÇ¢
9.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
9.8 í˛z_Ó˚üy°y
9.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
xy˛õöyÓ˚y !öÿ˛Î˚•z §Óy•z ï˛yˆÏÓ˚Ó˚ Óyçöy ÷öˆÏï˛ û˛y°ÓyˆÏ§ö ~ÓÇ xˆÏöˆÏܲ •Î˚ï˛ !öˆÏçÓ˚y Óyçöy Óy!çˆÏÎ˚Ä
ÌyˆÏܲö– ï˛yˆÏÓ˚Ó˚ ÓyçöyÓ˚ ˆ«˛ˆÏe xy˛õöyÓ˚y xÓ¢ƒ•z °«˛ ܲˆÏÓ˚ ÌyܲˆÏÓö ˆÎñ §ÓÓ˚ܲü ï˛yˆÏÓ˚Ó˚ ÓyòƒÎsf ÓyçyÓyÓ˚
û˛!D ~ܲÓ˚ܲü öÎ˚– ˆÜ˛yöÄ ˆÜ˛yöÄ ÎˆÏsf ï˛yˆÏÓ˚Ó˚ ܲï˛Ü˛=!° !ӈϢ£Ï !Ó®%ˆÏï˛ xyâyï˛ Ü˛ˆÏÓ˚ ~ܲ!ê˛ !ӈϢ£Ï §%Ó˚ñ
§,!‹T ܲÓ˚y •Î˚– !˛õÎ˚yˆÏöy Ä §vÓ˚ ~ çyï˛#Î˚ ÓyòƒÎsf– xyÓyÓ˚ xyüÓ˚y Îáö ˆÜ˛yöÄ ˆ§ï˛yÓ˚ ÓyòˆÏܲÓ˚ Óyçöy
÷!öñ ï˛áö ˆò!á ˆÎ !ï˛!ö ï˛yÓ˚!ê˛ˆÏܲ !ӈϢ£Ï !ӈϢ£Ï fliyˆÏö ˆã˛ˆÏ˛õ ôÓ˚ˆÏåÈö ~ÓÇ xöƒ •yˆÏï˛ xöƒ ~ܲ!ê˛ !ӈϢ£Ï
fliyˆÏö ï˛yÓ˚!ê˛ˆÏܲ ˛õyˆÏ¢Ó˚ !òˆÏܲ ˆê˛ˆÏö ôˆÏÓ˚ Ä ˆåȈÏí˛¸ !òˆÏÎ˚ §%Ó˚ §,!‹T ܲÓ˚ˆÏåÈö– ~!ê˛ Ü˛ï˛Ü˛ê˛y ôö%ˆÏܲÓ˚ ê˛ÇܲyÓ˚ ˆòÄÎ˚yÓ˚
üï˛– ~•z ôÓ˚ˆÏöÓ˚ ï˛yˆÏÓ˚ ê˛ÇܲyÓ˚ §%Ó˚ §,!‹T ܲÓ˚yÓ˚ ΈÏsfÓ˚ üˆÏôƒ !àê˛yÓ˚Ä ˛õˆÏí˛¸– °«˛ƒ ܲˆÏÓ˚ ˆòáˆÏÓöñ !àê˛yˆÏÓ˚ ˆÎ
åÈÎ˚!ê˛ ï˛yÓ˚ xyˆÏåÈñ ˆ§=!°Ó˚ ≤ÃfliˆÏFåÈò !Ó!û˛ß¨– xyÓyÓ˚ Óyí˛zˆÏ°Ó˚ ~ܲï˛yÓ˚yÎ˚ ï˛yÓ˚ ~ܲ!ê˛ ÌyܲˆÏ°Ä ï˛yÓ˚ ê˛yö
ܲü Óy ˆÓ!¢ ܲÓ˚y ÎyÎ˚– ~åÈyí˛¸y xyÓ˚Ä ~ܲÓ˚ܲˆÏüÓ˚ ê˛yöy ï˛yˆÏÓ˚Ó˚ ÓyòƒÎsf xyˆÏåÈ– ˆÜ˛yöÄ ˆÓ•y°yÓyòܲ Îáö
ˆÓ•y°yÎ˚ §%Ó˚ §,!‹T ܲˆÏÓ˚ö ï˛áöñ !ï˛!ö ï˛yˆÏÓ˚ xyâyï˛ Ü˛ˆÏÓ˚ö öy Óy ï˛yÓ˚ ˆê˛ˆÏöÄ §%Ó˚ §,!‹T ܲˆÏÓ˚ö öy– !ï˛!ö
§%Ó˚ §,!‹T ܲˆÏÓ˚ö ï˛yˆÏÓ˚Ó˚ IJõÓ˚ ~ܲ!ê˛ åÈí˛¸ ˆê˛ˆÏö– ܲyˆÏç•z xyüÓ˚y ˆòáˆÏï˛ ˛õy!FåÈ ˆÎñ !Ó!û˛ß¨ í˛z˛õyˆÏÎ˚ ê˛yöy ï˛yˆÏÓ˚
flõ®ö §,!‹T ܲÓ˚y ÎyÎ˚– ~•z ~ܲˆÏܲ ˆ§•z !Ó!û˛ß¨ í˛z˛õyÎ˚=!° !öˆÏÎ˚ xyüÓ˚y !ÓhflÏy!Ó˚ï˛ ï˛y!_¥Ü˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠
~•z ~ܲܲ!ê˛ xy˛õöyˆÏܲ ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õö §¡∫ˆÏ¶˛ xˆÏöܲê˛y•z çyöˆÏï˛ §y•y΃ ܲÓ˚ˆÏÓ– ~!ê˛ ˛õí˛¸ˆÏ° xy˛õ!ö
ˆÎ §Ó ܲyç ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆ§=!° •° ≠
238
ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î ≤Ã!ï˛¤˛y ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛=!° ≤ÈÏÎ˚yà ܲˆÏÓ˚ ï˛Ó˚D §ü#ܲÓ˚ˆÏîÓ˚ §yôyÓ˚î §üyôyö !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ܲ¡õö¢#° ê˛yöy ï˛yˆÏÓ˚Ó˚ à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ Ó˚y!¢üy°y !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ܲ!£Ï≈ï˛ñ xy•ï˛ Ä åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î=!° ≤Ã!ï˛!¤˛ï˛ ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö ~ÓÇ ˆ§=!°Ó˚ §y•yˆÏ΃ ~•z !ï˛ö ˆ«˛ˆÏe ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ=!° !ã˛!•´ï˛ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
~=!° åÈyí˛¸yÄ ~•z ~ܲܲ!ê˛ ˛õí˛¸yÓ˚ ˛õÓ˚ xy˛õ!ö öyöy!Óô ï˛yˆÏÓ˚Ó˚ ÓyòƒÎˆÏsfÓ˚ àë˛ö Ä ¢∑˜ÏÓ!¢‹Tƒ §¡∫ˆÏ¶˛
xhsˇò≈,!‹T °yû˛ ܲÓ˚ˆÏÓö ~ÓÇ ˆ§=!° §¡∫ˆÏ¶˛ Ó%!é˛ˆÏÎ˚ Ó°yÓ˚ «˛üï˛y xç≈ö ܲÓ˚ˆÏÓö–
9.2 ê˛yöy ï˛yÓ˚ê˛yöy ï˛yÓ˚ê˛yöy ï˛yÓ˚ê˛yöy ï˛yÓ˚ê˛yöy ï˛yÓ˚
üˆÏö ܲÓ˚y Îyܲñ ~ܲ!ê˛ §%£Ïü ~ÓÇ §¡õ)î≈ öüö#Î˚ ï˛yÓ˚ ~üöû˛yˆÏÓ ê˛yö !òˆÏÎ˚ Ó˚yáy xyˆÏåÈ ÎyˆÏï˛ ï˛yÓ˚ ê˛yö
§Ó§üÎ˚•z T ÌyˆÏܲ– ï˛yˆÏÓ˚Ó˚ ò%•z≤Ãyhsˇ ~üöû˛yˆÏÓ xyê˛Ü˛yˆÏöy xyˆÏåÈ ˆÎñ ˆ§áyˆÏö ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ §Ó˚î §Ω˛Ó
öÎ˚ S!ã˛e 9.1V– ~ÓyÓ˚ ôÓ˚&öñ ï˛yÓ˚!ê˛ˆÏܲ xö%≤Ãfliû˛yˆÏÓ xyˆÏ®y!°ï˛ ܲÓ˚y •° ÎyˆÏï˛ §ü@ˇÃ ï˛yÓ˚!ê˛ ~ܲ §üï˛ˆÏ°
ÌyˆÏܲ–
!ã˛e 9.1-AB xyê˛Ü˛yˆÏöy !fliÓ˚ ≤Ãyhsˇ !ã˛e 9.2-ï˛yˆÏÓ˚Ó˚ x!ï˛«%˛o xLjϢÓ˚ ò%•z ≤ÃyˆÏhsˇ
ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ §üï˛ˆÏ° §#!üï˛ xyˆÏ®y°ö !e´Î˚y¢#° ê˛yö
ܲ¡õöÓ˚ï˛ ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ x!ï˛«%˛o xÇ¢ PQ !ã˛e 9.2 ÈÙÙÙÈˆï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ï˛yˆÏÓ˚Ó˚ ê˛yö ˜òâ≈ƒ ÓÓ˚yÓÓ˚
!e´Î˚y¢#°ñ §%ï˛Ó˚yÇ Óe´yÓfliyÎ˚ ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ x!B˛ï flõ¢≈ܲ ê˛yˆÏöÓ˚ !òܲ !öˆÏò≈¢ ܲˆÏÓ˚ xÌ≈yÍ P ≤ÃyˆÏhsˇ
ê˛yˆÏöÓ˚ !òܲ PP' ~ÓÇ Q ≤ÃyˆÏhsˇ QQ'– ï˛yˆÏÓ˚Ó˚ §yüƒyÓfliy AB ÓÓ˚yÓÓ˚ X x«˛ ~ÓÇ §Ó˚ˆÏîÓ˚ !òܲ ÓÓ˚yÓÓ˚
Y x«˛ ôˆÏÓ˚ !öˆÏÎ˚ P Ä Q !Ó®%ˆÏï˛ ê˛yö ò%!ê˛Ó˚ í˛z˛õyÇ¢ !öî≈Î˚ ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– xyüÓ˚y ôˆÏÓ˚ ˆöÓ ˆÎñ ï˛yÓ˚!ê˛ˆÏܲ
§yüƒyÓfliy ˆÌˆÏܲ á%Ó x“ ˛õ!Ó˚üyˆÏî•z ê˛yöy •ˆÏÎ˚ˆÏåÈ ÎyÓ˚ çöƒ PP' Ä QQ' flõ¢≈ܲ=!° AB ~Ó˚ §ˆÏD á%Ó•z
239
«%˛o ˆÜ˛yî Ó˚ã˛öy ܲˆÏÓ˚– ~ܲ•z ܲyÓ˚ˆÏî P Ä Q !Ó®% ò%!ê˛Ó˚ x fliyöyB˛ Î!ò ÎÌye´ˆÏü x ~ÓÇ x + xδ •Î˚ñ ï˛ˆÏÓ
PQ ˜òâ≈ƒ!ê˛
x δ
~Ó˚ §üyö ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚–
~áö PP' ÓÓ˚yÓÓ˚ ê˛yˆÏöÓ˚ X í˛z˛õyÇ¢ = – Tcos
θ
– T
~ÓÇ Y í˛z˛õyÇ¢ = – Tsin
θ
– Ttan
θ
= – T
P
yx
∂⎛⎞⎜⎟ ∂ ⎝⎠
~áyˆÏö
θ
ˆÜ˛yˆÏîÓ˚ üyö «%˛o ӈϰ cos
θ
ˆÜ˛ 1 ~Ó˚ §üyö ~ÓÇ sin
θ
ˆÜ˛ tan
θ
~Ó˚ §üyö ôÓ˚y •ˆÏÎ˚ˆÏåÈ–
~ܲ•zû˛yˆÏÓ
θ
+ d
θ
ˆÜ˛yî x!ï˛ «%˛o ôˆÏÓ˚ !öˆÏÎ˚
QQ' ÓÓ˚yÓÓ˚ ê˛yˆÏöÓ˚ X í˛z˛õyÇ¢ = T cos (
θ
+ d
θ
) = T
Y í˛z˛õyÇ¢ = T sin (
θ
+ d
θ
) = T tan (
θ
+ d
θ
)
= T
2
2·Qpp
yyy Txxxx
δ⎛⎞ ∂∂∂ ⎛⎞⎛⎞ =+⎜⎟ ⎜⎟⎜⎟⎜⎟ ∂∂∂ ⎝⎠⎝⎠⎝⎠
xï˛~Óñ PQ ~Ó˚ í˛z˛õÓ˚ ܲyÎ≈ܲÓ˚# °!∏˛ ӈϰÓ˚ X í˛z˛õyÇ¢ = –T + T= 0, xÌ≈yÍ x x«˛ ÓÓ˚yÓÓ˚ ˆÜ˛yöÄ
xö%˜Ïòâ≈ƒ Ó° ˆö•z–
Y í˛z˛õyÇ¢ = T2
2Q p
y y yT xx x x
δ⎛ ⎞∂ ∂ ∂⎛ ⎞ ⎛ ⎞− = ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠
xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈñ PQ xÇ¢!ê˛Ó˚ ˜òâ≈ƒ δx– ï˛yÓ˚!ê˛Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚ ρ •ˆÏ° ˆ§!ê˛Ó˚ δx ˜ò Ïâ≈ƒÓ˚
û˛Ó˚ •ˆÏÓ
ρδx– ~áö !öí˛zê˛ˆÏöÓ˚ àï˛#Î˚ §ü#ܲÓ˚î xö%ÎyÎ˚#
22
22yy xTxtx
ρδδ ∂∂ =∂∂
(
ρ
= ï˛yÓ˚!ê˛Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚V
xÌÓyñ
2222
222yTyy ctxx ρ
∂∂∂ ==∂∂∂
T cρ
⎛⎞= ⎜⎟ ⎜⎟ ⎝⎠
...9.1
9.1 §ü#ܲÓ˚ˆÏîÓ˚ ˆã˛•yÓ˚y!ê˛ •Î˚ï˛ xy˛õöyÓ˚ ܲyˆÏåÈ ˛õ!Ó˚!ã˛ï˛– ~!ê˛ ~üö ~ܲ!ê˛ ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î ÎyÓ˚ x x«˛
ÓÓ˚yÓÓ˚ ˆÓà c – §ü#ܲÓ˚î!ê˛ ˆÌˆÏܲ ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ê˛yöy ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ ˆÓà c =
Tρ
– ~•z §ü#ܲÓ˚î
ˆÌˆÏܲ ˆòáy ÎyˆÏFåÈ ˆÎñ ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ !ö!ò≈‹T !Ó®%ˆÏï˛ §Ó˚î y !öû≈˛Ó˚ ܲˆÏÓ˚ §üÎ˚ t ~ÓÇ !Ó®%!ê˛Ó˚ fliyöyB˛ x
~Ó˚ í˛z˛õÓ˚– ~áö xyüyˆÏòÓ˚ 9.1 §ü#ܲÓ˚î!ê˛Ó˚ §üyôyö ÓyÓ˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– !ܲv §ü#ܲÓ˚î!ê˛ˆÏï˛ x Ä t ~•z ò%•z
ã˛°Ó˚y!¢ ~ܲe ÌyܲyÎ˚ xyüyˆÏòÓ˚ ~ ò%!ê˛ˆÏܲ xy°yòy ܲÓ˚yÓ˚ ˆã˛‹Ty ܲÓ˚ˆÏï˛ •ˆÏÓ– ôÓ˚y Îyܲñ §Ó˚î y ò%!ê˛ xˆÏ˛õ«˛ˆÏܲÓ˚
=îú˛°ñ ÎyˆÏòÓ˚ ~ܲ!ê˛ x ~ÓÇ xöƒ!ê˛ t-~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#°– ˆ§ˆÏ«˛ˆÏe xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ
y(x, t) = X(x)·T(t) ...9.2
240
9.2 Ó˚y!¢!ê˛ˆÏܲ ~ÓyÓ˚ 9.1 §ü#ܲÓ˚ˆÏî Ó§yˆÏöy Îyܲ– ˆÎˆÏ•ï%˛
2
22 · yT Xtt
2 ∂∂ =∂∂
~ÓÇ
2
22yXxx
2 ∂∂ =∂∂
·T SˆÜ˛ööy X xˆÏ˛õ«˛Ü˛!ê˛ t ~Ó˚ í˛z˛õÓ˚ ~ÓÇ T xˆÏ˛õ«˛Ü˛!ê˛ x ~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚
öy–V
9.1 §ü#ܲÓ˚îˆÏܲ ˆ°áy ÎyÎ˚ñ
2
2221·TX Xctx
2 ∂∂ =∂∂
· T
xÌÓy í˛zû˛Î˚ !òˆÏܲ X T !òˆÏÎ˚ û˛yà ܲˆÏÓ˚
2
2221112 ∂∂
=∂∂
TXTX ctx
...9.3
9.3 §ü#ܲÓ˚î!ê˛ °«˛ƒ ܲÓ˚ˆÏ° Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ ~Ó˚ Óyü!òˆÏܲ ˆÜ˛ÏÓ°üye t-~Ó˚ í˛z˛õÓ˚ ~ÓÇ í˛yö!òˆÏܲ
ˆÜ˛Ó°üye x-~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#°– ~ xÓfliyÎ˚ í˛zû˛Î˚ !òܲˆÏܲ•z ô &Óܲ •ˆÏï˛ •ˆÏÓ– ôÓ˚y Îyܲñ ~•z ô &Óܲ!ê˛ •°
ÈÙÈ p2– ày!î!ï˛Ü˛ §%!ÓôyÓ˚ çöƒ xyüÓ˚y ~áyˆÏö ~ܲ!ê˛ ˆöˆÏà!ê˛û˛ Óà≈ Ó˚y!¢ !ö°yü– 9.3 §ü#ܲÓ˚î ˆÌˆÏܲ ~áö
xyüÓ˚y x Ä t - ~Ó˚ ò%!ê˛ ˛õ,Ìܲ §ü#ܲÓ˚î ˛õy•z≠
2
2Tt
∂∂
+ p2c2T = 0 ...9.4(a)
~ÓÇ
2
2Xx
∂∂
+ p2X = 0 ...9.4(b)
~Ó˚ üˆÏôƒ ≤ÃÌü!ê˛ñ xÌ≈yÍ 9.4(a) §ü#ܲÓ˚ˆÏîÓ˚ ~ܲ!ê˛ §Ω˛Ó˛õÓ˚ §üyôyö cos pct ~Ó˚ §üyö%˛õyï˛# ôÓ˚y ˆÎˆÏï˛
˛õyˆÏÓ˚– §ü#ܲÓ˚î 9.4(b) ~Ó˚ §yôyÓ˚î §üyôyö •° X = A1sin px + A2cos px
§%ï˛Ó˚yÇñ 9.1 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö≠
y(x, t) = X(x)·T(t) = (A1 sin px + A2 cos px) cos pct
~áyˆÏö A1 Ä A2 ò%!ê˛ x!ö!ò≈‹T ô &Óܲ–
!ܲv xyüÓ˚y 9.4(a) §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö sin pct ~Ó˚ §üyö%˛õyï˛# !•§yˆÏÓÄ ôÓ˚ˆÏï˛ ˛õy!Ó˚ ~ÓÇ ˆ§ˆÏ«˛ˆÏe
9.1 §ü#ܲÓ˚ˆÏîÓ˚ §üyôyö •ˆÏÓ ≠
y(x, t) = X(x)·T(t) = (B1 sin px + B
2 cos px) sin pct
ˆÎáyˆÏö B1 Ä B
2 ò%!ê˛ xöƒ x!ö!ò≈‹T ô &Óܲ– ~áö 9.1 §ü#ܲÓ˚ˆÏîÓ˚ ò%!ê˛ §üyôyö ˆÎyà ܲˆÏÓ˚ §yôyÓ˚î §üyôyö
˛õyÄÎ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚ ≠
241
y(x, t) = (A1 sin px + A
2 cos px) cos pct + (B
1 sin px + B
2 cos px) sin pct ...9.5
ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛ ≠ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛ ≠ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛ ≠ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛ ≠ê˛yöy ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛ ≠
xy˛õ!ö °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ 9.5 §ü#ܲÓ˚ˆÏî A1, A2, B1 Ä B2 ~•z ã˛yÓ˚!ê˛ x!ö!ò≈‹T ô &Óܲ Ó˚ˆÏÎ˚ˆÏåÈ– ï˛yˆÏÓ˚Ó˚
§#üy¢ï≈˛=!°Ó˚ §y•yˆÏ΃ xy˛õ!ö ~•z ô &Óܲ=!°Ó˚ üyö ÓyÓ˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– ≤Ã̈Ïü•z °«˛ƒ ܲÓ˚&öñ ï˛yˆÏÓ˚Ó˚
ò%•z ≤Ãyhsˇ xÌ≈yÍ x = 0 ~ÓÇ x = l !Ó®% ò%•z!ê˛ˆÏï˛ §Ó §üˆÏÎ˚•z y = 0– ~áyˆÏö l = ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒ– 9.5 §ü#ܲÓ˚ˆÏî
~•z §#üy¢ï≈˛=!° xyˆÏÓ˚y˛õ ܲÓ˚y Îyܲ–
(i) ≤ÃÌü!ê˛ SxÌ≈yÍ x = 0 •ˆÏ° y = 0) ˆÌˆÏܲ ≠
A2 cos pct + B
2 sin pct = 0
Î!ò ~•z §¡õÜ≈˛!ê˛ t-~Ó˚ §Ó üyˆÏö•z §ï˛ƒ •Î˚ ï˛ˆÏÓ A2 = B2 = 0– §%ï˛Ó˚yÇñ 9.5 §ü#ܲÓ˚ˆÏîÓ˚ §Ó˚°#Ü,˛ï˛
Ó˚*˛õ •° ≠
y(x, t) = (A1 cos pct + B
1 sin pct) sin px ...9.6 (a)
(ii) ~ÓyÓ˚ !mï˛#Î˚!ê˛ñ SxÌ≈yÍ x = l • Ï° y = 0) ˆÌ Ïܲ
(A1 cos pct + B1 sin pct) sin pl = 0
~•z §¡õÜ≈˛!ê˛ t ~Ó˚ §Ó üyˆÏöÓ˚ çöƒ §ï˛ƒ •ˆÏ° sin pl = 0 Óy pl = s
π
, ˆÎáyˆÏö s = ˛õ)î≈ §Çáƒy– §%ï˛Ó˚yÇ
p ~Ó˚ çyÎ˚àyÎ˚
slπ
ˆ°áy ˆÎˆÏï˛ ˛õyˆÏÓ˚– 9.6 (a) §ü#ܲÓ˚î!ê˛ ~ÓyÓ˚ ~û˛yˆÏÓ ˆ°áy ÎyÎ˚ ≠
y(x, t) = 11 cossinsin sctsctsx AB
lllπππ ⎛⎞ + ⎜⎟ ⎝⎠
...9.6(b)
ˆÎˆÏ•ï%˛ s = 0, 1, 2, 3 ≤Ãû,˛!ï˛ s ~Ó˚ §Ó üyˆÏöÓ˚ çöƒ•z 9.6(b) §üyôyö!ê˛ áyˆÏê˛ñ ~•z §üyôyö=!°Ó˚
ˆÎyàú˛°!ê˛ §ÓˆÏã˛ˆÏÎ˚ §yôyÓ˚î §üyôyö •ˆÏÓ– ܲyˆÏç•z 9.1 §ü#ܲÓ˚ˆÏîÓ˚ §yôyÓ˚î §üyôyö •° ≠
y (x, t ) = 1
cos sin sin∞
=
⎛ ⎞+⎜ ⎟⎝ ⎠∑ s ss
s ct s ct s xA Bl l l
π π π...9.7
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î!ê˛Ó˚ Ó˚*˛õ ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ x !öû˛≈Ó˚ ~ÓÇ t !öû≈˛Ó˚ ˛õÎ≈yÓ,_ Ó˚y!¢=!°
xy°yòy í˛zͲõyòܲ !•§yˆÏÓ Ó˚ˆÏÎ˚ˆÏåÈñ xÌ≈yÍ ~!ê˛ xö%≤Ãfli fliyî%ï˛Ó˚D !öˆÏò≈§ ܲˆÏÓ˚ˆÏåÈ– s ˛õ)î≈ §Çáƒy!ê˛ Ü˛¡õˆÏöÓ˚
!Ó!û˛ß¨ ôÓ˚öˆÏܲ ˆÓyé˛yÎ˚– ü)°§%ˆÏÓ˚Ó˚ çöƒ s = 1, !mï˛#Î˚ §üˆÏüˆÏ°Ó˚ çöƒ s = 2 •zï˛ƒy!ò ~ÓÇ 9.7 §ü#ܲÓ˚ˆÏîÓ˚
í˛yö!òˆÏܲÓ˚ Ó˚y!¢üy°yÎ˚ ~Ó˚ܲü x§Çრ§üˆÏüˆÏ°Ó˚ çöƒ §Ó˚î ˆÎyà •ˆÏÎ˚ˆÏåÈ– ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ §üˆÏüˆÏ°Ó˚
çöƒ ˆÜ˛Ô!îܲ ܲ¡õyB˛
sclπ
, xÌ≈yÍ Ü˛¡õyB˛ ó
1·22
= scscllπ
π
– §%ï˛Ó˚yÇñ §üˆÏü°=!°Ó˚ §¡õyˆÏB˛Ó˚ xö%˛õyï˛
1 : 2 : 3.... •zï˛ƒy!ò–
242
9.2.1 xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚xö%≤Ãfliû˛yˆÏÓ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ¢!_´ !öî≈Î˚
xyüÓ˚y 9.7 §ü#ܲÓ˚î ˆÌˆÏܲ ˆòˆÏá!åÈ ˆÎñ ~ܲ!ê˛ Ü˛!¡õï˛ ï˛yˆÏÓ˚Ó˚ ˆÎ ˆÜ˛yö !Ó®% x - ~ t §üˆÏÎ˚ §Ó˚î •ˆÏFåÈ
y = 1
sin cos sin∞
=
⎛ ⎞+⎜ ⎟⎝ ⎠∑ s ss
s x s ct s ctA Bl l lπ π π
= ( )1
sinss
s xy tlπ∞
=∑ ...9.8
SˆÎáyˆÏö y, (t) = Ascoss ct
lπ
+ Bssin
sctlπ
)
xï˛~Óñ
()1
sin·ss
dysxytdtl
π ∞
=
=∑
= 1
sin sin coss ss
s x s c s ct s ctA Bl l l lπ π π π∞
=
⎛ ⎞− +⎜ ⎟⎝ ⎠∑ ...9.9
ï˛yˆÏÓ˚Ó˚ «%˛o xÇ¢ dx ~Ó˚ û˛Ó˚ pdx ~ÓÇ ï˛yÓ˚ à!ï˛¢˘!_´
= 12 pdx
2 dydt
⎛⎞⎜⎟ ⎝⎠
(ρ
= ≤Ã!ï˛ ~ܲܲ ˜òˆÏâ≈ƒ ï˛yˆÏÓ˚Ó˚ û˛Ó˚V
§%ï˛Ó˚yÇñ l ˜òˆÏâ≈ƒÓ˚ ï˛yˆÏÓ˚Ó˚ à!ï˛¢!_´ T =
0
2
2ldydx
dtρ⎛⎞ ∫⎜⎟ ⎝⎠
= ( )0
2
sin2
l
ss
s x y t dxl
ρ π⎡ ⎤∫ ⎢ ⎥
⎢ ⎥⎣ ⎦∑ ...9.10
~•z §üyôyö!ê˛Ó˚ üˆÏôƒ ˆÎ Óà≈Ó˚y!¢!ê˛ Ó˚ˆÏÎ˚ˆÏåÈ ˆ§!ê˛ xy˛õ!ö ~•zû˛yˆÏÓ !°áˆÏï˛ ˛õyˆÏÓ˚ö ≠
( ) ( )sin sins ns n
s x n xy t y tl lπ π⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑
2 2sin ( ) sin sin ( ) ( )≠
= +∑ ∑ ∑ s ss s n
s x s x n xy t y t y t dxl l lπ π π
~áyˆÏö ≤ÃÌü Ó˚y!¢üy°y!ê˛ ˆÎ =îú˛° Ó˚y!¢Ó˚ çöƒ s = n ~ÓÇ !mï˛#Î˚!ê˛ ˆÎ=!°Ó˚ çöƒ s
≠
n, ˆ§=!°
ˆÓyé˛yˆÏFåÈ– ~áö 9.10 §üyܲ°ö!ê˛ˆÏܲ ˆ°áy ÎyÎ˚ ≠
243
T =
()() 2 2
00sinsinsin()
22
llsn
ssn
sxsxsx ytdxytytdxlll
ρπρππ
≠
+∑ ∑∑ ∫∫
!ܲv xy˛õ!ö çyˆÏöö ˆÎñ 0
sin sinl m x n x dx
l lπ π∫ - ~Ó˚ üyö m
≠
n •ˆÏ° ¢)öƒ •ˆÏÓ– §%ï˛Ó˚yÇ í˛z˛õˆÏÓ˚Ó˚
Ó˚y!¢üy°yÓ˚ !mï˛#Î˚ §ü#ܲ°ö!ê˛ ¢)öƒ •ˆÏÓ– ܲyˆÏç•z ï˛yˆÏÓ˚ à!ï˛¢!_´ •ˆÏÓ ≠
222222
0sinsincos2cossin
2
lssss
s
sxscsctsctsctsct dxABABllllll
ρππππππ ⎛⎞⎛ =+− ⎜⎟⎜ ⎝⎠⎝ ∑∫
= 2
2 2 2 2sin cos 2 cos sin4 s s s sl s c s ct s ct s ct s ctA B A B
l l l l lρ π π π π π⎛ ⎞ ⎛ ⎞∑ + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
...9.11
ˆÜ˛ööy
2
0
1 sin2
lsxdxlπ= ∫
–
!fli!ï˛¢!_´ ≠ !fli!ï˛¢!_´ ≠ !fli!ï˛¢!_´ ≠ !fli!ï˛¢!_´ ≠ !fli!ï˛¢!_´ ≠ ~ÓyÓ˚ xyüÓ˚y §yüƒyÓfliy ˆÌˆÏܲ §ˆÏÓ˚ Ìyܲy xÓfliyÎ˚ ï˛yÓ˚!ê˛Ó˚ !fli!ï˛¢!_´ !öî≈Î˚ ܲÓ˚Ó– §ÇK˛y
xö%ÎyÎ˚# ˆÜ˛yöÄ Ó›Ó˚ !fli!ï˛¢!_´
= Ó›!ê˛Ó˚ xÓfliyö ˛õ!Ó˚Óï≈˛ö ܲÓ˚yÓ˚ çöƒ Ü,˛ï˛Ü˛yÎ≈
= ≤ÃÎ%_´ Ó° Ä Ó° ≤ÈÏÎ˚yˆÏàÓ˚ ú˛ˆÏ° §Ó˚ˆÏîÓ˚ =îú˛ˆÏ°ÏÓ˚ §üyܲ!°ï˛ üyö–
9.1 §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y ܲÓ˚yÓ˚ §üÎ˚ xyüÓ˚y ˆòˆÏá!åÈ ˆÎñ ï˛yˆÏÓ˚Ó˚ x!ï˛ «%˛o xÇ¢
δx
~Ó˚ IJõˆÏÓ˚ °!∏˛ Ó°
Y x«˛ ÓÓ˚yÓÓ˚ ܲyç ܲˆÏÓ˚ ~ÓÇ ï˛yÓ˚ üyö T
2
2yxx
δ ∂∂
– ~áö Î!ò §Ó˚î
y δ
˛õ!Ó˚üyî Óyí˛¸ˆÏöy •Î˚ñ ï˛yÓ˚ çöƒ
Ü,˛ï˛Ü˛yÎ≈ •ˆÏÓ ≠ ÈÙÙÙÈT
2
2yxyx
δδ ∂∂
– §ü@ˇÃ ï˛yˆÏÓ˚Ó˚ çöƒ Ü,˛ï˛Ü˛yÎ≈ ~ÓÇ ˆ§ˆÏ•ï%˛ !fli!ï˛¢!_´Ó˚ Ó,!k˛ ≠
2
2 0δV = δ
ly Tydxx
∂ −∂ ∫
~áyˆÏö δy •ˆÏÓ x ~Ó˚ xˆÏ˛õ«˛Ü˛– !ӈϢ£Ïï˛ §#üy¢ˆÏï≈˛Ó˚ çöƒ ò%•z ≤ÃyˆÏhsˇ
δy
~Ó˚ üyö ¢)öƒ •ˆÏï˛ •ˆÏÓ– í˛z˛õˆÏÓ˚Ó˚
§ü#ܲÓ˚îˆÏܲ xyÇ!¢Ü˛ §üyܲ° ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎ˚
()0
δVδ|δ0
l l yy TyTydxxxx
∂∂∂ =−+∂∂∂ ∫
§üyܲ!°ï˛ ≤ÃÌü xÇ¢!ê˛ í˛zû˛Î˚ §#üyˆÏï˛•z ¢)öƒ ܲyÓ˚î δy ~Ó˚ üyö ~•z ò%•z !Ó®%ˆÏï˛ ¢)öƒ– xï˛~Óñ
244
0δVδ
lyy Tdxxx∂∂ ⎛⎞⎛⎞ =⎜⎟⎜⎟ ∂∂ ⎝⎠⎝⎠ ∫
§¡õ)î≈ !fli!ï˛¢!_´ •ˆÏÓ §yüƒyÓfliy SÎáö ï˛yˆÏÓ˚Ó˚ §Ó≈e•z y = 0 ~ÓÇ yx
∂∂ = 0) ˆÌˆÏܲ
yx∂∂
~Ó˚ ~ܲ!ê˛
!ö!ò≈‹T üyˆÏö xyöyÓ˚ çöƒ Ü,˛ï˛Ü˛yÎ≈– xÌ≈yÍñ
V =
00
δ
xxl
yx
yy Tdxxx
∂∂
∂=∂
∂∂ ⎛⎞⎛⎞⎜⎟⎜⎟ ∂∂ ⎝⎠⎝⎠ ∫∫
= 2
0
12
l yT dxx
∂⎛ ⎞⎜ ⎟∂⎝ ⎠∫ ...9.12
9.8 öÇ §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y ˆ˛õˆÏÎ˚!åÈñ
( )sinss
s xy y tlπ= ∑
∴
() cosss
dyssxytdxll
ππ =∑xï˛~Óñ §ü@ˇÃ ï˛yˆÏÓ˚Ó˚ !fli!ï˛¢!_´ ≠
= 2
2
0cos sin cos
2⎛ ⎞⎛ ⎞+ ×⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑ ∫
ls s
s
T s s ct s ct s xA B dxl l l lπ π π π
ˆÎˆÏ•ï%˛ ê˛yö T = ρc2 ~ÓÇ
2
0
1 cos2
lsxdxlπ= ∫
xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ §ü@ˇÃ ï˛yˆÏÓ˚Ó˚ !fli!ï˛¢!_´
22222 cossin2cossin
4sssss
lscsctsctsctsct VABABlllll
ρπππππ ⎛⎞⎛⎞ =++ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ∑
...9.13
ܲ¡õüyö ï˛yˆÏÓ˚Ó˚ ˆüyê˛ ¢!_´ à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ ˆÎyàú˛°–
§%ï˛Ó˚yÇñ 9.11 ~ÓÇ 9.13 §ü#ܲÓ˚î ˆÌˆÏܲ ˆüyê˛ ¢!_´
= ( )2
2 2 2
2 s ss
scl A Bl
ρ π ⎛ ⎞ +⎜ ⎟⎝ ⎠∑
= ( )2 2 2 2s s s
s
M n A Bπ +∑ ...9.14
245
ˆÎáyˆÏö M =
ρ
l = §ü@ˇÃ ï˛yˆÏÓ˚Ó˚ û˛Ó˚ ~ÓÇ
2sc
l
= ns= ܲ¡õˆÏöÓ˚ s-ï˛ü §üˆÏüˆÏ°Ó˚ ܲ¡õyB˛– ~ÓyÓ˚ Îy
˛õí˛¸ˆÏ°ö ï˛yÓ˚ §¡∫ˆÏ¶˛ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1
ôÓ˚&öñ kg Äçö !òˆÏÎ˚ ˆê˛ˆÏö Ó˚yáy 1.1 !üê˛yÓ˚ °¡∫y 2.2gm û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ §%£Ïü ï˛yÓ˚ ˆÜ˛Ó°üye !mï˛#Î˚
§üˆÏüˆÏ° ܲ!¡õï˛ •ˆÏFåÈ– ï˛yÓ˚!ê˛Ó˚ §ˆÏÓù≈yFã˛ §Ó˚î 2mm ~ÓÇ t = 0 §üˆÏÎ˚ ï˛yÓ˚!ê˛Ó˚ ˆÜ˛yÌyÄ ˆÜ˛yöÄ §Ó˚î
!åÈ° öy– ˛õÓ˚Óï˛#≈ §üˆÏÎ˚ ï˛yˆÏÓ˚Ó˚ §Ó˚îÈÙȧüÎ˚ §¡õÜ≈˛!ê˛ !öî≈Î˚ ܲÓ˚&ö–
9.3 ܲ!£Ï ≈ï˛ ï˛yÓ˚ܲ!£Ï ≈ï˛ ï˛yÓ˚ܲ!£Ï ≈ï˛ ï˛yÓ˚ܲ!£Ï ≈ï˛ ï˛yÓ˚ܲ!£Ï ≈ï˛ ï˛yÓ˚ (Plucked string)
ˆ§ï˛yÓ˚ñ !àê˛yÓ˚ ≤Ãû,˛!ï˛ ÓyòƒÎˆÏsf ï˛yÓ˚!ê˛ˆÏܲ ê˛ˆÏö ôˆÏÓ˚ åȈÏí˛¸ òÄÎ˚y •Î˚– åÈyí˛¸yÓ˚ õ)Ó≈ ü%•)ˆÏï≈˛ ï˛yÓ˚!ê˛ˆÏï˛ Ü˛yöÄ
à!ï˛ ÌyˆÏܲ öy– ~Ó˚ xyÜ,˛!ï˛ 9.3 !ã˛ˆÏe ˆòáyˆÏöy APB ~Ó˚ üï˛ •Î˚– ~•z çyï˛#Î˚ ï˛yÓ˚ˆÏܲ xyüÓ˚y ܲ!£Ï≈ï˛ ï˛yÓ˚
Ó!°–
9.1 §ü#ܲÓ˚ˆÏî xyüÓ˚y ˆ˛õˆÏÎ˚!åÈ xö%˜Ïòâ≈ƒû˛yˆÏÓ Ü˛!¡õï˛ ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚ xÓܲ° §ü#ܲÓ˚î ≠
222
22yy ctx
∂∂ =∂∂
~ÓÇ 9.7 §ü#ܲÓ˚î xö%ÎyÎ˚# ~!ê˛Ó˚ §üyôyö ≠
2 sincossin ⎛⎞ =+ ⎜⎟ ⎝⎠ ∑ss
sxsctsct yABlllπππ
...9.15
üˆÏö ܲÓ˚y Îyܲñ ï˛yÓ˚!ê˛ˆÏܲ x = a !Ó®%ˆÏï˛ ˛õyˆÏ¢Ó˚ !òˆÏܲ h ò)Ó˚ˆÏc ˆê˛ˆÏö ôÓ˚y •°– ~Ó˚
!ã˛e 9.3-!Ó®%üye fliyˆÏö ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ≤ÃyÌ!üܲ xÓfliyö–
ú˛ˆÏ° ï˛yÓ˚!ê˛Ó˚ xyÜ,˛!ï˛ òyÑí˛¸y° ò%!ê˛ §Ó˚°ˆÏÓ˚áyñ ÎyÓ˚y ܲ!£Ï≈ï˛ !Ó®%ˆÏï˛ !ü!°ï˛ •ˆÏFåÈ S!ã˛e 9.3 ˆòá%öV– t
= 0 §üˆÏÎ˚ ܲ!£Ï≈ï˛ !Ó®% p ~Ó˚ y fliyöyB˛ = h.
246
x = 0 ~ÓÇ x = a fliyöyˆÏB˛Ó˚ üˆÏôƒ ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ ï˛yˆÏÓ˚Ó˚ xö%≤Ãfli §Ó˚î •ˆÏÓ
y0 = ha x (0
≤
x
≤
a) ...9.15(a)
xyÓyÓ˚ñ x = a Ä x = l ~Ó˚ üˆÏôƒ ˆÎÈÙÈˆÜ˛yöÄ fliyöyˆÏB˛ ï˛yˆÏÓ˚Ó˚ §Ó˚î
y0 = h
lxla
−−
(a
≤
x
≤
l) ...9.15(b)
t = 0 §üˆÏÎ˚ ï˛yˆÏÓ˚Ó˚ !Ó!û˛ß¨ !Ó®%ˆÏï˛ ˆÎ ≤ÃyÌ!üܲ ¢ï≈˛=!° ˛õy!°ï˛ •Î˚ ˆ§=!° •°
i) y = y0
...9.16(a)
~ÓÇ ii)
0 yy =
...9.16(b)
~áö xyüÓ˚y ~•z ≤ÃyÓ˚!Ω˛Ü˛ ¢ï≈˛=!°Ó˚ §y•yˆÏ΃ 9.15 §ü#ܲÓ˚ˆÏîÓ˚ x!ö!ò≈‹T ô &Óܲ As Ä B
2 ~Ó˚ üyö !öî≈Î˚
ܲÓ˚ˆÏï˛ ˆã˛‹Ty ܲÓ˚Ó– 9.15 §ü#ܲÓ˚ˆÏî xyüÓ˚y ˆ˛õˆÏÎ˚!åÈñ
y (x, t) =
1
cossinsin sss
sctsctsx ABlllπππ ∞
=
⎛⎞ + ⎜⎟ ⎝⎠ ∑
~!ê˛ xy§ˆÏ° ~ܲ!ê˛ ˛ú%˛!Ó˚ˆÏÎ˚ ˆ◊î# Fourier Series ~ÓÇ As Ä B
s ˛ú%˛!Ó˚ˆÏÎ˚ =îyB˛– ~•z §ü#ܲÓ˚î ˆÌˆÏܲ
t = 0 §üˆÏÎ˚ xyüÓ˚y ˛õy•z ≠
y(x, 0) = y0=
1
sinss
s xAlπ∞
=∑ ...9.17(a)
~ÓÇ ( ) 01
,0 sinss
s c s xy x y Bl lπ π∞
=
= = ∑ ...9.17(b)
Î!ò 9.17(a) Ä (b) §ü#ܲÓ˚î ò%!ê˛Ó˚ í˛zû˛Î˚ !òܲˆÏܲ sins x
lπ
dx !òˆÏÎ˚ =î ܲˆÏÓ˚ö ~ÓÇ x = 0 ˆÌˆÏܲ x
= l §#üyÓ˚ üˆÏôƒ §üyܲ°ö ܲˆÏÓ˚öñ ï˛ˆÏÓ xy˛õ!ö ˛õyˆÏÓö ≠
20
00
1 sinsin·2
llss
sxsx ydxAdxAllππ == ∫∫
~ÓÇ
20
00
1 sinsin····22
=== ∫∫ll
ssssxsxscscsc ydxBdxBB
llllπππππ
xÌ≈yÍñ
00
2sinl
ssx Aydx
llπ =∫
~ÓÇ
00
2sin.l
ssx Bydx
sclπ
π=∫
~ÓyÓ˚ 9.15(a) Ä (b) ˆÌˆÏܲ y0 ~ÓÇ 9.16(b) 0y ~Ó˚ üyö Ó§yˆÏöy ˆÎˆÏï˛ ˛õyˆÏÓ˚–
247
0
2·sinsin − ⎡⎤ =+ ⎢⎥ − ⎣⎦ ∫∫al
sa
hsxlxsx Axdxhdxlallal
ππ
xÌÓyñ
0
1 sin sin sin2
a l l
sa a
h s x hl s x h s xA x dx dx x dxa l l a l l a l
π π π= + −− −∫ ∫ ∫
= 0
sin sin sin+ −− −∫ ∫ ∫
a l l
a a
h hl hx pxdx pxdx x px dxa l a l a
ˆÎáyˆÏö s plπ = ˆ°áy •ˆÏÎ˚ˆÏåÈ–
=
211 cossincoscos
⎡⎤⎡ ⎡⎤−++−−−+ ⎢⎥⎢ ⎢⎥ −− ⎢⎥⎢ ⎣⎦ ⎣⎦⎣
al
a a
hxhlhx pxpxpxpxaplaplap p
= ( ) ( ) ( )21 1 1cos sin cos cos cos sin cos sinh a hl hpa pa pa pl l pl pl a pa pa
a p p l a p l a p pp⎛ ⎞ ⎡ ⎤
− + + − − − + + −⎜ ⎟ ⎢ ⎥− − ⎣ ⎦⎝ ⎠
= ( )2 sin ,hl pap a l a−
ˆÜ˛ööy sin pl= 0–
∴
()()2
22222 ·sinsin ==
−− shlhlsa Apa
alal plsalaπ
π
xyÓyÓ˚ñ 00
2 sin 0= =∫l
ss xB y dx
s l lπ
π
ˆÜ˛ööy ï˛yˆÏÓ˚Ó˚ §Ó !Ó®%Ó˚ çöƒ•z 0 0y =
As~ÓÇ B
2 ~Ó˚ ˆÎ üyö=!° xyüÓ˚y !öî≈Î˚ ܲÓ˚°yü ˆ§=!° ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ 9.15
§ü#ܲÓ˚î!ê˛ˆÏܲ ˆ°áy ÎyÎ˚≠
()2
2221sinsincos
s
hlsasxsct ylll alasπππ
π=
−∑
...9.18
= cosss
s ctal
π∑
248
~áyˆÏöñ as =
( )2
2 22 sin sinhl s a s x
l ls a l aπ π
π −...9.18(a)
~ÓÇ ~!ê˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ s-ï˛ü §üˆÏüˆÏ°Ó˚ !ÓhflÏyÓ˚– ˆÎˆÏ•ï%˛ ï˛yˆÏÓ˚Ó˚ xö%≤Ãfli ï˛Ó˚D!ê˛ fliyî%ï˛Ó˚Dñ ~•z !ÓhflÏyÓ˚
ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒ ÓyÓ˚ÓÓ˚ x fliyöyˆÏB˛Ó˚ §ˆÏD ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚–
9.18 §ü#ܲÓ˚î ˆÌˆÏܲ ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ §ühflÏ ï˛Ìƒ•z xyüÓ˚y ˆ˛õˆÏï˛ ˛õy!Ó˚– ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ üˆÏôƒ
ˆÎ í˛z˛õ§%Ó˚=!° ÌyˆÏܲñ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° í˛zͲõߨ ¢∑ï˛Ó˚ˆÏDÄ ˆ§•z í˛z˛õ§%Ó˚=!° ÌyˆÏܲ ~ÓÇ ˆ§=!°Ó˚ xyö%˛õy!ï˛Ü˛
ï˛#Ó ï˛yÄ Ü˛¡õˆÏöÓ˚ í˛z˛õ§%Ó˚=!°Ó˚ xö%Ó˚*˛õ •Î˚– §%ï˛Ó˚yÇñ 9.18 §ü#ܲÓ˚î!ê˛Ó˚ ˛õÎ≈yˆÏ°yã˛öy ܲˆÏÓ˚ xyüÓ˚y ܲ!£Ï≈ï˛
ï˛yÓ˚ ˆÌˆÏܲ í˛zͲõߨ ¢∑ï˛Ó˚ˆÏDÓ˚ ˜Ó!¢‹Tƒ §¡∫ˆÏ¶˛Ä !ܲå%Èê˛y Ó%é˛ˆÏï˛ ˛õyÓ˚Ó–
ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠ܲ!£Ï ≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠ܲ!£Ï ≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠ܲ!£Ï ≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠ܲ!£Ï ≈ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠
(i) ܲ£Ï≈ˆÏîÓ˚ ú˛ˆÏ° §%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ÈÙÙÙÈ~ܲ!ê˛ ï˛yÓ˚ˆÏܲ x = a !Ó®%ˆÏï˛ Ü˛£Ï≈î ܲÓ˚ˆÏ°ñ ˆÎ fl∫Ó˚ §,!‹T •Î˚ ï˛yˆÏï˛
§Óܲ!ê˛ §%Ó˚ í˛z˛õ!fliï˛ öyÄ ÌyܲˆÏï˛ ˛õyˆÏÓ˚– ܲyÓ˚î ܲ!£Ï≈ï˛ !Ó®%!ê˛ Ü˛áöÄ•z !öflõ® !Ó®% •ˆÏï˛˛˛õyˆÏÓ˚ öy ˆÜ˛ööy
!öflõ® !Ó®%ˆÏï˛ y ~Ó˚ üyö §ü§üˆÏÎ˚•z ¢)öƒ •Î˚– ܲyˆÏç•z ˆÎ §%Ó˚à=!°ˆÏï˛ Ü˛!£Ï≈ï˛ !Ó®%ˆÏï˛ ~ܲ!ê˛ !öflõ®
!Ó®% ÌyˆÏܲñ ˆ§•z §%Ó˚=!° fl∫ˆÏÓ˚ í˛z˛õ!fliï˛ ÌyܲˆÏÓ öy– §%ï˛Ó˚yÇñ xyüyˆÏòÓ˚ ˆòáˆÏï˛ •ˆÏÓ s-~Ó˚ ˆÜ˛yöÄ ˆÜ˛yöÄ
üyˆÏöÓ˚ çöƒ x =a • Ï° as = 0 •ˆÏÓ– ~Ó˚ ¢ï≈˛ •° ≠
sin sin 0x a
s x s al lπ π
=
⎡ ⎤ = =⎢ ⎥⎣ ⎦Óyñ ,s a n
lπ π= ˆÎáyˆÏö n = ˛õ)î≈§Çáƒy 1, 2, 3, •zï˛ƒy!ò xÌÓy ˛õ)î≈ §Çáƒy
s = nla ...9.19
§%ï˛Ó˚yÇñ ï˛yÓ˚!ê˛ˆÏܲ Î!ò r-§Çáƒyܲ §üyö û˛yˆÏà û˛yà ܲÓ˚y ÎyÎ˚ ~ÓÇ ~Ó˚ m-ï˛ü û˛yà !Ó®%ˆÏï˛ Ü˛£Ï≈î ܲÓ˚y
•Î˚ SxÌ≈yÍ a Î!ò
mlr
•Î˚Vñï˛y•ˆÏ° ˆÎ fl∫Ó˚ §,!‹T •ˆÏÓ ï˛yˆÏï˛ s =
nlr nam
=
Î!ò ˛õ)î≈ §Çáƒy •Î˚ ï˛ˆÏÓ s-ï˛ü
§üˆÏü°!ê˛ xö%˛õ!fliï˛ ÌyܲˆÏÓ– xyÓyÓ˚ ˆÎˆÏ•ï%˛ n ˆÎ ˆÜ˛yöÄ ˛õ)î≈ §Çáƒy •ˆÏï˛ ˛õyˆÏÓ˚ñ s-ï˛ü §%ˆÏÓ˚Ó˚ çöƒ !öflõ®
!Ó®% 2s-ï˛üñ 3s-ï˛ü ≤Ãû,˛!ï˛ §%ˆÏÓ˚Ó˚ çöƒÄ !öflõ® !Ó®% •ˆÏÓ– ܲyˆÏçܲyˆÏç•z ~•z§Ó í˛zFã˛ï˛Ó˚ §üˆÏü°=!°
ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚ xö%˛õ!fliï˛ ÌyܲˆÏÓ– ~!ê˛•z •° •zÎ˚Ç ˆ•°‰ü‰ˆÏ•y°Í§‰ÈÙÈ~Ó˚ (Young Helmholtz) §)e– ï˛yˆÏÓ˚Ó˚
ÓyòƒÎsf !öü≈yˆÏî •zÎ˚Ç ˆ•°ü‰ˆÏ•y°Í§‰ §)ˆÏeÓ˚ ≤ÈÏÎ˚yà âˆÏê˛– ˆÜ˛yöÄ ï˛yÓ˚ üôƒ!Ó®%ˆÏï˛ Ü˛!£Ï≈ï˛ •ˆÏ° !mï˛#Î˚ñ ã˛ï%˛Ì≈
≤Ãû,˛!ï˛ Î%@¬ §üˆÏü°=!° §¡õ)î≈ Ó!ç≈ï˛ •Î˚– ~ˆÏï˛ fl∫ˆÏÓ˚Ó˚ üyô%ˆÏÎ≈Ó˚ •y!ö •Î˚– ~çöƒ ÓyòƒÎˆÏsf ï˛yˆÏÓ˚Ó˚ ~ܲ
≤ÃyˆÏhsˇÓ˚ ܲyåÈyܲy!åÈ !Ó®%ˆÏï˛ ï˛yÓ˚!ê˛ Ü˛!£Ï≈ï˛ •Î˚ñ ÎyˆÏï˛ ≤ÃyÎ˚ §Ó §üˆÏü° í˛zͲõߨ •ˆÏï˛ ˛õyˆÏÓ˚–
ï˛ˆÏÓ §Ó ÓyòƒÎˆÏsfÓ˚ ˆ«˛ˆÏe •zÎ˚Ç ˆ•°‰ü‰ˆÏ•y°Í§‰ §)e §¡õ)î≈ áyˆÏê˛ öy– ~Ó˚ ܲyÓ˚îñ ÓyòƒÎˆÏsf ÓƒÓ•*ï˛ ôyï˛Ó
ï˛yÓ˚ §¡õ)î≈ öüö#Î˚ •Î˚ öy ~ÓÇ Ü˛£Ï≈ˆÏîÓ˚ §üÎ˚ ܲ!£Ï≈ï˛ !Ó®%ˆÏï˛ ï˛yÓ˚!ê˛ ~ܲ!ê˛ ˆÜ˛yˆÏîÓ˚ ˛õ!Ó˚ÓˆÏï≈˛ ~ܲ!ê˛ «%˛o
Óe´ˆÏÓ˚áy Ó˚ã˛öy ܲˆÏÓ˚– ~åÈyí˛¸y ï˛yˆÏÓ˚Ó˚ Ó¶˛ö# Óy ˆÎ ˆ§ï%˛Ó˚ í˛z˛õÓ˚ !òˆÏÎ˚ ï˛yÓ˚!ê˛ ê˛yöy ÌyˆÏܲ ˆ§!ê˛ §¡õ)î≈ ò,ì˛¸
•Î˚ öy–
249
í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ ôÓ˚y ÎyÜñ ï˛yÓ˚!ê˛ˆÏܲ ≤Ãyhsˇ ˆÌˆÏܲ ˜òˆÏâ≈ƒÓ˚ ~ܲÈÙÈï,˛ï˛#Î˚yÇ¢ ò)ˆÏÓ˚ ܲ£Ï≈≈î ܲÓ˚y •°– ~ˆÏ«˛ˆÏe r =
3, m = 1– ~áö ˆÎ §üˆÏü°=!° xö%˛õ!fliï˛ ÌyܲˆÏÓñ ˆ§=!° e´!üܲ §Çáƒy •° s = n·rm = n·3 xÌ≈yÍ
s = 3, 6, 9........– í˛zͲõߨ fl∫ˆÏÓ˚ ï,˛ï˛#Î˚ñ £Ï¤˛ñ öÓü ≤Ãû,˛!ï˛ §üˆÏü°=!° ÌyܲˆÏÓ öy–
(ii) §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ 9.18 §ü#ܲÓ˚î ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ !ö!ò≈‹T ˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ Ü˛!£Ï≈ï˛
ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ ~ܲ!ê˛ ˛õ!Ó˚°!«˛ï˛ !Ó®%ˆÏï˛ s - ï˛ü §%ˆÏÓ˚Ó˚ !ÓhflÏyÓ˚ ò%!ê˛ !ӣψÏÎ˚Ó˚ IJõÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ≤ÃÌü!ê˛
•° ܲ!£Ï≈ï˛ !Ó®%Ó˚ fliyöyB˛ Ä ≤ÃyÌ!üܲ !Óã%˛ƒ!ï˛Ó˚ ˛õ!Ó˚üyî xÌ≈yÍ a Ä h ~ÓÇ !mï˛#Î˚!ê˛ •° ï˛yˆÏÓ˚Ó˚ xyÓk˛ ≤Ãyhsˇ
ˆÌˆÏܲ ˛õ!Ó˚°!«˛ï˛ !Ó®%Ó˚ ò)Ó˚cñ x–
≤ÃÌüï˛ñ Î!ò ܲ!£Ï≈ï˛ !Ó®%Ó˚ xÓfliyö Ä ≤ÃyÌ!üܲ !Óã%˛ƒ!ï˛ˆÏܲ !fliÓ˚ Ó˚yáy ÎyÎ˚ñ ï˛y•ˆÏ° s - ï˛ü §%ˆÏÓ˚Ó˚ !ÓhflÏyÓ˚
xyÓk˛ ≤Ãyhsˇ ˆÌˆÏܲ !Ó®%!ê˛Ó˚ ò)Ó˚c x-~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲÓ˚ˆÏÓ– ~•z !ÓhflÏyÓ˚ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ (asm
) •ˆÏÓ Îáö
sin
1 sxlπ=
xÌ≈yÍñ
12
sxjlππ ⎛⎞ =+ ⎜⎟ ⎝⎠
xÌÓyñ x =
11,2
js
⎛⎞ + ⎜⎟ ⎝⎠
j = ˛õ)î≈ §Çáƒy–
xï˛~Óñ x-~Ó˚ í˛z˛õ!Ó˚!°!áï˛ üyˆÏöÓ˚ çöƒ as= a
sm=
()2
2221sin hlsa
l alasπ
π×
−
...9.20
í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ í˛zòy•Ó˚î ≠ ï˛yÓ˚!ê˛ˆÏܲ Î!ò a = 12 !Ó®%ˆÏï˛ Ü˛£Ï≈î ܲÓ˚y •Î˚ ï˛y•ˆÏ° s - ï˛ü §%ˆÏÓ˚ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ §ˆÏÓù≈yFã˛
!ÓhflÏyÓ˚ •ˆÏÓ
asm=
22228118 sin·sin
22×= hshs
l ssππ
ππ
...9.21
s Îáö Î%@¬ ˛õ)î≈ §Çáƒy (2, 4, 6,...) ï˛áö sin
2sπ
= 0, §%ï˛Ó˚yÇ
θ
sm = 0– ~!ê˛ âê˛ˆÏÓ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏö
!mï˛#Î˚ñ ã˛ï%˛Ì≈ñ £Ï¤˛ •zï˛ƒy!ò §üˆÏüˆÏ°Ó˚ ˆ«˛ˆÏeñ xÌ≈yÍ ~•z §üˆÏü°=!° ~ˆÏܲÓyˆÏÓ˚•z ÌyܲˆÏÓ öy– xyÓyÓ˚ asm
ÈÙÈ ~Ó˚ üyö §ˆÏÓ≈yFã˛ •ˆÏÓ Îáö s ~ܲ!ê˛ xÎ%@¬ ˛õ)î≈ §Çáƒy (1, 3, 5,....), ˆÜ˛ööy ï˛áö sin
2sπ
= 1~ÓÇ asm
=
228h
s π
–
!mï˛#Î˚ï˛ñ ܲ!£Ï≈ï˛ !Ó®%Ó˚ fliyöyB˛ ~ÓÇ xyê˛Ü˛yˆÏöy ≤Ãyhsˇ ˆÌˆÏܲ ˛õ!Ó˚°!«˛ï˛ !Ó®%!ê˛Ó˚ ò)Ó˚c x Î!ò !fliÓ˚ ÌyˆÏܲñ
ï˛y•ˆÏ° !ÓhflÏyÓ˚ (as)s-~Ó˚ ÓˆÏà≈Ó˚ ÓƒhflÏyö%˛õyˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– xÌ≈yÍñ as
21s
∝
– ˆÎˆÏ•ï%˛ ï˛#Ó ï˛yÓ˚ !ÓhflÏyˆÏÓ˚
250
ÓˆÏà≈Ó˚ §üyö%˛õy!ï˛Ü˛ñ ܲyˆÏç•z ï˛#Ó ï˛y (Is) - ˆÜ˛ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ I
s 4
1s
∝ – ~Ó˚ ˆÌˆÏܲ flõ‹T •ˆÏFåÈ ˆÎñ
í˛zFã˛ï˛Ó˚ §%Ó˚=!°Ó˚ ï˛#Ó ï˛y á%Ó o&ï˛à!ï˛ˆÏï˛ •…y§ ˛õyˆÏÓ– ôÓ˚&öñ ï˛yÓ˚!ê˛ Î!ò üôƒ!Ó®%ˆÏï˛ Ü˛!£Ï≈ï˛ •Î˚ ≤ÃÌüñ ï,˛ï˛#Î˚ñ
˛õMÈ˛ü §%Ó˚=!°Ó˚ ï˛#Ó ï˛yÓ˚ xö%˛õyï˛ •ˆÏÓ I1: I2 : I3 = 1 : 4 41 1:3 5
(iii) fl∫ˆÏÓ˚Ó˚ çy!ï˛ ≠ fl∫ˆÏÓ˚Ó˚ çy!ï˛ ≠ fl∫ˆÏÓ˚Ó˚ çy!ï˛ ≠ fl∫ˆÏÓ˚Ó˚ çy!ï˛ ≠ fl∫ˆÏÓ˚Ó˚ çy!ï˛ ≠ ˆÎ fl∫ˆÏÓ˚ ü)°§%Ó˚ åÈyí˛¸y xöƒ §üˆÏü° ÌyˆÏܲ öy ï˛y ◊&!ï˛üô%Ó˚ •Î˚ öy– ˆÜ˛yöÄ
§%Ó˚¢°yܲyˆÏܲ ÓyòƒÎsf !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚ öy ˆÜ˛ööy ~=!°Ó˚ ~ܲ!ê˛üye ܲ¡õyB˛ ÌyˆÏܲ– ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚
ˆÎ fl∫Ó˚ §,!‹T •Î˚ ï˛yÓ˚ çy!ï˛ !öî#≈ï˛ •Î˚ ˆ§•z fl∫ˆÏÓ˚ í˛z˛õ!fliï˛ Ü˛ï˛Ü˛=!° §%Ó˚ xyˆÏåÈ ï˛yÓ˚ §Çáƒy ~ÓÇ §%Ó˚=!°Ó˚
xyˆÏ˛õ!«˛Ü˛ (relative) ï˛#Ó ï˛yÓ˚ myÓ˚y– ï˛yÓ˚!ê˛ˆÏܲ a ò)Ó˚ˆÏc ܲ£Ï≈î ܲÓ˚ˆÏ° §Ω˛yÓƒ §%Ó˚=!°Ó˚ §Çáƒy
al
xö%˛õyï˛
myÓ˚y !öî≈Î˚ ܲÓ˚y ÎyÎ˚– §%ï˛Ó˚yÇñ §,‹T fl∫Ó˚ í˛zß¨ï˛ çy!ï˛Ó˚ (good quality) ~ÓÇ ◊&!ï˛üô%Ó˚ •ˆÏÓ Î!ò ï˛yÓ˚!ê˛ˆÏܲ
Ó,•Í §ÇáƒÜ˛ §üû˛yˆÏà û˛yà ܲÓ˚y ÎyÎ˚ ~ÓÇ ~ܲ ≤Ãyhsˇ ˆÌˆÏܲ !öܲê˛ï˛ü !Óû˛yçö !Ó®%ˆÏï˛ Ü˛£Ï≈î ܲÓ˚y •Î˚– ~ˆÏï˛˛
˛õÓ˚˛ ˛õÓ˚ xˆÏöܲ=!° §üˆÏü° í˛zͲõߨ •ˆÏÓ– ˆÎ §%Ó˚=!° xö%˛õ!fliï˛ ÌyܲˆÏÓ ï˛yˆÏòÓ˚ ï˛#Ó ï˛y öàîƒ •ï˛ñ ÎyÓ˚
ú˛ˆÏ° ˆ§=!°Ó˚ xö%˛õ!fli!ï˛ˆÏï˛ í˛zͲõߨ ¢ˆÏ∑Ó˚ çy!ï˛Ó˚ ˆÜ˛yöÄ •y!ö âê˛ˆÏÓ öy– í˛z˛õ!fliï˛ §%Ó˚=!°Ó˚ §Çáƒy ˆÎ
Ó›!ê˛ !òˆÏÎ˚ ܲ£Ï≈î ܲÓ˚y •Î˚ ï˛yÓ˚ ≤ÃÜ,˛!ï˛Ó˚ IJõÓ˚Ä !öû≈˛Ó˚ ܲˆÏÓ˚–
~ÓyÓ˚ xy˛õ!ö !öˆÏç ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ ˆòÄÎ˚yÓ˚ ˆã˛‹Ty ܲÓ˚&ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 2
~ܲ!ê˛ ê˛yöy ï˛yÓ˚ˆÏܲ xyÓk˛˛ ≤Ãyhsˇ ˆÌˆÏܲ
4l
ò)Ó˚ˆÏc ܲ£Ï≈î ܲÓ˚y •°– ~ˆÏï˛ ˆÎ fl∫Ó˚ §,!‹T •ˆÏÓ˛ ï˛yˆÏï˛ ˆÜ˛yö
ˆÜ˛yö §%Ó˚ xö%˛õ!fliï˛ ÌyܲˆÏÓ⁄
9.4 xy•ï˛ ï˛yÓ˚xy•ï˛ ï˛yÓ˚xy•ï˛ ï˛yÓ˚xy•ï˛ ï˛yÓ˚xy•ï˛ ï˛yÓ˚ (Struck string)
!˛õÎ˚yˆÏöy ~ܲ!ê˛ çö!≤ÃÎ˚ ÓyòƒÎsf– ~Ó˚ ï˛yÓ˚ˆÏܲ ~ܲ!ê˛ ˆú˛ˆÏŒê˛ (Felt) xyÓ,ï˛ •yï%˛!í˛¸Ó˚ myÓ˚y xyâyï˛ Ü˛Ó˚y
•Î˚ ~ÓÇ xyâyï˛ Ü˛Ó˚yÓ˚ ˛õÓ˚ •yï%˛!í˛¸!ê˛ xyÓyÓ˚ !öˆÏçÓ˚ fliyˆÏö ˆú˛Ó˚ï˛ xyˆÏ§– üˆÏö ܲÓ˚y Îyܲñ l ˜òâ≈ƒ ~ÓÇ m
˜Ó˚!áܲ âöˆÏcÓ˚ ~ܲ!ê˛ ï˛yÓ˚ˆÏܲ T ê˛yö myÓ˚y ˆê˛ˆÏö Ó˚yáy •ˆÏÎ˚ˆÏåÈ– x˛õ!Ó˚Ó!ï≈˛ï˛ xÓfliyÎ˚ ï˛yÓ˚!ê˛Ó˚ xyÓk˛ !òܲ
ˆÌˆÏܲ (x = 0) x = a ò)Ó˚ˆÏc ï˛yÓ˚!ê˛ˆÏܲ ~ܲ!ê˛ ˛õƒyí˛Î%_´ •yï%˛!í˛¸ !òˆÏÎ˚ ~üöû˛yˆÏÓ xyâyï˛ Ü˛Ó˚y •° ÎyˆÏï˛
~•z xyâyˆÏï˛Ó˚ §üÎ˚ x = a ~ÓÇ x = a + da ~•z «%˛o ˜òˆÏâ≈ƒÓ˚ üˆÏôƒ•z ï˛yˆÏÓ˚Ó˚ §ˆÏD •yï%˛!í˛¸Ó˚ §ÇˆÏÎyà âˆÏê˛–
~áö xyüÓ˚y ê˛yöy ï˛yˆÏÓ˚Ó˚ §yôyÓ˚î §Ó˚ˆÏîÓ˚ 9.7 §ü#ܲÓ˚î ÓƒÓ•yÓ˚ ܲÓ˚Ó– ~•z §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y ˛õy•z
y (x, t) =
cossinsin sss
sctsctsx ABlllπππ ⎡⎤ + ⎢⎥ ⎣⎦ ∑
251
~áö xyüÓ˚y xy•ï˛ ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ≤ÈÏÎyçƒ ê˛yöy ¢ï≈˛=!° xyˆÏÓ˚y˛õ ܲÓ˚Ó–
(i) ≤ÃÌü §#üy¢ï≈˛!ê˛ •°ñ ≤ÃyÓ˚!Ω˛Ü˛ xÓfliyÎ˚ñ Îáö t = 0, ï˛yˆÏÓ˚Ó˚ §Ó !Ó®%ˆÏï˛ §Ó˚î ¢)öƒ •ˆÏÓ–
y(x, t)-~Ó˚ Ó˚y!¢üy°yÎ˚ t = 0 Ó§yˆÏ° xy˛õ!ö ˛õyˆÏÓöñ
y (x, 0) = sinss
s xAlπ∑
í˛zû˛Î˚!òܲˆÏܲ sin r x dx
lπ
!òˆÏÎ˚ =î ܲˆÏÓ˚ x = 0 ˆÌˆÏܲ x = l ˛õyÕ‘yÎ˚ §üyܲ°ö ܲÓ˚ˆÏ° ˛õyÄÎ˚y ÎyÎ˚ S~áyˆÏö
r = ˛õ)î≈ §ÇáƒyV
()00
,0sinsinsinll
ss xx
rxsxrx yxdxAdxlllπππ
==
=∑ ∫∫
Óyü!òˆÏܲÓ˚ Ó˚y!¢!ê˛Ó˚ üyö ¢)öƒñ ˆÜ˛ööy y (x, 0) = 0– í˛yö!òˆÏܲÓ˚ §üyܲ°!ê˛Ó˚ üyö Ar·12 ˆÜ˛ööy
0sinsinlsxrxdx
llππ ∫
- ~Ó˚ üyö s
≠
r •ˆÏ° ¢)öƒ •ˆÏÓ–
§%ï˛Ó˚yÇñ Ar
12
= 0 Óy Ar = 0– xÌ≈yÍ A
1, A
2, A
3 ≤Ãû,˛!ï˛ §Ó=!° ô &ÓˆÏܲÓ˚ üyö•z ¢)öƒ– §%ï˛Ó˚yÇñ
y(x, t) =
sinsin ss
sctsx Bllππ ∑
...9.22
(ii) !mï˛#Î˚ §#üy¢ï≈˛!ê˛ •°ñ ≤ÃyÓ˚!Ω˛Ü˛ xÓfliyÎ˚ ï˛yˆÏÓ˚Ó˚ ˆÓà
y (x, t) = u0 Îáöñ a
≤
x < a +
Δ
(
Δ
= «%˛o ˜òâ≈ƒV
= 0 Îáö x ~Ó˚ üyö !û˛ß¨–
xÌ≈yÍñ •yï%˛!í˛¸Ó˚ xyâyˆÏï˛Ó˚ ú˛ˆÏ° ï˛yˆÏÓ˚Ó˚ x = a !Ó®%Ó˚ §!ߨ!•ï˛
Δ
˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ á%Ó ˆåÈyê˛ xÇ¢ u0 ˆÓà
°yû˛ ܲˆÏÓ˚ !ܲv ≤ÃyÓ˚!Ω˛Ü˛ xÓfliyÎ˚ ï˛yˆÏÓ˚Ó˚ Óy!ܲ xÇ¢ §yüƒyÓfliyˆÏö ÌyˆÏܲ– ~ÓyÓ˚ §#üy¢ï≈˛!ê˛ ≤ÈÏÎ˚yà ܲÓ˚y Îyܲ–
y (x, t) - ~Ó˚ Ó˚y!¢üy°y 9.22 ˆÌˆÏܲ xyüÓ˚y ˛õy•z ≠
(,0)sin ss
scsx yxBllππ =∑
~•z §ü#ܲÓ˚ˆÏîÓ˚ ò%•z !òܲˆÏܲ sin r x dx
lπ
!òˆÏÎ˚ =î ܲˆÏÓ˚ x = 0 Ä x = l §#üyÓ˚ üˆÏôƒ §üyܲ°ö ܲÓ˚ˆÏ°
252
()00
,0sinsinsinll
ss
rxscsxrx yxdxBdxllllππππ =∑ ∫∫
~•z §ü#ܲÓ˚ˆÏîÓ˚ Óyü!òˆÏܲÓ˚ Ó˚y!¢!ê˛Ó˚ üyö≠
00
0·sin sin 0·sina a l
a a
r x r x r xdx u dx dxl l lπ π π
+Δ
+Δ
+ +∫ ∫ ∫
= u0sin
r alπ
·
Δ
ˆÜ˛ööy
Δ
˜òˆÏâ≈ƒÓ˚ í˛z˛õÓ˚ x-~Ó˚ üyö a ӈϰ ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚–
í˛yö!òˆÏܲÓ˚ Ó˚y!¢!ê˛Ó˚ üyö Br
122 r
rcrc Blππ =
§%ï˛Ó˚yÇ ˆ°áy ÎyÎ˚ñ Bs = 0
2 sin s aus c l
ππ
Δ –
ï˛yˆÏÓ˚Ó˚ Δ ˜òˆÏâ≈ƒÓ˚ xÇ¢ê%%˛Ü%˛ ˆÎ ≤ÃyÌ!üܲ û˛Ó˚ˆÏÓà °yû˛ ܲˆÏÓ˚ ï˛yÓ˚ üyö
p =
ρΔ
·u0, ˆÎáyˆÏö
ρ
= ï˛yˆÏÓ˚Ó˚ ~ܲܲ ˜òˆÏâ≈ƒÓ˚ û˛Ó˚–
∴u0 Δ = p
ρ
~ÓÇñ Bs =
2sin psascpl
ππ
Bs-~Ó˚ ~•z üyö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ 9.22 §ü#ܲÓ˚î!ê˛ ˆ°áy ÎyÎ˚ ≠
y(x, t) = 2 1 sin sin sin∑
s
p s a s x s ctcp s l l l
π π ππ
...9.23
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î ˆÌˆÏܲ flõ‹T•z ˆÓyé˛y ÎyÎ˚ ˆÎñ xy•ï˛ ï˛yˆÏÓ˚ ˆÎ fliyî% ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚ñ ï˛yÓ˚ !ÓhflÏyÓ˚ ï˛yˆÏÓ˚Ó˚
~ܲ ~ܲ fliyˆÏö ~ܲ ~ܲ ˛õ!Ó˚üyˆÏî •Î˚– xyÓk˛ ≤Ãyhsˇ ˆÌˆÏܲ x ò)Ó˚ˆÏc s-ï˛ü §%ˆÏÓ˚ ˆÎ ܲ¡õö •Î˚ ï˛yÓ˚ !ÓhflÏyÓ˚
as =
2 1· ·sin sinp s a s xc s l l
π ππ ρ
...9.24
9.23 §ü#ܲÓ˚î!ê˛ˆÏܲ ï˛áö ~û˛yˆÏÓÄ ˆ°áy ÎyÎ˚ ≠
y (x, t) = sinss
s ctal
π∑ ...9.25
253
xy•ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠xy•ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠xy•ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠xy•ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠xy•ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ˜Ó!¢‹Tƒ ≠
(i) §%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ ≠§%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ ≠§%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ ≠§%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ ≠§%ˆÏÓ˚Ó˚ xö%˛õ!fli!ï˛ ≠ 9.23 §ü#ܲÓ˚î ˆÌˆÏܲ xy˛õ!ö §•ˆÏç•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ï˛yˆÏÓ˚Ó˚ xyÓk˛
≤Ãyhsˇ ˆÌˆÏܲ xy•ï˛ !Ó®%Ó˚ (struck point) ò)Ó˚c a Î!ò ~üö •Î˚ ˆÎ sin s a
lπ
~Ó˚ üyö ¢)öƒ •Î˚ñ ï˛ˆÏÓ ï˛yˆÏÓ˚Ó˚
ܲ¡õˆÏö s-ï˛ü §%Ó˚ xö%˛õ!fliï˛ ÌyܲˆÏÓ– ~•z ¢ˆÏï≈˛Ó˚ xÌ≈
salπ
= r
π
(r = ˛õ)î≈ §ÇáƒyV Óy a = l·
rs
–
Î!ò ï˛yÓ˚!ê˛ˆÏܲ s §üyöû˛yˆÏà û˛yà ܲÓ˚y ÎyÎ˚ ~ÓÇ ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ !Óû˛yçö !Ó®%ˆÏï˛ xyâyï˛ Ü˛Ó˚y •Î˚
ï˛y•ˆÏ° ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏö s-ï˛ü §%Ó˚!ê˛ xö%˛õ!fliï˛ •ˆÏÓ– ~áyˆÏö ü)°ö#!ï˛!ê˛ ~•z ˆÎñ xy•ï˛ !Ó®%ˆÏï˛ ˆÎ §%ˆÏÓ˚Ó˚
~ܲ!ê˛ !öflõ® !Ó®% Ìyܲï˛ñ ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏö ˆ§•z §%Ó˚ ÌyܲˆÏï˛ ˛õyˆÏÓ˚ öy– xyÓyÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ !Ó®% s-ï˛ü
§%ˆÏÓ˚Ó˚ !öflõ® !Ó®% •ˆÏ° ˆ§!ê˛ 2s-ï˛üñ 3s-ï˛ü •zï˛ƒy!ò §%Ó˚=!°Ó˚Ä !öflõ® !Ó®% •ˆÏÓ– xÌ≈yÍñ 2s-ï˛üñ
3s-ï˛ü •zï˛ƒy!ò §%Ó˚=!°Ä ˆ§ˆÏ«˛ˆÏe xö%˛õ!fliï˛ •ˆÏÓ–
(ii) §%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠§%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠§%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠§%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠§%ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒ ≠ 9.23 §ü#ܲÓ˚î ˆÌˆÏܲ xy˛õ!ö xyˆÏà•z ˆòˆÏáˆÏåÈö ˆÎñ ï˛yˆÏÓ˚Ó˚ xyÓk˛ ≤Ãyhsˇ ˆÌˆÏܲ
x ò)Ó˚ˆÏc s-ï˛ü §%ˆÏÓ˚Ó˚ !ÓhflÏyÓ˚ as xy•ï˛ !Ó®%Ó˚ ‘a’ ~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– xyÓyÓ˚ ï˛yÓ˚!ê˛ˆÏܲ !ö!ò≈‹T ~ܲ!ê˛
!Ó®%ˆÏï˛ xyâyï˛ Ü˛Ó˚ˆÏ°ñ xÌ≈yÍ xy•ï˛ !Ó®%Ó˚ a ˆÜ˛ !fliÓ˚ Ó˚yáˆÏ°ñ s-ï˛ü §%ˆÏÓ˚Ó˚ !ÓhflÏyÓ˚ Ók˛ ≤Ãyhsˇ ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚
(= x) !Ó!û˛ß¨ üyˆÏö !Ó!û˛ß¨ •Î˚– !ÓhflÏyÓ˚ as §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ •ˆÏÓ Îáö sin1 srx
l=±
xÌÓyñ x = 12s
(2n +
1), n = ˛õ)î≈ §Çáƒy– !ÓhflÏyˆÏÓ˚Ó˚ §ˆÏÓ≈yFã˛ üyö •ˆÏÓ
asm=
21··sin psacsl
ππρ
...9.26
~•z §ˆÏÓ≈yFã˛ !ÓhflÏyÓ˚ §üˆÏüˆÏ°Ó˚ e´ü ‘s’ ÈÙÈ ~Ó˚ §ˆÏD ÓƒyhflÏyö%˛õyˆÏï˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– °«˛ƒ ܲÓ˚&öñ ~ˆÏ«˛ˆÏe
§%ˆÏÓ˚Ó˚ ï˛#Ó ï˛y I 21s
∝ – xÌ≈yÍ §%ˆÏÓ˚Ó˚ e´ˆÏüÓ˚ (order) Ó,!k˛Ó˚ §ˆÏD ï˛#Ó ï˛y •…y§ ˆ˛õˆÏ°Ä ~•z •…yˆÏ§Ó˚ •yÓ˚
ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ï%˛°öyÎ˚ ܲü o&ï˛–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 3
(a) l ˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ ï˛yÓ˚ˆÏܲ xyÓk˛ ≤Ãyhsˇ ˆÌˆÏܲ 3l
ò)ˆÏÓ˚ xyâyï˛ Ü˛Ó˚y •°– 9.18a §ü#ܲÓ˚ˆÏîÓ˚ §y•yˆÏ΃
ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ s-ï˛ü §üˆÏüˆÏ°Ó˚ !ÓhflÏyˆÏÓ˚Ó˚ Ó˚y!¢üy°y!ê˛ !°á%ö– ˆÜ˛yö ˆÜ˛yö §%ˆÏÓ˚ !ÓhflÏyÓ˚ §Ó≈!ö¡¨ SxÌ≈yÍ
¢)öƒV xyÓ˚ ˆÜ˛yö ˆÜ˛yö §%ˆÏÓ˚Ó˚ çöƒ•z Óy !ÓhflÏyÓ˚ §ˆÏÓ≈yFã˛ •ˆÏÓ⁄ §ˆÏÓ≈yFã˛ !ÓhflÏyˆÏÓ˚Ó˚ Ó˚y!¢üy°y!ê˛ !öî≈Î˚ ܲÓ˚&ö–
254
(b) xy•ï˛ ï˛yˆÏÓ˚Ó˚ ày!î!ï˛Ü˛ !ӈϟ’£ÏˆÏî ôˆÏÓ˚ ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈ ˆÎñ •yï%˛!í˛¸!ê˛ ï˛yˆÏÓ˚Ó˚ xï˛ƒhsˇ «%˛o ˜òˆÏâ≈ƒ xyâyï˛
ܲˆÏÓ˚– ~!ê˛ §ï˛ƒ öy •ˆÏ° !ӈϟ’£ÏˆÏîÓ˚ ˆÜ˛yöÄ ˛õyÌ≈ܲƒ •ˆÏÓ !ܲ⁄
9.5 åÈí˛¸ê˛yöy ï˛yÓ˚åÈí˛¸ê˛yöy ï˛yÓ˚åÈí˛¸ê˛yöy ï˛yÓ˚åÈí˛¸ê˛yöy ï˛yÓ˚åÈí˛¸ê˛yöy ï˛yÓ˚ (Bowed string)
xyüyˆÏòÓ˚ §%˛õ!Ó˚!ã˛ï˛ ˆÓ•y°y Ä ~§Ó˚yç •° åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ÓyòƒÎˆÏsfÓ˚ í˛zòy•Ó˚î– åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏö
~ܲ!ê˛ x!û˛öÓc xyˆÏåÈ– ï˛yÓ˚!ê˛ !öçfl∫ ܲ¡õyˆÏB˛ ܲ!¡õï˛ •ˆÏ°Ä ~!ê˛ ï˛yˆÏÓ˚Ó˚ fl∫yû˛y!Óܲ ܲ¡õö öÎ˚ñ ܲyÓ˚î
ܲ¡õˆÏöÓ˚ çöƒ åȈÏí˛¸Ó˚ ≤Ãû˛yÓ xÓ¢ƒ ≤ÈÏÎ˚yçö#Î˚– ï˛yÓ˚!ê˛ à!ï˛¢#° åÈí˛¸ ˆÌˆÏܲ•z ܲ¡õˆÏöÓ˚ çöƒ ï˛yÓ˚ ≤ÈÏÎ˚yçö#Î˚
¢!_´ ˆ˛õˆÏÎ˚ ÌyˆÏܲ– !ܲv ï˛y•ˆÏ°Ä ~!ê˛ ˛õÓ˚Ó¢ ܲ¡õö (forced vibration) öÎ˚ñ ܲyÓ˚î ï˛yÓ˚!ê˛ ï˛yÓ˚ fl∫yû˛y!Óܲ
ܲ¡õyˆÏB˛•z xyˆÏ®y!°ï˛ •Î˚ñ åÈí˛¸!ê˛Ó˚ ˆÜ˛yöÄ !öçfl∫ ܲ¡õyB˛ ˆö•z– ~çöƒ åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õöˆÏܲ Ó°y •Î˚
ˆ˛õy!£Ïï˛ Ü˛¡õö ˆ˛õy!£Ïï˛ Ü˛¡õö ˆ˛õy!£Ïï˛ Ü˛¡õö ˆ˛õy!£Ïï˛ Ü˛¡õö ˆ˛õy!£Ïï˛ Ü˛¡õö (maintained vibration)–
åȈÏí˛¸Ó˚ à!ï˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õöˆÏܲ ÓçyÎ˚ Ó˚yˆÏá– ~ܲ!ê˛ åÈí˛¸ˆÏܲ ï˛yˆÏÓ˚Ó˚ IJõÓ˚ ã˛y°yˆÏ° åÈí˛¸!ê˛ ï˛yÓ˚!ê˛ˆÏܲ ï˛yÓ˚
§ˆÏD ˆê˛ˆÏö !öˆÏÎ˚ ÎyÎ˚– ~!ê˛ §Ω˛Ó •Î˚ ï˛yÓ˚ Ä åȈÏí˛¸Ó˚ üˆÏôƒ !fliï˛#Î˚ â£Ï≈ˆÏîÓ˚ ú˛ˆÏ°– åȈÏí˛¸Ó˚ ò%Û!òˆÏܲÓ˚ ï˛yˆÏÓ˚Ó˚
ò%!ê˛ xÇ¢ åȈÏí˛¸Ó˚ !òˆÏܲ e´ˆÏü e´ˆÏü xyöï˛ •ˆÏï˛ ÌyˆÏܲ– ú˛ˆÏ° ê˛yöy ï˛yˆÏÓ˚Ó˚ ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ üyö e´ü¢ Ó,!k˛
˛õyÎ˚– ~ܲ§üÎ˚ ~•z ≤Ãï˛ƒyöÎ˚ܲ ӈϰÓ˚ üyö !fliï˛#Î˚ â£Ï≈î ӈϰÓ˚ ˆÌˆÏܲ ˆÓ!¢ •Î˚ ~ÓÇ ï˛yÓ˚!ê˛ !˛õåȈϰ !àˆÏÎ˚
!öçfl∫ xÓfliyˆÏö ˆú˛Ó˚ï˛ xy§ˆÏï˛ ÌyˆÏܲ– çyˆÏí˛ƒÓ˚ çöƒ ˆ§!ê˛ !öçfl∫ xÓfliyö åÈy!í˛¸ˆÏÎ˚ ã˛ˆÏ° ÎyÎ˚– ï˛yÓ˚!ê˛ Îï˛«˛î
åˆÏí˛¸Ó˚ §yˆÏ˛õˆÏ«˛ à!ï˛¢#° ÌyˆÏܲ ï˛ï˛«˛î ï˛yˆÏÓ˚Ó˚ í˛z˛õÓ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° åÈyí˛¸yÄ àï˛#Î˚ â£Ï≈î Ó° ܲyç ܲˆÏÓ˚–
~•z ˛ú˛ˆÏ°•z ï˛yÓ˚!ê˛ ü!®ï˛ •ˆÏÎ˚ e´ü¢ !fliÓ˚ xÓfliyÎ˚ xyˆÏ§– ï˛áö åÈí˛¸ !òˆÏÎ˚ xyÓyÓ˚ ï˛yÓ˚!ê˛ˆÏܲ ˆê˛ˆÏö !öˆÏÎ˚ ÎyÄÎ˚y
•Î˚– Îï˛«˛î åÈí˛¸!ê˛ ï˛yˆÏÓ˚Ó˚ §ÇflõˆÏ¢≈ à!ï˛¢#° ÌyˆÏܲ ï˛ï˛«˛î ~•z åÈí˛¸ !òˆÏÎ˚ ï˛yÓ˚!ê˛ §yüˆÏö ˆê˛ˆÏö !öˆÏÎ˚
ÎyÄÎ˚y ~ÓÇ ï˛yˆÏÓ˚Ó˚ !˛õåȈϰ ˛õ)ˆÏÓ≈Ó˚ xÓfliyˆÏö !ú˛ˆÏÓ˚ xy§y ã˛°ˆÏï˛ ÌyˆÏܲ– ï˛yˆÏÓ˚Ó˚ x@ˇÃà!ï˛Ó˚ §üÎ˚ !fliï˛#Î˚
â£Ï≈ˆÏîÓ˚ üyˆÏöÓ˚ ˛õyÌ≈ˆÏܲƒÓ˚ çöƒ•z ï˛yˆÏÓ˚Ó˚ ܲ¡õö ÓçyÎ˚ ÌyˆÏܲñ ܲyÓ˚î !fliï˛#Î˚ â£Ï≈î Ó° ï˛yˆÏÓ˚Ó˚ IJõÓ˚ ˆÓ!¢
ܲyç ܲˆÏÓ˚–
ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc=!° ~ÓyÓ˚ xyˆÏ°yã˛öy ܲÓ˚y Îyܲ–
ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc ≠ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc ≠ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc ≠ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc ≠ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ !ӈϢ£Ïc ≠
(i) ≤ÃÎ%_´ ӈϰÓ˚ ܲ¡õyB˛ ï˛yˆÏÓ˚Ó˚ !öçfl∫ ܲ¡õyˆÏB˛Ó˚ myÓ˚y•z !öÎ˚!sfï˛ •Î˚–
(ii) ≤ÃÎ%_´ Ó° ï˛sfˆÏܲ ˆÎ •yˆÏÓ˚ ¢!_´Ó˚ ˆÎyàyö ˆòÎ˚ ï˛y ¢!_´Ó˚ xÓ«˛ˆÏÎ˚Ó˚ •yˆÏÓ˚Ó˚ §üyö–
(iii) x!Ó!FåÈߨï˛yÓ˚ §)e ˆÌˆÏܲ Ó°ˆÏï˛ ˛õyÓ˚y ÎyÎ˚ ˆÎñ ï˛ˆÏsfÓ˚ ܲ¡õö ã˛°yܲy°#ö ¢!_´Ó˚ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yˆÏÓ˚
•ë˛yÍ ˆÜ˛yöÄ ˛õ!Ó˚Óï≈˛ö âê˛ˆÏï˛ ˛õyˆÏÓ˚ öy–
255
9.5.1 ˆ•°‰ü‰ˆÏ•y°Í§‰ÙÈ~Ó˚ §)e ≠ˆ•°‰ü‰ˆÏ•y°Í§‰ÙÈ~Ó˚ §)e ≠ˆ•°‰ü‰ˆÏ•y°Í§‰ÙÈ~Ó˚ §)e ≠ˆ•°‰ü‰ˆÏ•y°Í§‰ÙÈ~Ó˚ §)e ≠ˆ•°‰ü‰ˆÏ•y°Í§‰ÙÈ~Ó˚ §)e ≠
åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ à!ï˛Ó˚ !ӣψÏÎ˚ !ÓK˛yö# ˆ•°ü‰ˆÏ•y°Í§‰ÈÙÈ~Ó˚ !ï˛ö!ê˛ §)e xyˆÏåÈ– §)e=!° ~•z ≤ÃܲyÓ˚≠
≤ÃÌü §)e ÈÙÙÙÈ ≤ÃÌü §)e ÈÙÙÙÈ ≤ÃÌü §)e ÈÙÙÙÈ ≤ÃÌü §)e ÈÙÙÙÈ ≤ÃÌü §)e ÈÙÙÙÈ åȈÏí˛¸Ó˚ §ÇflõˆÏ¢≈ ï˛yˆÏÓ˚Ó˚ !Ó®%!ê˛Ó˚ x@ˇÃà!ï˛Ó˚ ˆÓà åȈÏí˛¸Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ §üyö–
!ÓK˛yö# Ó˚üˆÏîÓ˚ Î%!_´ xö%ÎyÎ˚#ñ ˆÎˆÏ•ï%˛ ï˛yˆÏÓ˚Ó˚ ܲ¡õö åȈÏí˛¸Ó˚ myÓ˚y §,‹Tñ xï˛~Ó ï˛yˆÏÓ˚Ó˚ §ß√%á à!ï˛Ó˚ §üˆÏÎ˚
åÈí˛¸ ˆÌˆÏܲ ï˛yˆÏÓ˚ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yÓ˚ ï˛yÓ˚!ê˛ !˛õåȈϰ !˛õåȈÏöÓ˚ !òˆÏܲ ÎyÄÎ˚yÓ˚ §üˆÏÎ˚Ó˚ •yˆÏÓ˚Ó˚ §üyö– ï˛yˆÏÓ˚Ó˚
ˆÎ !Ó®%!ê˛ åȈÏí˛¸Ó˚ §ÇflõˆÏ¢≈ ÌyˆÏܲ ï˛yÓ˚ §ß√%á à!ï˛ˆÏÓà Î!ò åȈÏí˛¸Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ ܲü •ï˛ ï˛y•ˆÏ° ï˛yˆÏòÓ˚ üˆÏôƒ
~ܲ!ê˛ xyˆÏ˛õ!«˛Ü˛ ˆÓà §,!‹T •ï˛ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ò%•z ~Ó˚ üˆÏôƒ àï˛#Î˚ â£Ï≈î ˜ï˛!Ó˚ •ï˛– ï˛yˆÏÓ˚Ó˚ !˛õåȈÏöÓ˚ !òˆÏܲ
!˛õåȈϰ ˛õí˛¸yÓ˚ §üÎ˚ ~•z xyˆÏ˛õ!«˛Ü˛ ˆÓˆÏàÓ˚ •ë˛yÍ ˛õ!Ó˚Óï≈˛ö âê˛ï˛ ~ÓÇ â£Ï≈î ӈϰÓ˚Ä •ë˛yÍ â£Ï≈î ӈϰÓ˚Ä
•ë˛yÍ ˛õ!Ó˚Óï≈˛ö •ï˛– ú˛ˆÏ° ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yˆÏÓ˚Ó˚ §üï˛y ÓçyÎ˚ ÌyÜ˛ï˛ öy– ~ˆÏï˛ ˆ˛õy!£Ïï˛ Ü˛¡õˆÏöÓ˚ ü)°ö#!ï˛
xÌ≈yÍ ¢!_´ §Ó˚ÓÓ˚yˆÏ•Ó˚ •yˆÏÓ˚Ó˚ §üï˛y °!Aâï˛ •ˆÏÓ– §%ï˛Ó˚yÇñ ï˛yˆÏÓ˚Ó˚ x@ˇÃà!ï˛Ó˚ §üˆÏÎ˚ åȈÏí˛¸Ó˚ à!ï˛ˆÏÓà Ä
ï˛yˆÏÓ˚Ó˚ flõ¢≈!Ó®%Ó˚ à!ï˛ˆÏÓà §üyö •ˆÏÓ ÎyˆÏï˛ í˛zû˛ˆÏÎ˚Ó˚ üˆÏôƒ !fliï˛#Î˚ â£Ï≈î ܲyç ܲˆÏÓ˚–
!mï˛#Î˚ §)e ÈÙÙÙÈ !mï˛#Î˚ §)e ÈÙÙÙÈ !mï˛#Î˚ §)e ÈÙÙÙÈ !mï˛#Î˚ §)e ÈÙÙÙÈ !mï˛#Î˚ §)e ÈÙÙÙÈ Ü˛yöÄ ê˛yöy ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ ~ܲ!ê˛ ≤ÃyˆÏhsˇ Î!ò åÈí˛¸ !òˆÏÎ˚ ܲ¡õö âê˛yˆÏöy ÎyÎ˚ ï˛y•ˆÏ° ܲ!¡õï˛
!Ó®%Ó˚ x@ˇÃ Ä ˛õÿ˛yò‰àyü# à!ï˛ˆÏÓàñ ò%!ê˛•z !fliÓ˚ ÌyˆÏܲ ~ÓÇ ˆÓà ò%!ê˛Ó˚ xö%˛õyï˛ åÈí˛¸!ê˛ ï˛yÓ˚!ê˛ˆÏܲ ˆÎ ò%!ê˛ áˆÏ[˛
!Óû˛_´ ܲˆÏÓ˚ˆÏåÈ ï˛yˆÏòÓ˚ ˜òˆÏâ≈ƒÓ˚ xö%˛õyˆÏï˛Ó˚ §üyö •Î˚– §Çflõ¢≈ !Ó®%Ó˚ §Ó˚î §üÎ˚ ˆ°á!ã˛e!ê˛ ˛õÎ≈yÓ,_ ~ÓÇ
ܲÓ˚yˆÏï˛Ó˚ òyшÏï˛Ó˚ üï˛ ò%!ê˛ ôyˆÏ˛õ xÑyܲyÓyÑܲy •Î˚ S!ã˛e 9.4V– ˆ°á!ã˛ˆÏeÓ˚ AB xÇ¢ §Çflõ¢≈ !Ó®%Ó˚ x@ˇÃà!ï˛
~ÓÇ BC xÇ¢ ˙ !Ó®%Ó˚ ˛õÿ˛yÍà!ï˛ !öˆÏò≈¢ ܲˆÏÓ˚ˆÏåÈ– !ã˛ˆÏe x@ˇÃà!ï˛Ó˚ ˆÓà v1 = = tan
θ
1 ~ÓÇ
˛õÿ˛yÍà!ï˛Ó˚ ˆÓà v2 = = tan
θ
2
ò%•z ˆÓˆÏàÓ˚ xö%˛õyï˛ñ
1
2
//
vBDADBEvECBEAD
==
ˆÜ˛ööy BD = EC
ò)Ó˚c
§üÎ˚ò)Ó˚c
§üÎ˚
256
!ã˛e 9.4 - ï˛yˆÏÓ˚Ó˚ §Çflõ¢≈ ï˛yˆÏÓ˚Ó˚ !Ó®%Ó˚ §üÎ˚ §Ó˚î ˆ°á–
Î!ò ~ܲ≤Ãyhsˇ ˆÌˆÏܲ ï˛yˆÏÓ˚Ó˚ §Çflõ¢≈ !Ó®%Ó˚ ò)Ó˚c a ~ÓÇ xöƒ≤Ãyhsˇ ˆÌˆÏܲ l – a (l = ï˛yˆÏÓ˚Ó˚ ˜òâ≈ƒV •Î˚ñ
ï˛ˆÏÓ ˆ•°‰ü‰ˆÏ•y°Í§ ÈÙÈ ~Ó˚ !mï˛#Î˚ §)e xö%ÎyÎ˚#1
2
v av l a
=− – ~•z xö%˛õyï˛ 9.4 !ã˛ˆÏeÓ˚
BEAD
xö%˛õyˆÏï˛Ó˚ §üyö
•Î˚–
ï,˛ï˛#Î˚ §)e ÈÙÙÙÈ ï,˛ï˛#Î˚ §)e ÈÙÙÙÈ ï,˛ï˛#Î˚ §)e ÈÙÙÙÈ ï,˛ï˛#Î˚ §)e ÈÙÙÙÈ ï,˛ï˛#Î˚ §)e ÈÙÙÙÈ ~ܲ•zû˛yˆÏÓ åÈí˛¸ ê˛yöy •ˆÏ° ï˛yˆÏÓ˚Ó˚ ˆÎ ˆÜ˛yöÄ !Ó®%Ó˚ x@ˇÃà!ï˛ ~ÓÇ ˛õÿ˛yÍà!ï˛Ó˚ ˆÓˆÏàÓ˚
ˆÎyàú˛° v1 + v2 ~ܲ!ê˛ !fliÓ Ó˚y!¢ ~ÓÇ ~•z §Çáƒy!ê˛ x-~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° öÎ˚–
9.5.2 åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°ü‰ˆÏ•y°Í§‰ ÈÙÈ ~Ó˚ ï˛_¥åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°ü‰ˆÏ•y°Í§‰ ÈÙÈ ~Ó˚ ï˛_¥åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°ü‰ˆÏ•y°Í§‰ ÈÙÈ ~Ó˚ ï˛_¥åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°ü‰ˆÏ•y°Í§‰ ÈÙÈ ~Ó˚ ï˛_¥åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ çöƒ ˆ•°ü‰ˆÏ•y°Í§‰ ÈÙÈ ~Ó˚ ï˛_¥
ôÓ˚y Îyܲñ l ˜òˆÏâ≈ƒÓ˚ ~ÓÇ
ρ
˜Ó˚!áܲ âöˆÏcÓ˚ ~ܲ!ê˛ ï˛yÓ˚ˆÏܲ ò%•z≤Ãyhsˇ x = 0 ~ÓÇ x = l - ~ xyê˛Ü˛yˆÏöy
•° ~ÓÇ T ӈϰ ê˛yö ܲˆÏÓ˚ Ó˚yáy •°– ~áö ~•z ï˛yÓ˚!ê˛ˆÏܲ x = a !Ó®%ˆÏï˛ åÈí˛¸ ˆê˛ˆÏö flõ!®ï˛ ܲÓ˚y •°–
~Ó˚ ú˛ˆÏ° ˆÎ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ §,!‹T •ˆÏÓ ï˛yÓ˚ ˆÓà •ˆÏÓ ≠
T cρ
=
(9.1 §ü#ܲÓ˚î xö%ÎyÎ˚#)
~ˆÏ«˛ˆÏeÄ xyüÓ˚y ܲ!¡õï˛ ï˛yˆÏÓ˚Ó˚ ˆÎ xÓܲ° §ü#ܲÓ˚î ˛õyÓ S§ü#ܲÓ˚î 9.1), ï˛yÓ˚ §yôyÓ˚î §üyôyö 9.7
§ü#ܲÓ˚ˆÏîÓ˚ xö%Ó˚*˛õ •ˆÏÓ ≠
sin cos sins ss
s x s ct s cty A Bl l lπ π π⎡ ⎤= +⎢ ⎥⎣ ⎦∑
ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ ï˛yˆÏÓ˚Ó˚ xö%≤Ãfli ˆÓà •ˆÏÓ ≠
D
257
sin · · sin cos⎛ ⎞= − +⎜ ⎟⎝ ⎠∑ s ss
s x s c s ct s cty A Bl l l lπ π π π
...9.26
~áö xyüyˆÏòÓ˚ As Ä B
s ô &Óܲ=!°Ó˚ üyö çyöˆÏï˛ •ˆÏÓ– 9.26 §ü#ܲÓ˚ˆÏîÓ˚ í˛zû˛Î˚ !òܲˆÏܲ sin
s ctl
πdt
!òˆÏÎ˚ =î ܲˆÏÓ˚ ï˛yˆÏÓ˚Ó˚ ü)°§%ˆÏÓ˚Ó˚ ~ܲ ˛õÎ≈yÎ˚ܲy° xÌ≈yÍ t = 0 ˆÌ Ïܲ t = T §#üyÓ˚ üˆÏôƒ §üyܲ°ö ܲÓ˚ˆÏ°
˛õyÄÎ˚y ÎyˆÏÓ ≠
00
sinsin·sincossin ⎛⎞ =−+ ⎜⎟ ⎝⎠ ∑ ∫∫T T
sss
sctsxscsctsctsct ydtABllllllππππππ
= – As sin ·
2⎛ ⎞⎜ ⎟⎝ ⎠
s x s c Tl lπ π
= – As sπ sin
sxlπ
ˆÜ˛ööy cT = 2l
Î!ò ôÓ˚y ÎyÎ˚ñ ï˛yˆÏÓ˚Ó˚ x@ˇÃà!ï˛Ó˚ §üˆÏÎ˚ t = 0 ˆÌˆÏܲ t =
τ
˛õÎ≈hsˇ
1 yv =
~ÓÇ ˛õ¢˘ã˛yÍà!ï˛Ó˚ §üˆÏÎ˚ t
=
τ
ˆÌˆÏܲ t =T ˛õÎ≈hsˇ
2 yv =−
, ï˛ˆÏÓ
1200
sinsinsinTT
r
sctsctsct ydtvdtvdtlll
τπππ =− ∫∫∫
= 1 2cos 1 1 cosv l v ls s cs c l s c l
πτ π τπ π
⎛ ⎞ ⎛ ⎞− − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
=
() 121cosvvlsc
sclπτ
π+⎛⎞ − ⎜⎟ ⎝⎠
=
() 122 2sin
2vvlsc
sclπτ
π+
∴ As sin
() 12222sin
2vvT sxsc
ll sππτ
π+
=−
...9.27(a)
~ÓyÓ˚ y ˆÜ˛ cos
sctlπ
dt !òˆÏÎ˚ =î ܲˆÏÓ˚ xyˆÏàÓ˚ üï˛ §üyܲ°ö ܲÏÓ˚ˆÏ° ˛õyÄÎ˚y ÎyˆÏÓ ≠
00
cossin·sincoscos ⎛⎞ =−+ ⎜⎟ ⎝⎠ ∑ ∫∫TT
sss
sctsxscsctscts ydtABllllllππππππ
258
= Bs sin · ·2
s x s c Tl lπ π
= Bs sπ sin
sxlπ
xyˆÏàÓ˚ üï˛
y
~Ó˚ üyö ÓƒÓ•yÓ˚ ܲˆÏÓ˚ñ
1200
coscoscosTT
sctsctsct ydtvdtvdtlll
τ
τ
πππ =− ∫∫∫
= 1 2sin sinv l v ls c s cs c l s c l
π τ π τπ π
+
= ( )1 22
sin cos2 2
v v l s c s cs c l l
π τ π τπ+
∴ Bs sin
() 1222
2sincos
22vvl sxscsc
lll scππτπτ
π+
=
= ( )1 2
2 2 sin cos2 2
v v T s c s cl ls
π τ π τπ
+...9.27(b)
§%ï˛Ó˚yÇñ y = ( ) ( )1 2 1 22
2 2 2 2sin · cos sin cos sin2 · 2 2
⎡ ⎤+ +− +⎢ ⎥
⎢ ⎥⎣ ⎦∑
s
v v T v v Ts c s ct s c s c s ctl l l l ls s
π τ π π τ π τ ππ π
= ( )1 2
2 21 · sin sin cos cos sin
2 2 2+ ⎛ ⎞−⎜ ⎟⎝ ⎠∑
s
v v T s c s ct s c s ct s cl l l l ls
π τ π π τ π π τπ
xÌ≈yÍ y = ( )1 2
2 21 sin sin
2 2s
v v s c s cT tl ls
π τ π τπ+ ⎛ ⎞−⎜ ⎟⎝ ⎠∑ ...9.28
ˆ•°‰ü‰ˆÏ•y°ÍˆÏ§Ó˚ §)e ˆÌˆÏܲ xyüÓ˚y çy!ö ˆÎñ ï˛yˆÏÓ˚Ó˚ ˛õ!Ó˚°!«˛ï˛ !Ó®%Ó˚ fliyöyB˛ x ~ÓÇ ï˛yˆÏÓ˚Ó˚ x@ˇÃà!ï˛Ó˚
§üÎ˚τ ˛õÓ˚flõÓ˚ ˜Ó˚!áܲû˛yˆÏÓ §¡õ!Ü≈˛ï˛– Îáö
sxlπ
= 0,
π
, 2
π
•zï˛ƒy!òÓ˚ §üyö •Î˚ ï˛áö As sin
sxlπ
~ÓÇ Bs s sin
sxlπ
Ó˚y!¢=!°Ó˚ üyö ¢)öƒ •Î˚– xyÓyÓ˚ Îáö
2sc
lπτ
= 0,
π
, 2
π
•zï˛ƒy!ò •Î˚ ï˛áöÄ 9.28
§ü#ܲÓ˚ˆÏîÓ˚ í˛yö!òܲ ¢)öƒ •ˆÏÓ– ˆÎˆÏ•ï%˛ ~=!° ˆÜ˛yöÄ !û˛ß¨ ¢ï≈˛ •ˆÏï˛ ˛õyˆÏÓ˚ öyñ xï˛~Ó xyüÓ˚y
2sc
lπτ
~Ó˚
259
fliyˆÏö
sxlπ
!°áˆÏï˛ ˛õy!Ó˚– ~åÈyí˛¸y xyüÓ˚y Î!ò §üÎ˚ àîöyÓ˚ ü%•)ï≈˛!ê˛
2τ
˛õ!Ó˚üyˆÏî ~!àˆÏÎ˚ !ò•z ï˛ˆÏÓ t –
2τ
~Ó˚
fliyˆÏö t ˆ°áy ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~•z ˛õ!Ó˚Óï≈˛ö=!°Ó˚ ˛õÓ˚ 9.28 §ü#ܲÓ˚îˆÏܲ ˆ°áy ÎyˆÏÓ ≠
y =
() 1222
1sinsins
vvTsxsctll sππ
π+∑
...9.29
xÌÓyñ y = sinss
s ctal
π∑ ...9.30
ˆÎáyˆÏöñ as =
( )1 22 2
1· · sin+v v T s x
lsπ
π...9.31
~áö xy˛õ!ö ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ 9.18 §ü#ܲÓ˚î Ä xy•ï˛ ï˛yˆÏÓ˚Ó˚ 9.23 §ü#ܲÓ˚ˆÏîÓ˚ §ˆÏD åÈí˛¸ ê˛yöy ï˛yˆÏÓ˚Ó˚
9.29 §ü#ܲÓ˚ˆÏîÓ˚ ï%˛°öy ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– °«˛ƒ ܲˆÏÓ˚ ˆòá%öñ åÈí˛¸ ê˛yöy ï˛yÓ˚ ~ÓÇ Ü˛!£Ï≈ï˛ ï˛yÓ˚ÈÙÙÙÈí˛zû˛Î˚ ˆ«˛ˆÏe•z
§üˆÏü°=!°Ó˚ !ÓhflÏyÓ˚
21s
~Ó˚ §üyö%˛õyï˛#– ï˛ˆÏÓ Ü˛!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ܲ£Ï≈ˆÏîÓ˚ !Ó®%!ê˛Ó˚ fliyöyB˛ Îï˛ê˛y =Ó˚&c˛õ)î≈
åÈí˛¸ ê˛yöy ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe åȈÏí˛¸Ó˚ §ÇflõˆÏ¢≈ Ìyܲy !Ó®%!ê˛Ó˚ fliyöyB˛ ï˛ï˛ê˛y =Ó˚&c˛õ)î≈ öÎ˚–
~ÓyÓ˚ åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ !ӣψÏÎ˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 4
!öˆÏã˛Ó˚ í˛z!_´=!°Ó˚ üˆÏôƒ ˆÎ=!° §ï˛ƒ ˆ§=!°Ó˚ ˛õyˆÏ¢ Ú§Û ~ÓÇ ˆÎ=!° !ü̃y ˆ§=!°Ó˚ ˛õyˆÏ¢ Ú!üÛ !°á%ö–
(a) åí˛¸ê˛yöy ï˛yˆÏÓ˚ !fliï˛#Î˚ â£Ï≈ˆÏîÓ˚ ˆÜ˛yöÄ û)˛!üܲy ˆö•z–
(b) åÈí˛¸!ê˛ o&ï˛à!ï˛ˆÏï˛ ê˛yöy •ˆÏ° ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ ÓyˆÏí˛¸–
(c) åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ s ï˛ü §üˆÏüˆÏ°Ó˚ !ÓhflÏyÓ˚ s–2 ~Ó˚ §üyö%˛õyï˛#–
(d) åÈí˛¸ê˛yöyÓ˚ §üÎ˚ åȈÏí˛¸Ó˚ §ÇflõˆÏ¢≈ Ìyܲy !Ó®%Ó˚ x@ˇÃà!ï˛ Ä ˛õÿ˛yÍà!ï˛ §Ó≈òy•z §üyö •Î˚–
(e) åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õöˆÏܲ ≤ÈÏîy!òï˛ Ü˛¡õö Ó°y ÎyÎ˚–
(f) ˆ˛õy!£Ïï˛ Ü˛¡õˆÏö Óy•zˆÏÓ˚ ˆÌˆÏܲ ¢!_´Ó˚ ˆÎyàyö ≤ÈÏÎ˚yçö •Î˚–
(g) ~ܲ•z ï˛yÓ˚ ܲ!£Ï≈ï˛ •ˆÏ° ï˛yÓ˚ ˆÎ ü%°§%Ó˚ •Î˚ñ åÈí˛¸ê˛yöy •ˆÏ° ˆ§•z ü)°§%Ó˚ •Î˚ öy–
9.6 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~•z ~ܲˆÏܲ xyüÓ˚y ò%•z ≤Ãyhsˇ xyÓk˛ ~üö ê˛yö !òˆÏÎ˚ Ó˚yáy ï˛yˆÏÓ˚Ó˚ ܲ¡õö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ–
≤Ã̈Ïü•zñ ~ ôÓ˚ˆÏöÓ˚ ï˛yˆÏÓ˚ xö%≤Ãfli §Ó˚ˆÏîÓ˚ ï˛Ó˚D §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y ܲˆÏÓ˚ ï˛yÓ˚ §yôyÓ˚î §üyôyö ÓyÓ˚ ܲÓ˚y
•ˆÏÎ˚ˆÏåÈ ~ÓÇ ~•z §üyôyˆÏöÓ˚ §y•yˆÏ΃ ï˛Ó˚D!ê˛Ó˚ à!ï˛¢!_´ ~ÓÇ !fli!ï˛¢!_´Ó˚ üyö !öî≈Î˚ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
260
~Ó˚˛õÓ˚ xyüÓ˚y ê˛yö !òˆÏÎ˚ Ó˚yáy ï˛yˆÏÓ˚ !Ó!û˛ß¨ ˛õk˛!ï˛ˆÏï˛ í˛zͲõߨ ܲ¡õˆÏöÓ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ܲˆÏÓ˚!åÈ ~ÓÇ
ˆ§=!°Ó˚ ã˛!Ó˚eàï˛ ˜Ó!¢ˆÏ‹TƒÓ˚ xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ày!î!ï˛Ü˛ !ӈϟ’£ÏˆÏîÓ˚ ü)° °«˛ƒ ï˛yˆÏÓ˚Ó˚ §#üy¢ï≈˛=!°Ó˚
≤ÈÏÎ˚yà ~ÓÇ ï˛yÓ˚ myÓ˚y ï˛yˆÏÓ˚Ó˚ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î !öî≈Î˚–
ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õö !ï˛ö ≤ÃܲyˆÏÓ˚ §,!‹T ܲÓ˚y ÎyÎ˚– ~=!° •° (i) ܲ£Ï≈ˆÏöÓ˚ myÓ˚yñ xÌ≈yÍ ï˛yˆÏÓ˚Ó˚ ~ܲ!ê˛ !Ó®%ˆÏܲ
˛õyˆÏ¢Ó˚ !òˆÏܲ ˆê˛ˆÏö ôˆÏÓ˚ ~ÓÇ ï˛yÓ˚˛õÓ˚ !fliÓ˚ xÓfliy ˆÌˆÏܲ ˆåȈÏí˛¸ !òˆÏÎ˚ó (ii) xyâyˆÏï˛Ó˚ myÓ˚y xÌ≈yÍ •yï%˛!í˛¸
Óy ˙ çyï˛#Î˚ ˆÜ˛yöÄ Ó›Ó˚ myÓ˚y ï˛yˆÏÓ˚Ó˚ ˆÜ˛yöÄ !Ó®%ˆÏï˛ xyâyï˛ Ü˛ˆÏÓ˚ ~ÓÇ (iii) åÈí˛¸ ˆê˛ˆÏöñ xÌ≈yÍ ˆÓ•y°y
≤Ãû,˛!ï˛ ÓyòƒÎˆÏsf ÓƒÓ•*ï˛ åÈí˛¸ˆÏܲ ï˛yˆÏÓ˚Ó˚ í˛z˛õÓ˚ ã˛y˛õ !òˆÏÎ˚ ôˆÏÓ˚ ï˛yˆÏÓ˚Ó˚ §ˆÏD §üˆÏܲyˆÏî ˆê˛ˆÏö– ~•z !ï˛ö ˛õk˛!ï˛Ó˚
ˆ«˛ˆÏe xyüÓ˚y ˆÎ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î ˆ˛õˆÏÎ˚!åÈ ˆ§=!° •° ≠
ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ≠ y =
()2
2221sinsincos
s
hlsasxsctlll alasπππ
π−∑
...(§ü#ܲÓ˚î 9.18)
xy•ï˛ ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ≠ y = 2 1 sin sin sin
s
p s a s x s ctc s l l l
π π ππ ρ ∑ ...(§ü#ܲÓ˚î 9.23)
åÈí˛¸ê˛yöy ï˛yˆÏÓ˚Ó˚ ˆ«˛ˆÏe ≠ y = ( )1 2
2 21 sin sin
s
v v T s x s ctl lsπ π
π+ ∑ ...(§ü#ܲÓ˚î 9.29)
!ï˛ö!ê˛ ˆ«˛ˆÏeÓ˚ Ó˚y!¢üy°y=!°ˆÏï˛ ˆÎ ≤Ãï˛#ܲ=!° ÓƒÓ•*ï˛ •ˆÏÎ˚ˆÏåÈ ˆ§=!°Ó˚ xÌ≈ ˛õyë˛ƒyLjϢÓ˚ üˆÏôƒ ˆòÄÎ˚y
•ˆÏÎ˚ˆÏåÈ– ~•z §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î=!°Ó˚ §y•yˆÏ΃ ê˛yöy ï˛yˆÏÓ˚Ó˚ !Ó!û˛ß¨ í˛z˛õyˆÏÎ˚ §,‹T ܲ¡õˆÏöÓ˚ !ܲå%È ˜Ó!¢‹Tƒ
xyüÓ˚y xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ– ~=!°Ó˚ §y•yˆÏ΃ xy˛õ!ö !ï˛ö ˛õk˛!ï˛ˆÏï˛ í˛zj#!˛õï˛ Ü˛¡õˆÏöÓ˚ ï%˛°öyü)°Ü˛
xyˆÏ°yã˛öyÄ Ü˛Ó˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
9.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ê˛yö ˆòÄÎ˚y ï˛yˆÏÓ˚ xö%≤Ãfli ï˛Ó˚ˆÏDÓ˚ §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛¤˛y ܲÓ˚&ö ~ÓÇ ˆ§!ê˛Ó˚ §yôyÓ˚î §üyôyö !öî≈Î˚ ܲÓ˚&ö–
2. ê˛yöy ï˛yˆÏÓ˚Ó˚ à!ï˛¢!_´ Ä !fli!ï˛¢!_´Ó˚ Ó˚y!¢üy°y !öî≈Î˚ ܲÓ˚&ö– ˆòáyö ˆÎñ ˆüyê˛ Îy!sfï˛ ¢!_´ §üˆÏÎ˚Ó˚
§ˆÏD ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ öy–
3. ܲ!£Ï≈ï˛ ï˛yˆÏÓ˚Ó˚ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î !öî≈Î˚ ܲÓ˚&ö ~ÓÇ ~Ó˚ ˆÌˆÏܲ •zÎ˚Ç ˆ•°‰ü‰ˆÏ•y°Í§‰ §)e!ê˛ Ó%!é˛ˆÏÎ˚
!òö–
4. ê˛yöy ï˛yˆÏÓ˚Ó˚ §Ó˚ˆÏîÓ˚ §yôyÓ˚î §ü#ܲÓ˚î ≠
y (x, t) = cos sin sins ss
s ct s ct s xA Bl l l
π π π⎡ ⎤+⎢ ⎥⎣ ⎦∑
ôˆÏÓ˚ !öˆÏÎ˚ xy•ï˛ ï˛yˆÏÓ˚Ó˚ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛!¤˛ï˛ ܲÓ˚&ö–
261
5. åÈí˛¸ ê˛yöy ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏö åÈí˛¸ Ä ï˛yˆÏÓ˚Ó˚ üˆÏôƒ àï˛#Î˚ Ä !fliï˛#Î˚ â£Ï≈î ӈϰÓ˚ û)˛!üܲy xyˆÏ°yã˛öy ܲÓ˚&ö–
6. åÈí˛¸ ê˛yöy ï˛yˆÏÓ˚Ó˚ §üÎ˚ §Ó˚î §ü#ܲÓ˚î!ê˛ ≤Ã!ï˛!¤˛ï˛ ܲÓ˚&ö–
9.8 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#
1. 9.7 §ü#ܲÓ˚ˆÏî !mï˛#Î˚ §üˆÏüˆÏ°Ó˚ çöƒ s-~Ó˚ üyö ˆÜ˛Ó°üye 2 ôÓ˚ˆÏ° ˆòáy ÎyÎ˚ñ
y = 2 22 2 2cos sin sinct ct xA B
l l lπ π π⎛ ⎞+⎜ ⎟⎝ ⎠
ˆÎˆÏ•ï%˛ t = 0 §üˆÏÎ˚ §Ó˚ˆÏîÓ˚ üyö §Ó≈e ¢)öƒñ A2 = 0
~ˆÏ«˛ˆÏe ê˛yö T = 1 × 9·8 = 9·8N
ï˛yˆÏÓ˚Ó˚ ˜Ó˚!áܲ âöc ρ =
3 2·2101·1
− ×
2 × 10–3kgm–1
∴ ï˛yˆÏÓ˚Ó˚ ï˛Ó˚ˆÏDÓ˚ ˆÓà c =
12
39·8
210Tρ−
⎛⎞ =⎜⎟ × ⎝⎠
= 70ms–1
§ˆÏÓ≈yFã˛ §Ó˚î B2 = 2mm
§%ï˛Ó˚yÇñ y = 2sin
2·702 sin1·11·1
x t ππ
= 2sin400t sin5·71x mm
2. 9.18 §ü#ܲÓ˚î ˆÌˆÏܲ ˛õyÄÎ˚y ÎyÎ˚ñ x = a !Ó®%ˆÏï˛ ï˛yÓ˚!ê˛ˆÏܲ ܲ£Ï≈î ܲÓ˚y •ˆÏ° ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ !ÓhflÏyÓ˚
as =
()2
2221sinsin hlsasx
ll alasππ
π×
−
ï˛yÓ˚!ê˛ˆÏܲ 4l
ò)Ó˚ˆÏc ܲ£Ï≈î ܲÓ˚y •ˆÏ° a =
4l
xï˛~Óñ asm
=
22321sin
4hs
sπ
π×
– ˆÎˆÏ•ï%˛ s = 4, 8, 12,....•zï˛ƒy!ò •ˆÏ° sin4sπ
= 0 •Î˚ñ ܲyˆÏç•z asm
=
0 •ˆÏÓñ ã˛ï%˛Ì≈ñ x‹Tüñ myò¢ •zï˛ƒy!ò §%ˆÏÓ˚–
3(a) §ü#ܲÓ˚î 9.24 ˆÌˆÏܲ ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛ !ÓhflÏyÓ˚ ≠
as=
21··sinsin sasxcsll
ρπππρ
262
§ˆÏÓ≈yFã˛ !ÓhflÏyˆÏÓ˚Ó˚ üyö ≠
asm
= 2 1 sinp s arc s l
ππ ρ
(Îáö x = 3,
2 2l ls s
•zï˛ƒy!òV–
Î!ò ï˛yÓ˚!ê˛ˆÏܲ a =
3l
ò)Ó˚ˆÏc xyâyï˛ Ü˛Ó˚y ÎyÎ˚ñ ï˛y•ˆÏ°
asm
=
21·sin3
pscs
ππρ
ï,˛ï˛#Î˚ñ £Ï¤˛ñ öÓü •zï˛ƒy!ò §%ˆÏÓ˚ SÎáö s = 3, 6, 9 •zï˛ƒy!òV asm ~Ó˚ üyö ¢)öƒ •ˆÏÓ xÌ≈yÍ ~•z §%Ó˚=!°
xö%˛õ!fliï˛ ÌyܲˆÏÓ–
xyÓyÓ˚ asm
§Ó≈yˆÏ˛õ«˛y ˆÓ!¢ •ˆÏÓ ≤ÃÌüñ !mï˛#Î˚ñ ã˛ï%˛Ì≈ •zï˛ƒy!ò §%ˆÏÓ˚ SˆÎüö s = 1, 2, 4 •zï˛ƒy!ò) Îáö
x-~Ó˚ üyö ÎÌye´ˆÏü
,,248lll
•zï˛ƒy!ò– ܲyˆÏç•z xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚ñ §ˆÏÓ≈yFã˛ !ÓhflÏyÓ˚=!°Ó˚ üyö ≠
a1m =
24333 ,,
24 mmppp aa
cccm πρπρπ==−
•zï˛ƒy!ò–
3.(b) •yï%˛!í˛¸!ê˛ ï˛yˆÏÓ˚Ó˚ ˆÎ xLjϢ xyâyï˛ Ü˛ˆÏÓ˚ñ ï˛yÓ˚ ˜òâ≈ƒ x!ï˛«%˛o öy •ˆÏ° 9.23 §ü#ܲÓ˚ˆÏîÓ˚ ≤Ã!ï˛¤˛yÎ˚
0 sina
a
xu dxl
γπ+Δ
∫ Ó˚y!¢!ê˛Ó˚ üyö !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ x ~Ó˚ §ˆÏD u0 ~Ó˚ ˛õ!Ó˚Óï≈˛ö çyöy ≤ÈÏÎ˚yçö •ˆÏÓ– ï˛ˆÏÓ
u0 ~Ó˚ ˛õ!Ó˚Óï≈˛ö çyöy ÌyܲˆÏ°Ä §üyܲ°!ê˛Ó˚ üyö !öî≈Î˚ ܲÓ˚y ò)Ó˚*• •ˆÏÓ–
4. (a) !üó (b) §ó (c) §ó (d) !üó (e) !üó (f) §ó(g) !ü–
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. 9.2 xö%ˆÏFåȈÏò ~•z !Ó£ÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
2. 9.2.1 xLjϢ xy˛õ!ö ~ !ӣψÏÎ˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ˆ˛õˆÏÎ˚ˆÏåÈö–
3. 9.3 xLjϢ ~•z ≤Èϟ¿Ó˚ í˛z_Ó˚ xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ–
4. ~•z §ü#ܲÓ˚î 9.4 xLjϢ ≤Ã!ï˛!¤˛ï˛ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5. 9.5 xLjϢ åÈí˛¸ ê˛yöy ï˛yÓ˚ §¡∫¶˛#Î˚ xyˆÏ°yã˛öy!ê˛ ˆòˆÏá !öö–
6. 9.5 xLjϢ åÈí˛¸ ê˛yöy ï˛yˆÏÓ˚Ó˚ §üÎ˚ÈÙȧÓ˚î §ü#ܲÓ˚î ≤Ã!ï˛!¤˛ï˛ •ˆÏÎ˚ˆÏåÈ–
263
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 10 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y¢ˆÏ∑Ó˚ ï˛#Ó ï˛y¢ˆÏ∑Ó˚ ï˛#Ó ï˛y¢ˆÏ∑Ó˚ ï˛#Ó ï˛y¢ˆÏ∑Ó˚ ï˛#Ó ï˛y (intensity), ≤ÃyÓ°ƒ≤ÃyÓ°ƒ≤ÃyÓ°ƒ≤ÃyÓ°ƒ≤ÃyÓ°ƒ (loudness) ÄÄÄÄÄ
ܲ¡õyˆÏB˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ܲ¡õyˆÏB˛Ó˚ (frequency) ˛õ!Ó˚üy˛õ˛õ!Ó˚üy˛õ˛õ!Ó˚üy˛õ˛õ!Ó˚üy˛õ˛õ!Ó˚üy˛õ
àë˛öàë˛öàë˛öàë˛öàë˛ö
10.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
10.2 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§Ó° ~ܲܲ
10.3 ≤ÃyÓ°ƒ ~ÓÇ ï˛yÓ˚ ~ܲܲ
10.3.1 ï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öy
10. 4 ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õ
10.4.1 Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛ (Rayleighi’s Disc)
10.4.2 ï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yö (Hot wire microphone)
10.4.3 xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛ (optical method)
10.4.4 ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ (Sound radiometer)
10.4.5 !í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yˆÏÓ˚Ó˚ Ó˚*˛õˆÏÓ˚áy.
10.5 ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ
10.5.1 fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´ ˛õk˛!ï˛
10.5.2 Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´
10.5.3 ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö
10.5.4 ˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲ
10.5.5 !í˛!çê˛y° !ú ˛ˆÏܲyˆÏÎ˚!™!üê˛yÓ˚ñ
10.5.6 x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛.
10.6 §yÓ˚yÇ¢
10.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
10.8 í˛z_Ó˚üy°y
10.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
üyö%£Ï ï˛yÓ˚ çß√ ˆÌˆÏܲ öyöyçyï˛#Î˚ ¢ˆÏ∑Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •Î˚– Óy•zˆÏÓ˚Ó˚ ≤ÃÜ,˛!ï˛Ó˚ §ˆÏD ï˛Ìƒ xyòyöÈÙÈ≤ÃòyˆÏöÓ˚
~ܲ!ê˛ =Ó˚&c˛õ%î≈ í˛z˛õyÎ˚ •° ◊ÓˆÏî!wÎ˚Ó˚ §y•yˆÏ΃ ¢∑@ˇÃ•î ~ÓÇ !öçfl∫ Óyà‰Îsf Óy xöƒ ˆÜ˛yˆÏöy Îy!sfܲ ÓƒÓfliyÓ˚
myÓ˚y ¢∑ §,!‹T– ¢∑ ÷ô% òÓ˚ܲyÓ˚# ܲÌyÓyï≈˛y xyÓ˚ ö#Ó˚§ ï˛Ìƒ xyòyöÈÙÈ≤ÃòyˆÏöÓ˚ üyôƒü öÎ˚ñ ¢∑ üyö%ˆÏ£ÏÓ˚ !¢“§,!‹Tñ
û˛yÓ !Ó!öüÎ˚ ~ÓÇ !ÓˆÏöyòˆÏöÓ˚Ä üyôƒü– ˆ§ ܲyÓ˚ˆÏî ¢ˆÏ∑Ó˚ ˜Ó!ã˛eƒ Ä ≤ÃܲyÓ˚ˆÏû˛ò xyüyˆÏòÓ˚ ˆÜ˛Ôï)˛•° çy@ˇÃï˛
ܲˆÏÓ˚–
264
fl∫yû˛y!Óܲ «˛üï˛yÎ˚ üyö%£Ï !Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ ¢ˆÏ∑Ó˚ ˛õyÌ≈ܲƒ Ó%é˛ˆÏï˛ ˛õyˆÏÓ˚– ˆÎüöÈÙÙÙȈçyˆÏÓ˚ñ xyˆÏhflÏñ ï˛#«¯˛ñ àΩ˛#Ó˚
•zï˛ƒy!ò– !ܲv ˆû˛Ôï˛!ÓK˛yˆÏöÓ˚ ˆ«˛ˆÏe ~Ó˚ܲü =îàï˛ ˛õyÌ≈ܲƒ ΈÏÌ‹T öÎ˚– xy˛õ!ö !öÿ˛Î˚•z ~ܲüï˛ •ˆÏÓö ˆÎñ
ˆû˛Ôï˛!ÓK˛yˆÏö ˆÎ §Ó Ó˚y!¢ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ñ ˆ§=!° ˛õ!Ó˚üy˛õˆÏÎyàƒ •ÄÎ˚y ~ܲyhsˇ•z ≤ÈÏÎ˚yçö– xï˛~Óñ !Ó!û˛ß¨
í˛zͧ ˆÌˆÏܲ í˛zͲõy!òï˛ !Ó!û˛ß¨ ¢ˆÏ∑Ó˚ ≤ÃÜ,˛!ï˛ !öô≈yÓ˚î ܲÓ˚y •Î˚ ܲˆÏÎ˚ܲ!ê˛ !ö!ò≈‹T ˜Ó!¢‹Tƒ !Óã˛yÓ˚ ܲˆÏÓ˚– ~ܲ!ê˛
•° Úܲ¡õyB˛ÛÈÙÙÙÈxÌ≈yÍñ ¢∑ í˛zͧ!ê˛ ˆ§ˆÏܲˆÏu˛ ܲï˛ÓyÓ˚ ܲ!¡õï˛ •ˆÏFåÈ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ üyôƒˆÏü
ܲï˛=!° ï˛Ó˚D í˛zͲõߨ •ˆÏFåÈ– xyÓ˚ ~ܲ!ê˛ •° Úï˛#Ó ï˛yÛÈÙÙÙÈxÌ≈yÍ ¢ˆÏ∑Ó˚ à!ï˛ü%ˆÏá Óy °¡∫û˛yˆÏÓ !fliï˛ ~ܲܲ
ˆ«˛eú˛ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ~ܲܲ §üˆÏÎ˚ ≤ÃÓy!•ï˛ àí˛¸ ¢∑¢!_´Ó˚ ˛õ!Ó˚üyî– ~=!° ˆû˛Ôï˛ Ó˚y!¢ñ !ö!ò≈‹Tû˛yˆÏÓ üy˛õy
§Ω˛Ó– xyÓyÓ˚ ܲï˛Ü˛=!° ˜Ó!¢‹Tñ xyˆÏåÈñ Îy üyöÓ •z!wÎ˚ xö%û˛Ó ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ñ !ܲv §Ó˚y§!Ó˚ ˛õ!Ó˚üy˛õ ܲÓ˚y
ÎyÎ˚ öy– ˆÎüö ¢ˆÏ∑Ó˚ Ú≤ÃyÓ°ƒÛ üyˆÏö ¢∑ ܲï˛ê˛y ˆçyÓ˚yˆÏ°yñ ï˛yÓ˚ xö%û˛Ó– ï˛#Ó ï˛yˆÏܲ üy˛õyÓ˚ çöƒ Úˆí˛!§ˆÏÓ°Û
öyˆÏü ~ܲ!ê˛ !ö!ò≈‹T ~ܲܲ xyˆÏåÈñ Îy xy˛õ!ö ~ܲê%˛ ˛õˆÏÓ˚•z ˛õí˛¸ˆÏÓö– !ܲv ≤ÃyÓ°ƒ ~ܲ!ê˛ ˆ◊yï˛yÈÙÈ!öû≈˛Ó˚ xö%û)˛!ï˛ñ
§%ï˛Ó˚yÇ ï˛yˆÏܲ §Ó˚y§!Ó˚ üy˛õy ÎyÎ˚ öy– ≤ÃyÓ°ƒ ˛õˆÏÓ˚y«˛û˛yˆÏÓ ò%!ê˛ ï˛#Ó ï˛yÓ˚ ï%˛°öy !•§yˆÏÓ üy˛õy •Î˚ ~ÓÇ ú˛ö
(Phon) ~ܲˆÏܲ ≤Ãܲy¢ ܲÓ˚y •Î˚– ~•z ~ܲˆÏܲ ï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öyü)°Ü˛ !ӈϟ’£Ïî ܲÓ˚y •ˆÏÎ˚ˆÏåÈ ~ÓÇ
ï˛#Ó ï˛y üy˛õyÓ˚ !Ó!û˛ß¨ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ §ˆÏD §¡∫!¶˛ï˛ ~ܲ!ê˛ xö%û)˛!ï˛ •° ï˛#«¯˛ï˛yñ ¢∑ ܲï˛ê˛y ã˛í˛¸y ӈϰ üˆÏö •ˆÏFåÈ ˆ§•z xö%û˛Ó–
xyüÓ˚y ÷ˆÏö Ó%é˛ˆÏï˛ ˛õy!Ó˚ñ ‡•z§ˆÏ°Ó˚ ¢∑ ï˛#«¯˛ xyÓ˚ §yöy•zˆÏÎ˚Ó˚ áyˆÏòÓ˚ ˛õò≈yÓ˚ xyÄÎ˚yç û˛yÓ˚#– !ܲv ~•z Úï˛#«¯˛Û
xyÓ˚ Úû˛yÓ˚#Û û˛yÓ üy˛õy §Ω˛Ó öÎ˚– ˆÎê˛y üy˛õy ÎyÎ˚ñ ï˛y •° ܲ¡õyB˛– ¢∑ û˛yÓ˚# Óy ã˛í˛¸y •ˆÏÓ˛ ï˛yÓ˚ ܲ¡õyB˛
ܲü Óy ˆÓ!¢ •ˆÏ°– ܲ¡õyB˛ §Ó˚y§!Ó˚ üy˛õy ÎyÎ˚ñ xÌÓy ò%!ê˛ í˛zͧ ˆÌˆÏܲ !öà≈ï˛ ¢ˆÏ∑Ó˚ ï˛Ó˚D §ÇáƒyÓ˚ ï%˛°öy
ܲˆÏÓ˚ !öî≈Î˚ ܲÓ˚y ÎyÎ˚– ~•z ~ܲˆÏܲ ò%ÈÙÈÓ˚ܲü ˛õk˛!ï˛•z Óƒyáƒy ܲÓ˚y •ˆÏÓ– ¢ˆÏ∑Ó˚ xyÓ˚ ~ܲ!ê˛ !ӈϢ£Ïc •° ï˛yÓ˚
=î Óy çy!ï˛– ~Ó˚ üyôƒˆÏü ~ܲ•z ܲ¡õyB˛ Ä ï˛#Ó ï˛yÓ˚ ò%!ê˛ ¢∑ í˛zͧˆÏܲ ˛õ,Ìܲû˛yˆÏÓ ˆã˛öy ÎyÎ˚– çy!ï˛ !öû≈˛Ó˚
ܲˆÏÓ˚ ï˛Ó˚ˆÏDÓ˚ ≤ÃÜ,˛!ï˛ xÌ≈yÍ §üˆÏÎ˚Ó˚ §ˆÏD ܲ¡õyB˛ !ÓhflÏyÓ˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ≤ÃÜ,˛!ï˛Ó˚ í˛z˛õÓ˚– ~•z ~ܲˆÏܲ ˆ§ §¡∫ˆÏ¶˛
xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÓ– ~áyˆÏö ӈϰ Ó˚yáy û˛y°ñ ï˛#«¯˛ï˛y Ä çy!ï˛ ~ܲüye §%ˆÏÓ˚°y ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe•z !Óã˛yÎ≈–
ˆÜ˛y°y•° (noise)-~Ó˚ ï˛#«¯˛ï˛y Ä çy!ï˛ xÌ≈•#öñ ܲyÓ˚î ܲ¡õyB˛•z !ö!ò≈‹T öÎ˚–
í˛ zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒí˛zˆ Ïj¢ƒ
~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° xy˛õ!ö ˆÎ ܲyç=!° ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓöñ ï˛y •°ÈÙÙÙÈ
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ≤ÃyÓ°ƒ Ó°ˆÏï˛ Ü˛# ˆÓyé˛yÎ˚ñ ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
˛õ!Ó˚!ã˛ï˛ Úˆí˛!§ˆÏÓ°Û ~ܲܲ!ê˛Ó˚ xÌ≈ ܲ#ñ ï˛y ˆÓyé˛yˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ≤ÈÏÎ˚yçöüï˛ ~•z ~ܲܲ!ê˛Ó˚ ÓƒÓ•yÓ˚
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ï˛#Ó ï˛y˛ ˛õ!Ó˚üy˛õ ܲÓ˚yÓ˚ !Ó!û˛ß¨ ˛õk˛!ï˛ Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ܲ¡õyB˛ ܲ#û˛yˆÏÓ üy˛õy •Î˚ñ ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
§%ˆÏÓ˚°y ¢ˆÏ∑Ó˚ =î Óy çy!ï˛ Ü˛yˆÏܲ ӈϰñ ï˛y Ó%!é˛ˆÏÎ˚ !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
í˛z˛õˆÏÓ˚Ó˚ !Ó£ÏÎ˚=!°Ó˚ §ˆÏD Î%_´ !Ó!û˛ß¨ ΈÏsfÓ˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÎ˚ ≤ÃyˆÏÎ˚yçˆÏö ˆ§=!° ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
265
10.2 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§ˆÏÓ° ~ܲܲ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§ˆÏÓ° ~ܲܲ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§ˆÏÓ° ~ܲܲ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§ˆÏÓ° ~ܲܲ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yÓ˚ ˆí˛!§ˆÏÓ° ~ܲܲ
~ܲܲ !öÿ˛Î˚ ˆÜ˛yöÄ §üˆÏÎ˚ ˆüyê˛Ó˚ày!í˛¸Ó˚ •ö≈ Óy °yí˛zí˛!flõܲyÓ˚ ˆÌˆÏܲ !öà≈ï˛ ¢∑ xï˛ƒhsˇ ï˛#Ó ÓˆÏ° xfl∫!hflÏ
xö%û˛Ó ܲˆÏÓ˚ˆÏåÈö– xyÓyÓ˚ ˆê˛!°ˆÏú˛yˆÏö ܲÌy Ó°yÓ˚ §üÎ˚ ˆÎ ¢∑ ÷öˆÏï˛ ˆ˛õˆÏÎ˚ˆÏåÈöñ ï˛y ΈÏÌ‹T ï˛#Ó öÎ˚ ӈϰ
xy˛õöyÓ˚ •Î˚ï˛ x§%!Óôy •ˆÏÎ˚ˆÏåÈ– ¢ˆÏ∑Ó˚ ~•z Úï˛#«¯˛ï˛yÛ Ó°ˆÏï˛ ¢∑!ÓK˛yˆÏö !ë˛Ü˛ ܲ# ˆÓyé˛yÎ˚⁄
ˆÜ˛yˆÏöy ~ܲ!ê˛ !ö!ò‹T !Ó®%ˆÏï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó°ˆÏï˛ ˆÓyé˛yÎ˚ñ ˆ§áyˆÏö ¢ˆÏ∑Ó˚ à!ï˛ x!û˛ü%ˆÏáÓ˚ §ˆÏD
°¡∫û˛yˆÏÓ xÓ!fliï˛ ~ܲܲ ˆ«˛eú˛ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ àí˛¸ ¢∑«˛üï˛yÓ˚ ˛õ!Ó˚üyî– ï˛#Ó ï˛yÓ˚ ~ܲܲ ≤Ã!ï˛
Óà≈!üê˛yÓ˚ ˛!˛õå%È ÄÎ˚yê˛ xÌ≈yÍ W/m2– ˆòáyˆÏöy ÎyÎ˚ ˆÎñ ˆÜ˛yö üyôƒˆÏü ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ üyö
I = 2 20
12
c Aρ ω
ˆÎáyˆÏö ρ0 = üyôƒˆÏüÓ˚ àí˛¸ âöcñ c = üyôƒˆÏü ¢ˆÏ∑Ó˚ ˆÓà
ω
= ¢∑ï˛Ó˚ˆÏDÓ˚ ˆÜ˛Ô!îܲ ܲ¡õyB˛ ~ÓÇ
A = ¢∑ï˛ˆÏÓ˚DÓ˚ !ÓhflÏyÓ˚– l -~Ó˚ Ó˚y!¢üy°y ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y xˆÏöܲ=!°
!ӣψÏÎ˚Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#°– ~áö ~•z !Ó£ÏÎ˚=!° xy°yòy ܲˆÏÓ˚ ˆòáy ˆÎˆÏï˛ ˛õyˆÏÓ˚–
(i) ܲ¡õyˆÏB˛Ó˚ !ÓhflÏyÓ˚ ≠ ܲ¡õyˆÏB˛Ó˚ !ÓhflÏyÓ˚ ≠ ܲ¡õyˆÏB˛Ó˚ !ÓhflÏyÓ˚ ≠ ܲ¡õyˆÏB˛Ó˚ !ÓhflÏyÓ˚ ≠ ܲ¡õyˆÏB˛Ó˚ !ÓhflÏyÓ˚ ≠ ï˛#Ó ï˛yÓ˚ Ó˚y!¢üy°y ˆÌˆÏܲ ~ê˛y flõ‹T ˆÎñ !ö!ò≈‹T ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ çöƒ ï˛#Ó ï˛y
!ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õyï˛# •Î˚– xy˛õ!ö Îáö ˆÓ˚!í˛ÄÓ˚ û˛!°í˛zü öÓ!ê˛ ˆâyÓ˚yö ï˛áö !öà≈ï˛ ¢ˆÏ∑Ó˚ !ÓhflÏyÓ˚
ÓyˆÏí˛¸ ӈϰ•z ï˛#Ó ï˛yÄ ÓyˆÏí˛¸– ï˛ˆÏÓ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ xöƒ ܲˆÏÎ˚ܲ!ê˛ !ӣψÏÎ˚Ó˚ í˛z˛õÓ˚Ä !öû≈˛Ó˚ ܲˆÏÓ˚–
(a) íí˛z͈ϧÓ˚ xyܲyÓ˚ ≠ í˛z͈ϧÓ˚ xyܲyÓ˚ ≠ í˛z͈ϧÓ˚ xyܲyÓ˚ ≠ í˛z͈ϧÓ˚ xyܲyÓ˚ ≠ í˛z͈ϧÓ˚ xyܲyÓ˚ ≠ í˛z͈ϧÓ˚ xyܲyˆÏÓ˚Ó˚ §ˆÏD ¢∑ï˛Ó˚ˆÏDÓ˚ !ÓhflÏyÓ˚ Ó,!k˛ ˛õyÎ˚– ~çöƒ•z !àê˛yÓ˚ñ ˆÓ•y°y
≤Ãû,˛!ï˛ ÓyòƒÎˆÏsf ú˛yÑ˛õy §yí˛zu˛ Ó: ÌyˆÏܲ– ÷ô% ï˛yˆÏÓ˚Ó˚ ܲ¡õˆÏöÓ˚ ï%˛°öyÎ˚ §yí˛zu˛ ÓˆÏ:Ó˚ Óyï˛yˆÏ§Ó˚ ܲ¡õˆÏöÓ˚
ú˛ˆÏ° í˛zͲõߨ ¢ˆÏ∑Ó˚ !ÓhflÏyÓ˚ Ä ï˛#Ó ï˛y xˆÏöܲ ˆÓ¢# •Î˚–
(b) ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚c ≠ ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚c ≠ ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚c ≠ ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚c ≠ ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚c ≠ ¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ Îï˛ ò)ˆÏÓ˚ ÎyÄÎ˚y ÎyÎ˚ñ §yôyÓ˚îû˛yˆÏÓ
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ï˛ï˛ •…y§ ˛õyÎ˚– Î!ò ~ܲ!ê˛ !Ó®% í˛zͧ ˆÌˆÏܲ í˛zͲõߨ ¢∑ ã˛ï%˛!ò≈ܲ §üû˛yˆÏÓ åÈ!í˛¸ˆÏÎ˚ ˛õˆÏí˛¸ñ ï˛ˆÏÓ
!öà≈ï˛ ¢∑¢!_´ ˆÌˆÏܲ r ò)Ó˚ˆÏc ˆüyê˛ 4
π
r2 ˆ«˛eú˛ˆÏ°Ó˚ SxÌ≈yÍ r Óƒy§yˆÏô≈Ó˚ ˆày°ˆÏܲÓ˚ ï˛ˆÏ°Ó˚ ˆ«˛eú˛ˆÏ°V
!Óï˛!Ó˚ï˛ •Î˚– §%ï˛Ó˚yÇñ ~ܲܲ ˆ«˛eú˛ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ «˛üï˛y í˛zͧ ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚ ÓˆÏà≈Ó˚ ÓƒhflÏyö%˛õyï˛#
•Î˚– ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä í˛zͧ ˆÌˆÏܲ ò)Ó˚ˆÏcÓ˚ ~ çyï˛#Î˚ §¡õÜ≈˛ˆÏܲ xyüÓ˚y ÓƒhflÏÓà≈ (inverse square) !öÎ˚ü
ӈϰ Ìy!ܲ–
¢ˆÏ∑Ó˚ í˛zͧ ˆÌˆÏܲ ˆ◊yï˛yÓ˚ ò)Ó˚ˆÏcÓ˚ í˛z˛õÓ˚ ï˛#Ó ï˛y !öû≈˛Ó˚ ܲˆÏÓ˚ ӈϰ•z xˆÏöܲ í˛zFã˛ï˛yÎ˚ Äí˛¸y ˆçê˛ ˆ≤’ˆÏöÓ˚
¢∑ «˛#î ˆ¢yöyÎ˚– !ܲv ˆ§!ê˛ Îáö !öã˛ !òˆÏÎ˚ ĈÏí˛¸ñ ï˛áö ï˛yÓ˚ àç≈ö Ó˚#!ï˛üï˛ x§%!ÓôyÓ˚ §,!‹T ܲˆÏÓ˚–
(ii) ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ≠ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ≠ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ≠ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ≠ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ≠ ¢ˆÏ∑Ó˚ !ÓhflÏyÓ˚ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏ° ï˛#Ó ï˛y ܲ¡õyˆÏB˛Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õyï˛# •Î˚–
~!ê˛ xÓ¢ƒ ܲyˆÏö ÷ˆÏö §ü§üˆÏÎ˚ ˆÓyé˛y ÎyÎ˚ öyñ ˆÜ˛ööy xyüyˆÏòÓ˚ ܲyˆÏöÓ˚ §ÇˆÏÓòö¢#°ï˛yÄ Ü˛¡õyˆÏB˛Ó˚ í˛z˛õÓ˚
!öû˛≈Ó˚ ܲˆÏÓ˚–
266
(iii) üyôƒˆÏüÓ˚ âöc ≠ üyôƒˆÏüÓ˚ âöc ≠ üyôƒˆÏüÓ˚ âöc ≠ üyôƒˆÏüÓ˚ âöc ≠ üyôƒˆÏüÓ˚ âöc ≠ ï˛#Ó ï˛yÓ˚ Ó˚y!¢üy°y ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñï˛#Ó ï˛y üyôƒˆÏüÓ˚ âöˆÏcÓ˚
§üyö%˛õyï˛#– ~çöƒ ÓyÎ˚% xˆÏ˛õ«˛y çˆÏ° ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ˆÓ!¢ •Î˚– xyÓyÓ˚ x!ôܲ í˛zFã˛ï˛yÎ˚ ˆÎáyˆÏö ÓyÎ˚%Ó˚ âöc
§ü%oï˛ˆÏ°Ó˚ ˆã˛ˆÏÎ˚ ܲüñ ˆ§áyˆÏö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÄ xˆÏ˛õ«˛yÜ,˛ï˛ ܲü •Î˚–
~ÓyÓ˚ xyüÓ˚y ï˛#Ó ï˛yÓ˚ ~ܲܲ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó– xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ÚÚ≤Ã!ï˛ Óà≈!üê˛yÓ˚
!˛õå%È ÄÎ˚yê˛ÛÛ ˆÜ˛•z ï˛#Ó ï˛yÓ˚ ~ܲܲ !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~!ê˛ !ܲv xyüyˆÏòÓ˚ ܲyˆÏåÈ á%Ó §%!Óôyçöܲ
öÎ˚– §Ó≈!ö¡¨ ï˛#Ó ï˛yÓ˚ ˆÎ ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö ôÓ˚y ˛õˆÏí˛¸ñ ï˛yÓ˚ ï˛#Ó ï˛y I0 ~Ó˚ üyey 10–12 W/m2 – xyüyˆÏòÓ˚
§yôyÓ˚î ܲÌyÓyï≈˛yÓ˚ §üÎ˚ ï˛#Ó ï˛y ~Ó˚Ä °«˛=î xÌ≈yÍ 10–7W/m2 •ˆÏï˛ ˛õyˆÏÓ˚– xyÓyÓ˚ °yí˛zí˛!flõܲyˆÏÓ˚Ó˚
ˆçyÓ˚ ¢∑ Óy !ÓˆÏfl≥˛yÓ˚ˆÏîÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ~Ó˚Ä °«˛ =î xÌ≈yÍ 10–2 W/m2 x!ï˛e´ü ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– ï˛#Ó ï˛yÓ˚
~•z !Ó¢y° ˛õyÕ‘y xyüyˆÏòÓ˚ ܲyˆÏöÓ˚ !ÓfløÎ˚ܲÓ˚ ܲü≈«˛üï˛y §)!ã˛ï˛ ܲÓ˚ˆÏ°Ä ï˛#Ó ï˛yÓ˚ üyey §Çáƒy myÓ˚y ≤ÃܲyˆÏ¢Ó˚
ˆ«˛ˆÏe x§%!ÓôyÓ˚ §,!‹T ܲˆÏÓ˚– ï˛y•z ï˛#Ó ï˛yˆÏܲ §Ó˚y§!Ó˚ W/m2 ~ܲˆÏܲ ≤Ãܲy¢ öy ܲˆÏÓ˚ xyüÓ˚y !ö¡¨ï˛ü ◊ÓîˆÏÎyàƒ
ï˛#Ó ï˛yÓ˚ (I0) §ˆÏD xö%˛õyï˛ !•§yˆÏÓ °ày!Ó˚ò‰ü‰ ˆfl҈ϰ ≤Ãܲy¢ ܲ!Ó˚–
~Ó˚ ú˛ˆÏ° ˆÎ ~ܲܲ!ê˛Ó˚ í˛zͲõ!_ •Î˚ñ ˆ§ öyü!ê˛ xy˛õöyÓ˚ á%Ó•z ˆã˛öyÈÙÙÙȈí˛!§ˆÏÓ°– ≤ÃyÎ˚•z ˆ¢yöy ÎyÎ˚ñ
Óy!ç˛õê˛Ü˛yñ ày!í˛¸Ó˚ •ö≈ñ üy•zܲ ≤Ãû,˛!ï˛Ó˚ í˛z˛õÓ˚ !öˆÏ£ÏôyK˛yñ ÚÚ xü)ܲ˛ ˛õ!Ó˚üyî ˆí˛!§ˆÏÓ° åÈyí˛¸yˆÏöy ã˛°ˆÏÓ öyÛÛ–
xÌ≈yÍ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y üy˛õyÓ˚ ~ܲܲ •° ˆí˛!§ˆÏÓ°– ï˛#Ó ï˛y Ó,!k˛Ó˚ üy˛õܲy!ë˛ˆÏï˛ §üï˛y xyöyÓ˚ çöƒ ï˛#Ó ï˛y
Ó,!k˛ˆÏܲ í˛zû˛Î˚ ï˛#Ó ï˛yÓ˚ xö%˛õyˆÏï˛Ó˚ °ày!Ó˚ò‰ü‰Ó˚*ˆÏ˛õ Sû)˛!üÈÙÈ10V ≤Ãܲy¢ ܲÓ˚y •Î˚– ~•z xyˆÏ˛õ!«˛Ü˛ ï˛#Ó ï˛yˆÏܲ
!ÓK˛yö# xyˆÏ°Ü˛çyu˛yÓ˚ @ˇÃy•yü ˆÓˆÏ°Ó˚ (Bell) öyüyö%§yˆÏÓ˚ ˆÓ° (Bel) ~ܲܲ Ó°y •Î˚– Î!ò ¢ˆÏ∑Ó˚ «˛üï˛y
Óy ≤Ã!ï˛ ~ܲܲ §üˆÏÎ˚ !öà≈ï˛ ¢∑¢!_´Ó˚˛˛õ!Ó˚üyî P1 ˆÌˆÏܲ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ P2 •ˆÏÎ˚ ÎyÎ˚ñ ï˛yˆÏܲ ˆÓ° ~ܲˆÏܲ
~•zû˛yˆÏÓ ≤Ãܲy¢ ܲÓ˚y •ˆÏÓ≠
«˛üï˛yÓ˚ Ó,!k˛Ó˚ ˆÓ° §Çáƒy (number of bels) = log10
2
1
PP
⎛⎞⎜⎟⎝⎠
...10.1
ÓyhflψÏÓ ˆÓ° xÌ≈yÍ «˛üï˛yÓ˚ ò¢=î Ó,!k˛ ÓƒÓ•y!Ó˚ܲ ≤ÈÏÎ˚yˆÏàÓ˚ çöƒ á%Ó•z Óí˛¸ ӈϰ ï˛yÓ˚ ~ܲ ò¢üyÇ¢ˆÏܲ
ˆí˛!§ˆÏÓ° (decibel) öyü ˆòÄÎ˚y •Î˚ ~ÓÇ ~ܲܲ !•§yˆÏÓ dB ˆ°áy •Î˚– §%ï˛Ó˚yÇñ «˛üï˛yÓ˚ Ó,!k˛Ó˚ ˆí˛!§ˆÏÓ°
§Çáƒy (number of decibels) = 10log10
2
1
PP
⎛⎞⎜⎟⎝⎠
...10.2
ˆí˛!§ˆÏÓ°•z ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Óy «˛üï˛y ˆÓyé˛yˆÏöyÓ˚ ÓƒÓ•y!Ó˚ܲ ~ܲܲ– ~ÓyˆÏÓ˚ ˆòá%ö ˆí˛!§ˆÏÓ° (dB)
˛õ!Ó˚Óï≈˛ö ܲï˛ê˛y Ó,!k˛ Óy •…y§ !öˆÏò≈¢ ܲˆÏÓ˚– «˛üï˛yÓ˚ ~ܲ ˆí˛!§ˆÏÓ° Ó,!k˛ Ó°ˆÏï˛
1010
Óy ≤ÃyÎ˚ 1·26 =î
Ó,!k˛ ˆÓyé˛yÎ˚– üˆÏö ܲÓ˚&öñ P1 ~Ó˚ ï%˛°öyÎ˚ P2 ˛õ!Ó˚üyî 1000 =î ˆÓ!¢– ˆí˛!§ˆÏÓ° ~ܲˆÏܲ Ó,!k˛ •ˆÏÓ üye
30dB xÌ≈yÍñ ˆí˛!§ˆÏӈϰ §yüyöƒ Ó,!k˛ üyˆÏö ÓyhflψÏÓ xˆÏöܲ Ó,!k˛–
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y hflÏÓ˚ Ó°ˆÏï˛ ˆÓyé˛yÎ˚ñ ˆÜ˛yˆÏöy üyöܲ Ú¢)öƒÛ hflψÏÓ˚Ó˚ ï%˛°öyÎ˚ ï˛yÓ˚ xyˆÏ˛õ!«˛Ü˛ ï˛#Ó ï˛y– Úüyöܲ
¢)öƒ ï˛#Ó ï˛y hflÏÓ˚Û !•§yˆÏÓ ¢ˆÏ∑Ó˚ 10–12Wm–2 ï˛#Ó ï˛y ôˆÏÓ˚ ˆöÄÎ˚y •Î˚– ~áö Î!ò Ó°y •Î˚ Ú60 ˆí˛!§ˆÏÓ°
¢∑Û ï˛yÓ˚ üyˆÏö í˛z_´ hflψÏÓ˚Ó˚ §yˆÏ˛õˆÏ«˛ 60dB Ó,!k˛ Óy 106 =î Ó,!k˛–
267
10.3 ï˛#Ó ï˛y ~ÓÇ ï˛yÓ˚ ~ܲܲï˛#Ó ï˛y ~ÓÇ ï˛yÓ˚ ~ܲܲï˛#Ó ï˛y ~ÓÇ ï˛yÓ˚ ~ܲܲï˛#Ó ï˛y ~ÓÇ ï˛yÓ˚ ~ܲܲï˛#Ó ï˛y ~ÓÇ ï˛yÓ˚ ~ܲܲ
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y xyÓ˚ ≤ÃyÓ°ƒ Ó°ˆÏï˛ §yôyÓ˚î û˛y£ÏyÎ˚ ~ܲ•z !Ó£ÏÎ˚ ˆÓyé˛yÎ˚– !ܲv ˛õòyÌ≈!ÓòƒyÓ˚ û˛y£ÏyÎ˚ ï˛#Ó ï˛yÓ˚
~ܲ!ê˛ !ö!ò≈‹T §ÇK˛y xyˆÏåÈ– ≤ÃyӈϰƒÓ˚ ~Ó˚ܲü ˆÜ˛yˆÏöy §ÇK˛y ˆö•z– ~!ê˛ üyöÓ •z!wˆÏÎ˚Ó˚ ~ܲ!ê˛ xö%û˛Ó üye–
ˆ◊yï˛yÓ˚ ܲyˆÏöÓ˚ í˛z˛õˆÏÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ ¢∑ ܲï˛ê˛y ˆçyÓ˚yˆÏ°y ˆ¢yöy ÎyˆÏÓ–
ï˛#Ó ï˛yÓ˚ üˆÏï˛y ≤ÃyÓ°ƒ §Ó˚y§!Ó˚ ÎyÎ˚ öy– ï˛y åÈyí˛¸y ~ܲ•z ï˛#Ó ï˛yÓ˚ !ܲv !û˛ß¨ ܲ¡õyˆÏB˛Ó˚ ¢∑ ˆÜ˛yˆÏöy ~ܲ
Óƒ!_´Ó˚ ܲyˆÏåÈ §yôyÓ˚îï˛ §üyö ˆçyÓ˚yˆÏ°y üˆÏö •Î˚ öy– ˆ§ ܲyÓ˚ˆÏî ≤ÃyÓ°ƒ üy˛õyÓ˚ çöƒ ~ܲ!ê˛ !ö!ò≈‹T üyöܲ
ôˆÏÓ˚ !öˆÏÎ˚ ï˛yÓ˚ §yˆÏ˛õˆÏ«˛ xöƒ í˛zͧ=!°Ó˚ ï%˛ö°y ܲÓ˚y •Î˚– ôÓ˚y Îyܲ ~ܲ!ê˛ 1000Hz ܲ¡õyˆÏB˛Ó˚ 10–12
Wm–2 ï˛#Ó ï˛yÓ˚ ¢∑ í˛zͧ ˆöÄÎ˚y •°– ˆÎ ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒ üy˛õy •ˆÏÓñ ï˛yÓ˚ §ˆÏD üyöܲ í˛z͈ϧÓ˚ ¢∑ í˛zͲõyòö
ܲˆÏÓ˚ Ä•z üyöܲ í˛zͧ!ê˛Ó˚ ï˛#Ó ï˛y e´ü¢ ˛õ!Ó˚Óï≈˛ö ܲÓ˚y •Î˚ Îï˛«˛î öy í˛zû˛Î˚ í˛z͈ϧÓ˚ ≤ÃyÓ°ƒ §üyö •ˆÏÎ˚ ÎyÎ˚–
ôÓ˚y Îyܲñ ~•z ܲyç ܲÓ˚ˆÏï˛ !àˆÏÎ˚ üyöܲ í˛z͈ϧÓ˚ ï˛#Ó ï˛y ndB åÈí˛¸yˆÏöy •°– ï˛áö ˛õÓ˚#«˛yô#ö í˛zͧ!ê˛Ó˚
≤ÃyÓ°ƒüyeyˆÏܲ Ó°y •ˆÏÓ n ú˛ö (phon)–
§%ï˛Ó˚yÇñ ú˛ö •° ≤ÃyÓ°ƒüyeyÓ˚ ~ܲܲñ ˆÎüö ï˛#Ó ï˛yÓ˚ ~ܲܲ !åÈ° ˆí˛!§ˆÏÓ°– ~ܲüye 1000 Hz ܲ¡õyˆÏB˛Ó˚
çöƒ ú˛ö ˆflÒ° Ä ˆí˛!§ˆÏÓ° ˆflÒ° §üyö •ˆÏÓ– xÌ≈yÍñ ï˛#Ó ï˛y Îï˛ ˆí˛!§ˆÏÓ°ñ ≤ÃyÓ°ƒüyey !ë˛Ü˛ ï˛ï˛ ú˛ö
•ˆÏÓ–
10.3.1 ï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öyï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öyï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öyï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öyï˛#Ó ï˛y Ä ≤ÃyӈϰƒÓ˚ ï%˛°öy
xyˆÏàÓ˚ ò%!ê˛ xö%ˆÏFåÈò ˆÌˆÏܲ ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ï˛#Ó ï˛y xyÓ˚ ≤ÃyÓ°ƒ ò%!ê˛ ˛õ,Ìܲ ôyÓ˚îy– ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y
§Ó˚y§!Ó˚ üy˛õy ÎyÎ˚– ˛õˆÏÓ˚Ó˚ xö%ˆÏFåȈÏò•z ï˛y ˆòáy ÎyˆÏÓ– !ܲv ≤ÃyÓ°ƒ ~ܲ!ê˛ xö%û)˛!ï˛ÈÙÙÙÈ¢∑ ܲï˛ê˛y ˆçyÓ˚yˆÏ°y
ˆ¢yöyˆÏFåÈ ï˛y ≤Ãܲy¢ ܲÓ˚yÓ˚ í˛z˛õyÎ˚– ~•z ܲyÓ˚ˆÏî ≤ÃyÓ°ƒ ˛õˆÏÓ˚y«˛û˛yˆÏÓ üy˛õy •Î˚ ò%!ê˛ ï˛#Ó ï˛yÓ˚ ï%˛°öyÓ˚*ˆÏ˛õ–
ï˛ˆÏÓ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y xyÓ˚ ≤ÃyӈϰƒÓ˚ üˆÏôƒ ~ܲ!ê˛ §¡õÜ≈˛ xyˆÏåÈñ ÎyˆÏܲ ĈÏÎ˚ÓyÓ˚ ˆú˛Ü˛öyÓ˚ !öÎ˚ü Ó°y (Weber-
Fechner Law) •Î˚– ~•z !öÎ˚ü xö%§yˆÏÓ˚ñ ≤Ãyӈϰƒ ö)ƒöï˛ü xö%û˛ÓˆÏÎyàƒ Ó,!k˛ xyöyÓ˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ï˛#Ó ï˛y
Ó,!k˛ ü)° ï˛#Ó ï˛yÓ˚ §üyö%˛õyï˛# •Î˚– xÌ≈yÍñ ¢∑ ≤ÃyӈϰƒÓ˚ ܲyˆÏö ôÓ˚y ˛õí˛¸yÓ˚ üï˛ ö)ƒöï˛ü Ó,!k˛ Î!ò dL •Î˚ñ
xyÓ˚ ï˛yÓ˚ ˛≤ÈÏÎ˚yçö#Î˚ ï˛#Ó ï˛y Ó,!k˛ Î!ò dl •Î˚ñ ï˛ˆÏÓ
dl a dL·I ˆÎáyˆÏö I = ü)° ï˛#Ó ï˛y– xÌÓyñ
dL = k
dll
ˆÎáyˆÏö k ~ܲ!ê˛ ô &Óܲ– §üyܲ°ö ܲÓ˚ˆÏ° ˛õy•zñ
L = k logI + C
Óyñ L = k1log10l
ˆÎáyˆÏö k1 ~ܲ!ê˛ ô &Óܲ–
268
§ü#ܲÓ˚î 10.3 •ˆÏ° ĈÏÎ˚ÓyÓ˚ ˆú˛Ü˛öyÓ˚ !öÎ˚ˆÏüÓ˚ ày!î!ï˛Ü˛ Ó˚*˛õ– ~Ó˚ ˆÌˆÏÜ flõ‹T •ˆÏFåÈ ˆÎñ ï˛#Ó ï˛y Îáö
=ˆÏîy_Ó˚ ˆ◊î#ˆÏï˛ ÓyˆÏí˛¸ñ ≤ÃyÓ°ƒ ï˛áö ÓyˆÏí˛¸ §üyhsˇÓ˚ ˆ◊î#ˆÏï˛– ï˛y•z ï˛#Ó ï˛yÎ˚ §üyö xö%˛õyˆÏï˛ Ó,!k˛ âê˛ˆÏ°ñ
≤ÃyÓ°ƒÄ §üyö ôyˆÏ˛õ Óyí˛¸ˆÏÓ– ≤ÃyӈϰƒÓ˚ ~ܲܲ Ä ˛õ!Ó˚üyî !öˆÏÎ˚ ˛õˆÏÓ˚Ó˚ ~ܲˆÏܲ !ÓhflÏy!Ó˚ï˛ xyˆÏ°yã˛öy ܲÓ˚y
•ˆÏÓ– ~ÓyÓ˚ xy˛õ!ö ÓÓ˚Ç !öˆÏã˛Ó˚ xö%¢#°ö#!ê˛Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 1
(a) ~ܲ!ê˛ ÓƒhflÏ Ó˚yç˛õˆÏÌ üôƒyˆÏ•´ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y hflÏÓ˚ 80db !åÈ°– üôƒÓ˚y!eˆÏï˛ ï˛#Ó ï˛y ï%˛°öyÎ˚ 32000
=î ܲü •ˆÏ° ~•z ï˛#Ó ï˛y hflÏÓ˚ˆÏܲ ˆí˛!§ˆÏÓ° ~ܲˆÏܲ ˛≤Ãܲy¢ ܲÓ˚&ö–
(b) ¢)öƒfliyö=!° ˛õ)î≈ ܲÓ˚&ö S§yΩ˛yÓƒ ¢∑=!° Ó¶˛ö#ˆÏï˛ ˆòÄÎy xyˆÏåÈV ≠
¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ xyܲyÓ˚ Îï˛–––––––SÓyˆÏí˛¸†Ü˛ˆÏüV ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ï˛ï˛ ˆÓ!¢ •Î˚– í˛zͧ ˆÌˆÏܲ
ˆ◊yï˛yÓ˚ ò)Ó˚c xˆÏô≈ܲ •ˆÏ° ˆ◊yï˛yÓ˚ ܲyˆÏö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y––––––––Sò%•z=î†ã˛yÓ˚=îV ˆÓ!¢ üˆÏö
•Î˚– ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ ~ܲܲ––––––––(W/Wm–2/Wm2)– ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒ Ó°ˆÏï˛ ˆÓyé˛yÎ˚ ¢ˆÏ∑Ó˚–
–––––Sܲ¡õyB˛†ï˛#Ó ï˛y†≤ÃÓ°ï˛yÓ˚ xö%û)˛!ï˛V–
10.4 ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üy˛õ
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲ¡õˆÏöÓ˚ !ÓhflÏyˆÏÓ˚Ó˚ ÓˆÏà≈Ó˚ §üyö%˛õyï˛# •Î˚ ӈϰ öyöyû˛yˆÏÓ !ÓhflÏyÓ˚ ˛õ!Ó˚üy˛õ ܲˆÏÓ˚ ï˛#Ó ï˛y
àîöy ܲÓ˚y ÎyÎ˚– ˆÎüöñ ÓyÎ˚%Ó˚ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ï˛yÓ˚ üˆÏôƒÜ˛yÓ˚ ô)!°Ü˛îyÓ˚ xyˆÏ®y°ˆÏöÓ˚ !ÓhflÏyÓ˚ üy•zˆÏe´yˆÏflÒyˆÏ˛õÓ˚
§y•yˆÏ΃ ˆòáy ÎyÎ˚– xÌÓyñ ܲ¡õö Ä fliyî% ï˛Ó˚ˆÏDÓ˚ ú˛ˆÏ° ÓyÎ˚%Ó˚ âö#û˛Óö Ä ï˛ö%û˛ÓˆÏöÓ˚ òÓ˚&î ÓyÎ˚%Ó˚
≤Ã!ï˛§Ó˚yˆÏB˛Ó˚ ˛õÎ≈yÎ˚e´ˆÏü ˛õ!Ó˚Óï≈˛ö •Î˚– ï˛yÓ˚ üôƒ !òˆÏÎ˚ xyˆÏ°yÓ˚ Óƒ!ï˛ã˛yÓ˚ °«˛ƒ ܲˆÏÓ˚ ÓyÎ˚%Ó˚ ܲ¡õˆÏöÓ˚ ú˛ˆÏ°
ã˛yˆÏ˛õÓ˚ !ÓhflÏyÓ˚ !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚– ï˛ˆÏÓ ¢∑ï˛Ó˚ˆÏDÓ˚ ˛õ!Ó˚Óï˛#≈ ã˛y˛õ Óy ˆÓàˆÏܲ ï˛!í˛¸Í !Óû˛ˆÏÓ ˛õ!Ó˚îï˛ Ü˛ˆÏÓ˚
˙ ˛õ!Ó˚Óï˛#≈ ï˛!í˛¸Í !Óû˛ˆÏÓÓ˚ ˛õ!Ó˚üy˛õ•z Óï≈˛üyˆÏö ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üyˆÏ˛õÓ˚ ≤Ãã˛!°ï˛ í˛z˛õyÎ˚–
ˆÜ˛yöÄ ~ܲ!ê˛ !Ó®%ˆÏï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y üy˛õˆÏï˛ ˆÓ¢ ܲˆÏÎ˚ܲ!ê˛ ÓƒÓ•y!Ó˚ܲ x§%!ÓôyÓ˚ ü%ˆÏáyü%!á •ˆÏï˛ •Î˚–
~=!° xyüyˆÏòÓ˚ ˆçˆÏö Ó˚yáy û˛yˆÏ°y– x§%!ÓôyÓ˚ ܲyÓ˚î=!° •° É
1. ˆÎ ˆÜ˛yöÄ ¢∑üy˛õܲ Îsf ¢∑ˆÏ«˛eˆÏܲ ܲüˆÏÓ!¢ ˛õ!Ó˚üyˆÏî !ÓÜ,˛ï˛ ܲˆÏÓ˚ñ §%ï˛Ó˚yÇ Îsf!ê˛ ï˛yÓ˚ ã˛yÓ˚˛õyˆÏ¢
ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚Óï≈˛ö âê˛yÎ˚– ~çöƒ Îsf!ê˛Ó˚ üy˛õ ˛õ!Ó˚ˆÏüÎ˚ ¢ˆÏ∑Ó˚ ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ ï%˛°öyÎ˚ ˆåÈyê˛ Ó˚yáˆÏï˛ •Î˚–
2. ˆÜ˛yöÄ ¢∑üy˛õܲ Îsf•z §Ó ï˛#Ó ï˛yÎ˚ §üyöû˛yˆÏÓ §yí˛¸y ˆòÎ˚ öy– ~üö!ܲ ܲ¡õyˆÏB˛Ó˚ §ˆÏDÄ ¢∑üy˛õܲ
ΈÏsfÓ˚ ò«˛ï˛y ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– !ӈϢ£Ïï˛ñ ΈÏsfÓ˚ !öçfl∫ ˆÜ˛yöÄ Ü˛¡õyB˛ ÌyܲˆÏ° ˙ ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑ Îsf!ê˛ˆÏï˛
xö%öyò âˆÏê˛ ~ÓÇ §yí˛¸yÓ˚ ˛õ!Ó˚üyî xöƒ ܲ¡õyˆÏB˛Ó˚ ï%˛°öyÎ˚ xˆÏöܲ ˆÓ!¢ •Î˚– ~•z ܲyÓ˚ˆÏî ΈÏsfÓ˚ àë˛ö ~üö
•Î˚ ÎyˆÏï˛ Ü˛¡õyˆÏB˛Ó˚ ˆÎ ˛õyÕ‘yÎ˚ ˆ§!ê˛ ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÓ ï˛yÓ˚ üˆÏôƒ Óy !öܲˆÏê˛ ÎˆÏsfÓ˚ ˆÜ˛yöÄ xö%öyò# ܲ¡õyB˛
öy ÌyˆÏܲ–
269
3. ˆÜ˛yöÄ Îsf•z §Ó üyôƒˆÏü ÓƒÓ•yˆÏÓ˚Ó˚ í˛z˛õˆÏÎyà# •Î˚ öy–
4. §Ó !ܲå%ÈÓ˚ ˛õÓ˚ñ ¢ˆÏ∑Ó˚ «˛üï˛yÓ˚ ≤ÃÓy• ~ï˛ x“ ˆÎñ xï˛ƒhsˇ §%- §ÇˆÏÓòö¢#° Îsf•z ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚üyˆÏî
ÓƒÓ•yÓ˚ •Î˚–
xy§%öñ ~ÓyÓ˚ xyüÓ˚y ¢∑ï˛#Ó ï˛y üy˛õyÓ˚ ܲˆÏÎ˚ܲ!ê˛ ÎˆÏsfÓ˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ ••z–
10.4.1 Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛ (Rayleigh’s Disc)
≤Ã̈Ïü ~•z Îsf!ê˛ ˆÎ ö#!ï˛Ó˚ !û˛!_ˆÏï˛ Ü˛yç ܲˆÏÓ˚ñ ï˛y Ó°y Îyܲ– ~ܲ!ê˛ ˆåÈyê˛ñ •y°Ü˛y ã˛yܲ!ï˛ Î!ò ¢ˆÏ∑Ó˚
à!ï˛˛õˆÏÌ Ó˚yáy •Î˚ñ ï˛ˆÏÓ ¢∑ï˛Ó˚ˆÏDÓ˚ ã˛y˛õ ˆ§!ê˛ˆÏܲ ï˛Ó˚D§MÈ˛yˆÏÓ˚Ó˚ !òˆÏܲÓ˚ §ˆÏD °¡∫û˛yˆÏÓ Ó˚yáˆÏï˛ ã˛yÎ˚– ~ܲ!ê˛
§)ï˛y Óy ˛õyï˛°y ï˛yˆÏÓ˚ ˆé˛y°yˆÏöy •yÕÒy Ó,_ܲyÓ˚ ã˛yܲ!ï˛!ê˛ Î!ò ¢ˆÏ∑Ó˚ à!ï˛ü%ˆÏáÓ˚ §yˆÏ˛õˆÏ«˛ ~üöû˛yˆÏÓ xÓfliyö
ܲˆÏÓ˚ ÎyˆÏï˛ ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ °¡∫ ~ÓÇ ¢ˆÏ∑Ó˚ à!ï˛ü%ˆÏáÓ˚ üˆÏôƒ
θ
ˆÜ˛yî §,‹T •Î˚ñ ï˛áö ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ ~ܲ!ê˛
Ó°Î%@¬ ܲyç ܲˆÏÓ˚
ÎyÓ˚ üyö ≠ T =
32 4sin23
r ρνθ
...10.4(a)
xÌ≈yÍñ ~áyˆÏö
ρ
üyôƒˆÏüÓ˚ ÓyÎ˚%Ó˚ âöcñ r ã˛yܲ!ï˛Ó˚ Óƒy§yô≈ ~ÓÇ °
2 ν
•ˆÏFåÈ ¢ˆÏ∑Ó˚ ú˛ˆÏ° í˛zͲõߨ
ÓyÎ˚%%≤ÃyÓyˆÏ•Ó˚ àí˛¸ Óà≈ ˆÓà– ~•z §)ˆÏe xÓ¢ƒ öyöy ܲyÓ˚ˆÏî !ܲå%È e&!ê˛ ÌyˆÏܲ– ~•z ܲyÓ˚î=!° •°ñ ¢ˆÏ∑Ó˚
à!ï˛˛õˆÏÌ ã˛yܲ!ï˛!ê˛ ÌyܲyÓ˚ ú˛ˆÏ° ¢∑ï˛Ó˚ˆÏDÓ˚ ÓƒÓï≈˛öñ üyôƒˆÏüÓ˚ §ywï˛yñ ¢∑ã˛yˆÏ˛õÓ˚ ≤Ãû˛yˆÏÓ é%˛°hsˇ ã˛yܲ!ï˛Ó˚
ˆòy°ö •zï˛ƒy!ò– xyÓyÓ˚ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ Î!ò ã˛yܲ!ï˛Ó˚ ÓƒÓï≈˛ ˆòy°ˆÏöÓ˚ fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ §üyö •ˆÏÎ˚ ÎyÎ˚ñ
ï˛áö xö%öyˆÏòÓ˚ ú˛ˆÏ° ˛õ!Ó˚üyˆÏ˛õ Óƒyâyï˛ âˆÏê˛– ï˛ˆÏÓ ~ §ˆÏÓÓ˚ çöƒ §ÇˆÏ¢yôö# ÓƒÓfliy ˆöÄÎ˚y ÎyÎ˚ ~ÓÇ ~ˆÏï˛
Îsf!ê˛Ó˚ ü)° ö#!ï˛ ~ܲ•z ÌyˆÏܲ–
~ÓyˆÏÓ˚ Îsf!ê˛Ó˚ àë˛ö≤Ãîy°# ˆòáy Îyܲ– 10.1(a) !ã˛ˆÏe ~!ê˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~ܲ ˆ§!ü Óƒy§yˆÏô≈Ó˚ xˆÏºÓ˚
ã˛yܲ!ï˛ D ≤ÃyÎ˚ ò%•z ˆ§!ü ÓƒyˆÏ§Ó˚ °¡∫y ܲyã˛ö° A ~Ó˚ üˆÏôƒ ˆÜ˛yÎ˚yê≈˛ç §)e !òˆÏÎ˚ ˆé˛y°yˆÏöy xyˆÏåÈ– ˆÜ˛yÎ˚yê≈˛ç
§)ˆÏeÓ˚ §ˆÏD ~ܲ!ê˛ •y°Ü˛y xyÎ˚öy °yàyˆÏöy
xyˆÏåÈ ~ÓÇ §)e!ê˛Ó˚ í˛z˛õˆÏÓ˚Ó˚ ≤Ãyhsˇ ~ܲ!ê˛
ÓƒyÓï≈˛ÈÙÈ¢#ˆÏ£Ï≈Ó˚ (B) §ˆÏD xyÓk˛–
ã˛yܲ!ï˛!ê˛ ≤Ã̈Ïü ܲyã˛öˆÏ°Ó˚ xˆÏ«˛Ó˚ §ˆÏD 45°
ˆÜ˛yˆÏî Ó˚yáy •Î˚– ~ܲ!ê˛ xyˆÏ°yܲ í˛zͧ L ˆÌ Ïܲ
xyˆÏ°yܲ Ó˚!Ÿ¬ M xyÎ˚öyÎ˚˛≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ S
ˆfl҈ϰ ˛õˆÏí˛¸– H ~ܲ!ê˛ ˛õyï˛°y ܲyàˆÏçÓ˚ ˛õò≈y
ÎyÓ˚ üˆÏôƒ !òˆÏÎ˚ ¢∑ï˛Ó˚D §•ˆÏç•z §MÈ˛y!Ó˚ï˛ •Î˚
~ÓÇ ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ ˛õˆÏí˛¸– S ˆfl҈ϰ xyˆÏ°yܲ!Ó®%Ó˚ xÓfliyö ˆÌˆÏܲ ã˛yܲ!ï˛Ó˚ !ӈϫ˛˛õ °«˛ƒ ܲÓ˚y ÎyÎ˚ ~ÓÇ
ÓƒyÓï≈˛ÈÙÈ¢#£Ï≈!ê˛ˆÏܲ â%!Ó˚ˆÏÎ˚ ã˛yܲ!ï˛Ó˚ ˛õ)Ó≈yÓfliyÎ˚ !ú˛!Ó˚ˆÏÎ˚ xyöy ÎyÎ˚– ÓƒyÓï≈˛¢#ˆÏ£Ï≈Ó˚ ≤ÈÏÎ˚yçö#Î˚ â)î≈ö ˆÌˆÏܲ
¢∑ï˛Ó˚ˆÏDÓ˚ myÓ˚y ≤ÃÎ%_´ Ó°Î%ˆÏ@¬Ó˚ üyö ÓyÓ˚ ܲÓ˚y ÎyÎ˚–
270
ã˛yܲ!ï˛!ê˛ˆÏܲ ¢∑ï˛Ó˚ˆÏDÓ˚ à!ï˛ü%ˆÏáÓ˚ §ˆÏD 45° ˆÜ˛yˆÏî Ó˚yáyÎ˚ ˆ§!ê˛Ó˚ í˛z˛õÓ˚ Ó°Î%ˆÏ@¬Ó˚ üyö §Ó≈y!ôܲ •Î˚
ˆÜ˛ööy Îáö
θ
= 45°, sin 2
θ
= 1 ~ÓÇ T =
32 43
r ρν
– ...10.4(b)
~åÈyí˛¸y ¢ˆÏ∑Ó˚ ï˛Ó˚D
λ
•ˆÏ° AD =
4λ
~ÓÇ AH = 3
λ
/4 Ó˚yáy •Î˚ñ ÎyˆÏï˛ ï˛ˆÏ°Ó˚ üˆÏôƒ xö%öyò •ˆÏÎ˚
fliyî%ï˛Ó˚ˆÏDÓ˚ §,!‹T •Î˚ ~ÓÇ ã˛yܲ!ï˛!ê˛ ~ܲ!ê˛ §%flõ® !Ó®%ˆÏï˛ ÌyˆÏܲ– ~ˆÏï˛Ä Ó°Î%ˆÏ@¬Ó˚ üyö xˆÏöܲ ÓyˆÏí˛¸–
10.1(b) !ã˛ˆÏe ÓˆÏÎ˚çÈÙÈ~Ó˚ myÓ˚y í˛zqy!Óï˛ Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ ~ܲ!ê˛ !Óܲ“ Ä í˛zß¨ï˛ Îsf§Iy ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
~áyˆÏö A ~ÓÇ B ò%!ê˛ ú˛yÑ˛õy ôyï˛Ó ˛õyeñ ÎyˆÏòÓ˚ ˜òâ≈ƒ ܲüˆÏÓ!¢ ܲÓ˚y ÎyÎ˚– ~ˆÏòÓ˚ ˆÎyàyˆÏÎyà Ó˚ˆÏÎ˚ˆÏåÈ
C öˆÏ°Ó˚ üˆÏôƒ !òˆÏÎ˚– ~•z çyï˛#Î˚ ú˛yÑ˛õy ˛õye xö%öyòˆÏܲÓ˚ ܲyç ܲˆÏÓ˚– ~Ó˚ܲü ΈÏsfÓ˚ §¡∫ˆÏ¶˛ xyüÓ˚y xö%ˆÏFåÈò
10.7.2 ˆï˛ xyÓyÓ˚ ˛õí˛¸Ó– ~ˆÏòÓ˚ ܲyç ¢ˆÏ∑Ó˚ !ö!ò≈‹T ܲ¡õyˆÏB˛ xö%öy!òï˛ •ÄÎ˚y– A ˛õyˆÏeÓ˚ !˛õåȈÏö ˆÎ !åÈo
ÌyˆÏܲñ ï˛y !òˆÏÎ˚ xyˆÏ°y ˆú˛°yÓ˚ ÓƒÓfliy ÌyˆÏܲ ~ÓÇ B ˛õyˆÏeÓ˚ !˛õåȈÏöÓ˚ ˛õò≈yÓ˚ üôƒ !òˆÏÎ˚ ¢∑ï˛Ó˚D ≤ÈÏÓ¢
ܲˆÏÓ˚– C öˆÏ°Ó˚ üˆÏôƒ ~ܲ!ê˛ ˆÜ˛yÎ˚yê≈˛ç ï˛yˆÏÓ˚Ó˚ §y•yˆÏ΃ D ã˛yܲ!ï˛!ê˛ ¢∑ï˛Ó˚ˆÏDÓ˚ à!ï˛˛õˆÏÌÓ˚ §yˆÏ˛õˆÏ«˛ 45°
ˆÜ˛yˆÏî ˆé˛y°yˆÏöy ÌyˆÏܲ– ï˛yÓ˚ xyÎ˚öyÎ˚ xyˆÏ°y ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ ˆflÒ° S-~Ó˚ í˛z˛õÓ˚ ~ˆÏ§ ˛õˆÏí˛¸– !ã˛ˆÏe ~!ê˛
û˛@¿ˆÏÓ˚áy !òˆÏÎ˚ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˆÜ˛yÎ˚yê≈˛ç ï˛yÓ˚!ê˛ ~ܲ!ê˛ ÓƒyÓï≈˛ÈÙÈ¢#ˆÏ£Ï≈Ó˚ §ˆÏD °yàyˆÏöy ÌyˆÏܲ– ¢#£Ï≈!ê˛Ó˚ §ˆÏD
~ܲ!ê˛ Ó,_yܲyÓ˚ öÓÄ Î%_´ ÌyˆÏܲ ÎyˆÏï˛ ¢#£Ï≈!ê˛ˆÏܲ ܲï˛ê˛y ˆâyÓ˚yˆÏöy •°ñ ï˛y ˆÓyé˛y ÎyÎ˚– ÓˆÏÎ˚çÈÙÈ~Ó˚ Îsf!ê˛Ó˚
ܲyÎ≈˛õk˛!ï˛ Ó˚ƒyˆÏ°Ó˚ ΈÏsfÓ˚ xö%Ó˚*˛õ–
ôÓ˚y Îyܲñ ˆÜ˛yÎ˚yê˛≈ç ï˛yˆÏÓ˚ ~ܲܲ ˆüyã˛ˆÏí˛¸Ó˚ çöƒ ≤ÈÏÎ˚yçö#Î˚ ê˛Ü≈˛
τ
– Î!ò ÓƒyÓï≈˛ÈÙÈ¢#ˆÏ£Ï≈Ó˚ ≤ÈÏÎ˚yçö#Î˚ â)î≈≈ö
φ
•Î˚ ï˛ˆÏÓT =
τφ
– ~áö 10.4(b) §ü#ܲÓ˚ˆÏî ˆÌˆÏܲ
2 ν
ÈÙÈ~Ó˚ üyö !öî≈Î˚ ܲˆÏÓ˚ñ
I =
2c ρν
(c = ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓàV
§)e ˆÌˆÏܲ ï˛#Ó ï˛y 1-~Ó˚ üyö !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
271
˛10.4.2 ï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yöï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yöï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yöï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yöï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yö (Hot wire microphone)
xy˛õ!ö !öÿ˛Î˚ çyˆÏöö ˆÎñ ôyï˛Ó ï˛yˆÏÓ˚Ó˚ ˆÓ˚yô ï˛yÓ˚ í˛z£èï˛yÓ˚ §ˆÏD ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– Î!ò ~ܲ!ê˛ ï˛Æ ï˛yÓ˚
!ö!ò≈‹T í˛z£èï˛yÎ˚ Ó˚yáy •Î˚ñ ï˛yÓ˚ ˆÓ˚yô !fliÓ˚ ÌyܲˆÏÓ– !ܲv ï˛yˆÏܲ Î!ò ÓyÎ˚%≤ÃÓy• myÓ˚y ë˛y[˛y ܲÓ˚y •Î˚ñ ˆÓ˚yô
ܲˆÏü ÎyˆÏÓ– ~áöñ ÓyÎ˚%≤ÃÓy• Î!ò ¢mï˛Ó˚ˆÏDÓ˚ ú˛ˆÏ° §,‹T •Î˚ñ ï˛ˆÏÓ ï˛yˆÏÓ˚Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Îï˛
ˆÓ!¢ •ˆÏÓñ ÓyÎ˚%Ó˚ xyˆÏ®y°öÄ ï˛ï˛ Óyí˛¸ˆÏÓ ~ÓÇ ï˛yÓ˚!ê˛ ï˛y˛õ ˛õ!Ó˚ã˛y°ˆÏöÓ˚ ú˛ˆÏ° ë˛y[˛y •ˆÏÎ˚ ˆÓ˚yôÄ ï˛ï˛ ܲüˆÏÓ–
xÌ≈yÍ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛yˆÏÓ˚Ó˚ ˆÓ˚yˆÏôÓ˚ üˆÏôƒ ~ܲê˛y §¡õÜ≈˛ ÌyܲˆÏÓ ~ÓÇ ˆÓ˚yˆÏôÓ˚ ˛õ!Ó˚üy˛õ ˆÌˆÏܲ ï˛#Ó ï˛yÓ˚
üyö ˛õyÄÎ˚y ÎyˆÏÓ– ˆÓ˚yˆÏôÓ˚ ˛õ!Ó˚üyˆÏ˛õÓ˚ çöƒ xyüÓ˚y ‡•zê˛ˆÏfiê˛yö !Ó ç ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛˛ ˛õy!Ó˚– ~•z !Ó ˆÏçÓ˚
ˆÓ˚yô ÓƒÓfliyÓ˚ !Ó˛õÓ˚#ï˛ Óy‡ ò%!ê˛Ó˚ ˆÓ˚yˆÏôÓ˚ xö%˛õyï˛ Î!ò §üyö •Î˚ñ ò%•z !Ó˛õÓ˚#ï˛ §ÇˆÏÎyà !Ó®%Ó˚ üˆÏôƒÜ˛yÓ˚
àƒy°û˛yˆÏöy!üê˛yˆÏÓ˚ ˆÜ˛yöÄ !ӈϫ˛˛õ ˆòáy ÎyˆÏÓ öy– ~ˆÏܲ ӈϰ !Ó ˆÏçÓ˚ §yüƒyÓfliy– ~áö üˆÏö ܲÓ˚&öñ !ö!ò≈‹T
í˛z£èï˛yÎ˚ ï˛Æ ï˛yÓ˚!ê˛ ~•z !Ó ˆÏçÓ˚ ~ܲ Óy‡Ó˚ ˆÓ˚yô !•§yˆÏÓ Ó˚yáy xyˆÏåÈ– Î!ò í˛z£èï˛y ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ú˛ˆÏ° ˆÓ˚yô
ܲˆÏü ÎyÎ˚ñ !Ó ˆÏçÓ˚ §yüƒyÓfliy ö‹T •ˆÏÎ˚ àƒy°û˛yˆÏöy!üê˛yˆÏÓ˚ ï˛!í˛¸Í ≤ÃÓy• ˆòáy ˆòˆÏÓ– ~•zû˛yˆÏÓ xy˛õ!ï˛ï˛
¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y xyÓ˚ àƒy°û˛yˆÏöy!üê˛yˆÏÓ˚ ï˛!í˛¸Í ≤ÃÓy• §¡õ!Ü≈˛ï˛ ÌyˆÏܲ– ~áö ï˛#Ó ï˛y çyöy xyˆÏåÈ ~üö
ܲˆÏÎ˚ܲ!ê˛ ¢∑ï˛Ó˚ˆÏDÓ˚ §y•yˆÏ΃ ï˛!í˛¸Í ≤ÃÓy•ˆÏܲ xyÇ¢y!B˛ï˛ (calibrate) ܲÓ˚y ÎyÎ˚– ~•z xÇ¢yB˛ˆÏöÓ˚ §y•yˆÏ΃
xöƒ ˆÜ˛yöÄ ¢ˆÏ∑Ó˚ xçyöy ï˛#Ó ï˛y !öô≈yÓ˚î ܲÓ˚y §Ω˛Ó •Î˚–
!ã˛e ≠ 10.2
~ÓyˆÏÓ˚ 10.2 !ã˛ˆÏe ï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛ö≤Ãîy°# °«˛ƒ ܲÓ˚&ö– ~áyˆÏö M ~ܲ!ê˛ xº ã˛yܲ!ï˛ñ
ÎyÓ˚ üyé˛áyˆÏö ˆày°yÜ,˛!ï˛ !åÈo– ~Ó˚ í˛z˛õˆÏÓ˚ xyˆÏåÈ ~ܲ!ê˛ Ü˛yˆÏã˛Ó˚ Ó˚í˛ G ÎyÓ˚ àyˆÏÎ˚ 6 üy•ze´ö ÓƒyˆÏ§Ó˚ §)-
272
≤’ƒy!ê˛öyü ï˛yÓ˚ xyê˛Ü˛yˆÏöy ÌyˆÏܲ– ~•z ï˛yÓ˚•z ï˛Æ ï˛yˆÏÓ˚Ó˚ ܲyç ܲˆÏÓ˚– xº ã˛yܲ!ï˛!ê˛ ò%ÈÙÈ˛õyˆÏ¢ ò%!ê˛ ˆày°yܲyÓ˚
!åÈoÎ%_´ Ó˚&ˆÏ˛õyÓ˚ ˛õyï˛ s, s' !òˆÏÎ˚ ã˛y˛õy ÌyˆÏܲ– ≤’ƒy!ê˛öyü ï˛yˆÏÓ˚Ó˚ ~ܲ ≤Ãyhsˇ í˛z˛õˆÏÓ˚Ó˚ ˛õyï˛ ~ÓÇ xöƒ˛ ≤Ãyhsˇ
!öˆÏã˛Ó˚ ˛õyˆÏï˛Ó˚ §ˆÏD ÎÌye´ˆÏü T1 Ä T2 ê˛y!ü≈öyˆÏ° °yàyˆÏöy ÌyˆÏܲ– T1 Ä T2 ≤ÃyˆÏhsˇÓ˚ §y•yˆÏ΃ ï˛yˆÏÓ˚Ó˚ üˆÏôƒ
ï˛!í˛¸Í ≤ÃÓy• ˛õy!ë˛ˆÏÎ˚ ï˛yˆÏܲ àÓ˚ü ܲÓ˚y •Î˚– ~üöû˛yˆÏÓ ï˛yÓ˚!ê˛ !öü≈yî ܲÓ˚y •Î˚ñ ÎyˆÏï˛ 30mA ï˛!í˛¸Í ≤ÃÓyˆÏ••z
ï˛yˆÏÓ˚Ó˚ í˛z£èï˛y 400°C ˛õÎ≈hsˇ í˛zˆÏë˛ ÎyÎ˚–
‡•zê˛ˆÏfiê˛yö !Ó ç!ê˛ˆÏܲ ≤Ã̈Ïü §yüƒyÓfliyÎ˚ xyöy •Î˚– ~ÓyÓ˚ Îsf!ê˛ˆÏܲ ~ܲ!ê˛ xö%öyò ÏܲÓ˚ §yüˆÏö ~ˆÏö ˆÎ
¢∑ï˛Ó˚ˆÏDÓ˚ ï˛#Ó ï˛y üy˛õy •ˆÏÓñ ï˛yÓ˚ ܲ¡õyˆÏB˛Ó˚ §ˆÏD xö%öyò â!ê˛ˆÏÎ˚ ˆçyÓ˚yˆÏ°y ÓyÎ˚%ܲ¡õö ˜ï˛!Ó˚ ܲÓ˚y •Î˚–
ˆ§•z ÓyÎ˚%≤ÃÓyˆÏ• ï˛y˛õ ˛õ!Ó˚ã˛°ˆÏöÓ˚ ú˛ˆÏ° ï˛yˆÏÓ˚Ó˚ í˛z£èï˛y ܲˆÏü !àˆÏÎ˚ ˆÓ˚yô ܲ!üˆÏÎ˚ ˆòÎ˚ ~ÓÇ ‡•zê˛ˆÏfiê˛yö !Ó ˆÏçÓ˚
àƒy°û˛yˆÏöy!üê˛yˆÏÓ˚ ï˛!í˛¸Í ≤ÃÓy• ˆòáy ˆòÎ˚– ~•z ï˛!í˛¸Í ≤ÃÓy• ˆÌˆÏܲ ï˛Æ ï˛yˆÏÓ˚Ó˚ ˆÓ˚yô •…y§ ~ÓÇ ï˛yÓ˚ ˆÌˆÏܲ
xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
10.4.3 xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛ (Optical method)
~Ó˚ ü)° ö#!ï˛ •ˆÏFåÈ ò%!ê˛ xyˆÏ°yܲï˛Ó˚ˆÏDÓ˚ Óƒ!ï˛ã˛yÓ˚– ~ܲ!ê˛ xyˆÏ°yܲ Ó˚!Ÿ¬=FåÈ §yôyÓ˚î ÓyÎ˚%Ó˚ üôƒ !òˆÏÎ˚
ˆ≤ÃÓ˚î ܲÓ˚y •Î˚ ~ÓÇ xöƒ!ê˛ ˆ≤ÃÓ˚î ܲÓ˚y •Î˚ ¢∑ï˛Ó˚D xyˆÏÓ˚y!˛õï˛ ÓyÎ˚%Ó˚ üôƒ !òˆÏÎ˚– ¢∑ï˛Ó˚ˆÏDÓ˚ ≤Ãû˛yˆÏÓ ï˛yÓ˚
ï˛#Ó ï˛y xö%ÎyÎ˚# ÓyÎ˚%ˆÏï˛ ã˛yˆÏ˛õÓ˚ ˛õ!Ó˚Óï≈˛ö ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ÓyÎ˚%Ó˚ âöc ≤Ã!ï˛§Ó˚yˆÏB˛Ó˚ e´üyß∫ˆÏÎ˚ ˛õ!Ó˚Óï≈˛ö •ˆÏï˛
ÌyˆÏܲ– §%ï˛Ó˚yÇñ í˛zû˛Î˚ Ó˚!Ÿ¬=ˆÏFåÈÓ˚ üˆÏôƒ ˛õÌ ˛õyÌ≈ܲƒ (path difference) o&ï˛ Óò°yÎ˚ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° í˛zû˛Î˚
!ÓˆÏüÓ˚ (beam) Óƒ!ï˛ã˛yˆÏÓ˚ ˆÎ ˛õ!ê˛ (fringe) ˆòáy ÎyÎ˚ñ ï˛y §yôyÓ˚î xÓfliyˆÏöÓ˚ ˆÌˆÏܲ e´üyß∫ˆÏÎ˚ xyˆÏ®y!°ï˛
•ˆÏï˛ ÌyˆÏܲ– ò,!‹T fliy!Î˚ˆÏcÓ˚ çöƒ ~•z o&ï˛ xyˆÏ®y°ˆÏö Óƒ!ï˛ã˛yÓ˚ ˛õ!ê˛ ã˛Äí˛¸y •ˆÏÎ˚ ˆàˆÏåÈ ÓˆÏ° üˆÏö •Î˚– ˛õ!ê˛Ó˚
~•z ≤Ãfli ˛õ!Ó˚Óï≈˛ö ˆÌˆÏܲ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚– ï˛ˆÏÓ á%Ó ˆçyÓ˚yˆÏ°y ¢∑ öy •ˆÏ° ~•z ˛õ!Ó˚Óï≈˛ö
~ï˛•z öàîƒ •Î˚ ˆÎñ ~•z ˛õk˛!ï˛!ê˛ !ӈϢ£Ï ܲyÎ≈ܲÓ˚# •Î˚ öy–
10.4.4 ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ (Sound Radiometer)
ˆÜ˛yöÄ üyôƒˆÏü ¢∑ï˛Ó˚ˆÏDÓ˚ ú˛ˆÏ° í˛zͲõߨ ã˛y˛õ ~ܲ!ê˛ ˛õÎ≈yÓ,_ Ó˚y!¢– !ܲv ~ åÈyí˛¸yÄ ¢∑ ï˛Ó˚D Îáö ˆÜ˛yöÄ
ï˛ˆÏ° xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ˆ§!ê˛ ~ܲ!ê˛ «%˛o !fliÓ˚ ã˛y˛õÄ ≤ÈÏÎ˚yà ܲˆÏÓ˚– ~ˆÏܲ ôù!ö !Ó!ܲÓ˚î ã˛y˛õ (acoustic
radiation pressure) Ó°y •Î˚– ï˛!í˛¸Í ã%˛¡∫ܲ#Î˚ !Ó!ܲÓ˚îÄ (electromagnetic radiation) ~ôÓ˚ˆÏöÓ˚ ã˛y˛õ
≤ÈÏÎ˚yà ܲˆÏÓ˚– °yü≈Ó˚ (Larmor) ï˛yÓ˚ ~ܲ !§k˛yhsˇ ≤Ã!ï˛˛õyòö ܲˆÏÓ˚!åȈϰö– ˆÎˆÏ•ï%˛ ï˛yˆÏï˛ ï˛Ó˚D Óy üyôƒˆÏüÓ˚
≤ÃÜ,˛!ï˛ §¡∫ˆÏ¶˛ !ܲå%È !ö!ò≈‹T ˆö•zñ ˆ§•z ö#!ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏeÄ ≤ÈÏÎy烖 xyˆÏà °yü≈Ó˚ !§k˛yhsˇ xö%§yˆÏÓ˚
ôù!ö !Ó!ܲÓ˚î ã˛y˛õ ܲ#û˛yˆÏÓ í˛zͲõߨ •Î˚ ˆ§!ê˛ ˆòáy Îyܲ–
üˆÏö ܲÓ˚y Îyܲñ c à!ï˛ˆÏÓàÎ%_´ §üï˛° (plane) ¢∑ï˛Ó˚D ~ܲ!ê˛ xyò¢≈ ≤Ã!ï˛ú˛°Ü˛ ï˛ˆÏ° °¡∫û˛yˆÏÓ xy˛õ!ï˛ï˛
•ˆÏFåÈ– Î!ò ï˛°!ê˛ ¢ˆÏ∑Ó˚ !Ó˛õÓ˚#ï˛ !òˆÏܲ
ν
ˆÓˆÏà à!ï˛¢#° •Î˚ñ ¢∑ï˛Ó˚ˆÏDÓ˚ §yˆÏ˛õˆÏ«˛ ï˛yÓ˚ xyˆÏ˛õ!«˛Ü˛ ˆÓà
273
c +
ν
•ˆÏÓ– §%ï˛Ó˚yÇñ ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ ≤Ã!ï˛ú˛°Ü˛ ï˛° xy˛õ!ï˛ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ (c +
ν
) ˜òˆÏâ≈ƒÓ˚ ~ܲ!ê˛ hflψÏΩ˛Ó˚
üôƒ !òˆÏÎ˚ ÎyˆÏFåÈ– !ܲv ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ ˆ«˛ˆÏe xyˆÏ˛õ!«˛Ü˛ ˆÓà c –
ν
– §%ï˛Ó˚yÇñ ˆ§!ê˛Ó˚ çöƒ ≤Ã!ï˛ú˛°Ü˛
≤Ã!ï˛ ˆ§ˆÏܲu˛ (c –
ν
) ˜òˆÏâ≈ƒÓ˚ hflÏΩ˛ x!ï˛e´ü ܲÓ˚ˆÏÓ– Î!ò xy˛õ!ï˛ï˛ ï˛Ó˚ˆÏDÓ˚ ¢!_´ âöc E •Î˚ñ ≤Ã!ï˛ú˛°Ü˛
ï˛ˆÏ°Ó˚ ≤Ã!ï˛ ~ܲܲ ˆ«˛eú˛ˆÏ° ≤Ã!ï˛ ~ܲܲ §üˆÏÎ˚ xy˛õ!ï˛ï˛ ¢!_´ •ˆÏÓ (c +
ν
) E, Îy ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õÓ˚ (c
– u) ˜òˆÏâ≈ƒÓ˚ üˆÏôƒ §B%˛!ã˛ï˛ •ˆÏÎ˚ ÎyˆÏFåÈ– §%ï˛Ó˚yÇñ ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚ˆÏDÓ˚ ¢!_´ âöc ˆÓ!¢ SôÓ˚y Îyܲñ E +
δ
E) •ˆÏÓñ ˆÜ˛ööy ˆüyê˛ ˛õ!Ó˚üyî !öÿ˛Î˚•z !fliÓ˚ ÌyܲˆÏÓ– xï˛~Ó xyˆÏ°yã˛ƒ xÓfliy xö%ÎyÎ˚#ñ
(c +
ν
)E = (c –
ν
) (E +
δ
E)
Óyñ
1
1EEcc
Ecc
ν δνν ν
+ ++ ==−−
Óyñ 1 + 1
1 1EE c c
δ ν ν −⎛ ⎞⎛ ⎞= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
11,ccνν ⎛⎞⎛⎞ ≅++ ⎜⎟⎜⎟ ⎝⎠⎝⎠
Î!ò ν
<< c •Î˚–
= 1 + 2
vc
∴
c
δ
E = 2
ν
E ...10.5
~•z
cEδ
•° ˆ§ˆÏܲˆÏu˛ ≤Ã!ï˛ú˛°ˆÏܲÓ˚ ~ܲܲ ˆ«˛eú˛ˆÏ°Ó˚ çöƒ ¢∑ï˛Ó˚ˆÏDÓ˚ Óyí˛¸!ï˛ ¢!_´ ˆÎê˛y ≤Ã!ï˛ú˛°Ü˛
myÓ˚y Ü,˛ï˛Ü˛yÎ≈ ˆÌˆÏܲ•z ~ˆÏ§ˆÏåÈ– ôÓ˚y Îyܲñ xy˛õ!ï˛ï˛ ï˛Ó˚D ≤Ã!ï˛ú˛°ˆÏܲÓ˚ í˛z˛õÓ˚ P !Ó!ܲÓ˚î ã˛y˛õ ≤ÈÏÎ˚yà ܲÓ˚ˆÏåÈ
~ÓÇ ã˛yˆÏ˛õÓ˚ !ÓÓ˚&ˆÏk˛ ï˛°ˆÏܲ à!ï˛¢#° ܲÓ˚yˆÏï˛ •ˆÏFåÈ– §%ï˛Ó˚yÇñ ~ܲܲ §üˆÏÎ˚ ≤Ã!ï˛ú˛°ˆÏܲÓ˚ ~ܲܲ ˆ«˛eú˛ˆÏ°
Ü,˛ï˛Ü˛yÎ≈ P.
ν
– SˆÜ˛öñ ~ܲê%˛ !ã˛hsˇy ܲÓ˚&ö–V
§%ï˛Ó˚yÇñ P.
ν
=
cEδ
= 2
vE
§ü#ܲÓ˚î (10.5) xö%ÎyÎ˚#
∴
P = 2E ...10.6
274
ˆòáy ÎyˆÏFåÈ ˆÎñ ¢∑ ï˛Ó˚D ≤ÃÎ%_´ !Ó!ܲÓ˚î ã˛y˛õ (P) ≤Ã!ï˛ú˛°Ü˛ ï˛ˆÏ°Ó˚ §ß√%ˆÏá ¢!_´ âöˆÏcÓ˚ !m=î– §ü#ܲÓ˚î
10.6 ÈÙÈ ~ ≤Ã!ï˛ú˛°ˆÏܲÓ˚ à!ï˛Ó˚ (V) í˛zˆÏÕ‘á ˆö•z– xï˛~Óñ !fliÓ˚ ï˛ˆÏ°Ó˚ ˆ«˛ˆÏeÄ ~ܲ•z âê˛öy âê˛ˆÏÓ– xÓ¢ƒ
Î!ò ï˛°!ê˛ xyˆÏòÔ ≤Ã!ï˛ú˛°ö öy ܲˆÏÓ˚ñ P = E •ˆÏÓ–
ˆÎˆÏ•ï%˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y àí˛¸ ¢!_´ âöc Ä ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏàÓ˚ =îú˛ˆÏ°Ó˚ §üyöñ í˛z˛õˆÏÓ˚Ó˚ ï˛_¥ xö%ÎyÎ˚# àí˛¸
¢!_´ âöc ÓyÓ˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏ°•z ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y !öô≈yÓ˚î ܲÓ˚y ÎyˆÏÓ– ˆÎ ΈÏsfÓ˚ §y•yˆÏ΃ !Ó!ܲÓ˚î ã˛y˛õ !öî≈Î˚
ܲÓ˚y •Î˚ñ ï˛yˆÏܲ ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ˚ôù!ö ˆÓ˚!í˛Ä!üê˛yÓ Ó°y •Î˚– Îsf!ê˛Ó˚ àë˛ö ≤Ãîy°# 10.3 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö A
~ܲ!ê˛ ôyï˛Ó ˛õyï˛ ~ÓÇ B ï˛yÓ˚ §üyö û˛ˆÏÓ˚Ó˚ ~ܲ!ê˛ Äçö– í˛zû˛ˆÏÎ˚ ~ܲ!ê˛ ò,ì˛¸ xÌã˛ §Ó˚& Ó˚í˛ myÓ˚y Î%_´ ~ÓÇ
Ó˚ˆÏí˛Ó˚ üyé˛áyˆÏö ï˛sf!ê˛Ó˚ û˛Ó˚ˆÏܲˆÏw ~ܲ!ê˛ ï˛v (F) !òˆÏÎ˚ ˆé˛y°yˆÏöy ÌyˆÏܲ– ÓƒÓfliy!ê˛ ~ܲ!ê˛ §%@ˇÃy•# ÓƒyÓï≈˛
ï%˛°y (torsion balance) !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚– ï˛v!ê˛Ó˚ §ˆÏD Ó,_yܲyÓ˚ ˆflÒ°Î%_´ ~ܲ!ê˛ ÓƒyÓï≈˛ÈÙÈ¢#£Ï≈ (torsion
head) H ~ÓÇ ~ܲ!ê˛ «%˛o xyÎ˚öy M °yàyˆÏöy ÌyˆÏܲ– ÓƒyÓï≈˛ÈÙÈ¢#ˆÏ£Ï≈Ó˚ §y•yˆÏ΃ ï˛v!ê˛ˆÏï˛ ≤ÈÏÎ˚yçö üˆÏï˛y
ˆüyã˛í˛¸ ˆòÄÎ˚y ÎyÎ˚– ˆfl҈ϰÓ˚ §y•yˆÏ΃ xyÎ˚öyÎ˚ ˛≤Ã!ï˛ú˛!°ï˛ xyˆÏ°yܲÓ˚!Ÿ¬ â)î≈ö üy˛õyÓ˚ ÓƒÓfliy (lamp and
scale arrangement) ÌyˆÏܲñÎy !òˆÏÎ˚ Ó˚ˆÏí˛Ó˚ â)î≈ö ܲï˛ê˛y •°ñ ï˛y ˆÓyé˛y ÎyÎ˚– A ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ T öˆÏ°Ó˚
¢∑ï˛Ó˚D ˆÜ˛w#û)˛ï˛ ܲÓ˚y •Î˚– ôù!ö !Ó!ܲÓ˚î ã˛y˛õ ã˛yܲ!ï˛ˆÏܲ fl∫yû˛y!Óܲ xÓfliyö ˆÌˆÏܲ !Óã%˛ƒï˛ ܲˆÏÓ˚ ˆòÎ˚–
ï˛áö H ¢#£Ï≈!ê˛ â%!Ó˚ˆÏÎ˚ ã˛yܲ!ï˛ˆÏܲ xyÓyÓ˚ xyˆÏàܲyÓ˚ xÓfliyÎ˚ xyöy •Î˚– ~•z xÓfliyÎ˚ !Ó!ܲÓ˚îã˛y˛õ Î!ò P •Î˚ñ
ã˛yܲ!ï˛Ó˚ ˆ«˛eú˛° Î!ò a •Î˚ñ xyÓ˚
θ
Î!ò H ~Ó˚ ˆüyã˛í˛¸ (twist) •Î˚ñ ï˛ˆÏÓ
P
α
l =
τθ
...10.7
~áyˆÏö l •ï˛ AB Ó˚ˆÏí˛Ó˚ xô≈ ˜òâ≈ƒ xyÓ˚
τ
ï˛yˆÏÓ˚Ó˚ ÓƒyÓï≈˛ö ô &Óܲ (torsional constant)– §ü#ܲÓ˚î §ü#ܲÓ˚î §ü#ܲÓ˚î §ü#ܲÓ˚î §ü#ܲÓ˚î 10.7
ˆÌ Ïܲ P !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚–
275
10.4.5 !í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yÓ˚!í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yÓ˚!í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yÓ˚!í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yÓ˚!í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yÓ˚
x˜ÏÓò%ƒ!ï˛Ü˛ Ó˚y!¢ˆÏܲ ï˛!í˛¸Í ≤ÃÓy• Óy !Óû˛ˆÏÓÓ˚ !•§yˆÏÓ üy˛õˆÏï˛ •ˆÏ° xyˆÏà òÓ˚ܲyÓ˚ Ó˚y!¢!ê˛ˆÏܲ §üyö%˛õyï˛#
!Óû˛Ó Óy ï˛!í˛¸ˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚yÓ˚ ~ܲ!ê˛ í˛z˛õyÎ˚– ~•z ܲyˆÏç ˆÎ çyˆÏï˛Ó˚ Îsf ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ñ ï˛yˆÏòÓ˚
ê˛Δy™!í˛í˛z§yÓ˚ (Transducer) ӈϰ– ~=!°Ó˚ ܲyç ï˛!í˛¸Í ¢!_´Ó˚ xöƒ ¢!_´ˆÏï˛ Óy xöƒ ¢!_´ˆÏܲ ï˛!í˛¸Í ¢!_´ˆÏï˛
Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y– §ÓˆÏã˛ˆÏÎ˚ ˛õ!Ó˚!ã˛ï˛ ê˛Δy™!í˛í˛z§yÓ˚ •° üy•zˆÏe´yˆÏú˛ö Ä !flõܲyÓ˚ (speker)– üy•zˆÏe´yˆÏú˛yˆÏöÓ˚
§y•yˆÏ΃ Ó_´yÓ˚ ܲt˛fl∫Ó˚ˆÏܲ §üyö%˛õyï˛# ï˛!í˛¸Í §ˆÏB˛ˆÏï˛ ˛õ!Ó˚îï˛ Ü˛Ó˚y •Î˚– ï˛yˆÏܲ !ÓÓ!ô≈ï˛ Ü˛ˆÏÓ˚ xyÓyÓ˚
°yí˛zí˛!flõܲyˆÏÓ˚ ¢ˆÏ∑ Ó˚*˛õy!hsˇ!Ó˚ï˛ Ü˛Ó˚y •Î˚–
ˆÎ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y üy˛õy •ˆÏÓñ ï˛yˆÏܲ ≤Ã̈Ïü•z üy•zˆÏe´yˆÏú˛yö Óy xöƒ ˆÜ˛yöÄ ê˛Δy™!í˛í˛z§yˆÏÓ˚Ó˚ §y•yˆÏ΃ ï˛!í˛¸Í
§ˆÏB˛ˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y •Î˚– ~ÓyÓ˚ ~•z §ˆÏB˛ï˛ˆÏܲ öü%öy §Ç@ˇÃy•Ü˛ Ä ôyÓ˚ܲ (Sample-and-hold) öyüܲ
~ܲ!ê˛ Óï≈˛ö#ˆÏï˛ ˛õyë˛yˆÏöy •Î˚– !ã˛e 10.4(a) - ˆï˛ ~•z Óï≈˛ö#Ó˚ !e´Î˚y˛õk˛!ï˛ ˆÓyé˛yˆÏöy •ˆÏÎ˚ˆÏåÈ– ~ܲ!ê˛ §%•zã˛
(S) ~ÓÇ ~ܲ!ê˛ ôyÓ˚ܲ §•ˆÏÎyˆÏà ~•z Óï≈˛ö# à!ë˛ï˛ •Î˚– ~•z §%•zã˛!ê˛ xÓ¢ƒ §yôyÓ˚î xöÈÙÈxú˛ §%•zã˛ öÎ˚–
!ã˛e ≠ 10.4 : (a) !í˛!çê˛y° §yí˛zu˛ ˆ°ˆÏû˛° !üê˛yˆÏÓ˚Ó˚ àë˛ö (b) !Óû˛Ó ≤ÃÜ,˛!ï˛
ê˛Δyö!çfiê˛yÓ˚ Óy ü§ˆÏú˛ê˛ (MOSFET)-ˆÜ˛ ܲyˆÏç °y!àˆÏÎ˚ !ö!ò≈‹T §üÎ˚ xhsˇÓ˚ Óï≈˛ö#!ê˛ xöÈÙÈxú˛ ܲÓ˚yˆÏöy •Î˚–
xö •ˆÏ° •zö˛õ%ê˛ñ Óy ¢∑ï˛Ó˚D §üyö%˛õyï˛# !Óû˛Ó C ôyÓ˚ˆÏܲ ~ˆÏ§ ˛õˆÏí˛¸ ~ÓÇ ôyÓ˚ܲ!ê˛ o&ï˛ í˛z_´ !Óû˛Ó ˛õÎ≈hsˇ
xy!•ï˛ •Î˚– ~ÓyˆÏÓ˚ §%•zã˛!ê˛ xú˛ •ˆÏ° •zö˛õ%ˆÏê˛Ó˚ §ˆÏD §ÇˆÏÎyà !Ó!FåÈߨ •ˆÏÎ˚ ÎyÎ˚ ~ÓÇ ôyÓ˚ܲ!ê˛ ˆÓ˚yô R-~Ó˚
276
üôƒ !òˆÏÎ˚ xöy!•ï˛ •ˆÏï˛ ÌyˆÏܲ– R-~Ó˚ üyö Óy!í˛¸ˆÏÎ˚ «˛Ó˚î ܲy° (discharging time) Óyí˛¸yˆÏöy ÎyÎ˚– ï˛y
•ˆÏ° •zö˛õ%ê˛ §ÇˆÏÎyà !Ó!FåÈߨ •ÄÎ˚yÓ˚˛ ˛õˆÏÓ˚Ä C – R §üß∫Î˚ !ܲå%È«˛î !Óû˛Ó!ê˛ ôˆÏÓ˚ Ó˚yˆÏá– §%ï˛Ó˚yÇñ Óï≈˛ö#!ê˛
ˆÎö •zö˛õ%ê˛ !Óû˛ˆÏÓÓ˚ öü%öy (sample) @ˇÃ•î ܲˆÏÓ˚ ï˛y ôyÓ˚î (hold) ܲˆÏÓ˚ Ó˚yˆÏá– ~•z öü%öy !Óû˛Ó !ÓÓô≈ˆÏܲ
≤ÃÎ%_´ •Î˚– ~Ó˚ =Ó˚&c ~•z ˆÎñ •zö˛õ%ê˛ ï˛Ó˚D ¢ˆÏ∑Ó˚ o&ï˛ Äë˛yÈÙÈöyüyÓ˚ §ˆÏD Óyí˛¸ˆÏ°ÈÙÈܲüˆÏ°Ä !Óû˛ˆÏÓÓ˚ ~ܲ!ê˛
!fliÓ˚ üyö !ÓÓô≈ˆÏܲ ˆ˛õÔÑåÈÎ˚ ~ÓÇ ¢∑ ï˛#Ó ï˛yÓ˚ àí˛¸ xÌÓy x!ôܲï˛Ó˚ üyö çyöy ÎyÎ˚–
!ã˛e 10.4(b) ÈÙÈ ˆï˛ ~ܲ!ê˛ @ˇÃyˆÏú˛Ó˚ §y•yˆÏ΃ âê˛öy!ê˛ ˆÓyé˛yˆÏöy •ˆÏÎ˚ˆÏåÈ– ¢∑ï˛Ó˚ˆÏDÓ˚ §üyö%˛õyï˛# •zö˛õ%ê˛
!Óû˛Ó Óyí˛¸ˆÏåÈÈÙÈܲüˆÏåÈ– !ö!ò≈‹T §üÎ˚ ˛õˆÏÓ˚ ˛õˆÏÓ˚ A, B ~ÓÇ C ü%•)ˆÏï≈˛Ó˚ !Óû˛Ó @ˇÃ•î ܲÓ˚y •°– A ü%•)ˆÏï≈˛Ó˚ ˛õÓ˚
S ÈÙÈ ~Ó˚ §ÇˆÏÎyà !Ó!FåÈߨ •ˆÏÎ˚ ˆà° ~ÓÇ C – R §üß∫Î˚ Ä•z ü%•)ˆÏï≈˛Ó˚ !Óû˛Ó (a) ôˆÏÓ˚ Ó˚yá°– B-ˆï˛ !Óû˛Ó
xyÓ˚Ä Óyí˛¸°– ï˛áö ôyÓ˚ܲ C ˆ§•z Ó!ô≈ï˛ !Óû˛Ó (b) ˛õÎ≈hsˇ xy!•ï˛ •°– C ü%•)ˆÏï≈˛ !Óû˛Ó xyÓyÓ˚ ܲˆÏü ˆà°–
!ܲv ôyÓ˚ܲ ˆ§•z b !Óû˛Ó•z ôˆÏÓ˚ ˆÓ˚ˆÏá !ò°– xï˛~Óñ ¢∑ ï˛#Ó ï˛yÓ˚ í˛zFã˛ï˛ü üyö!ê˛ ˛õyÄÎ˚y ˆà°– Î!ò •zö˛õ%ê˛
ï˛Ó˚D ÷ô% A xÓfliyÓ˚ ܲyåyܲy!åÈ Äë˛y öyüy ܲÓ˚ï˛ñ ï˛ˆÏÓ ï˛yÓ˚•z ~ܲ!ê˛ üyö !fliÓ˚û˛yˆÏÓ ˛õyÄÎ˚y ˆÎï˛ ~ÓÇ ï˛y
¢∑ ï˛#Ó ï˛yÓ˚ àí˛¸ üyö !öˆÏò≈¢ ܲÓ˚ï˛– !ܲv Óï≈˛ö#!ê˛ ~ܲ!ê˛ !Óû˛Ó x!ö!ò≈‹T ܲy° ôˆÏÓ˚ Ó˚yáˆÏÓñ ï˛y ÓyN˛ö#Î˚
öÎ˚ ï˛y •ˆÏ° ˛õê˛Ü˛y ˆú˛ˆÏê˛ ã˛yÓ˚!òˆÏܲ hflÏ∏˛ •ˆÏÎ˚ ÎyÄÎ˚yÓ˚ ˛õÓ˚Ä Îsf!ê˛ ˆ§•z ˛õê˛Ü˛yÓ˚ ¢ˆÏ∑Ó˚ §üyö%˛õyï˛# í˛zFã˛ !Óû˛Ó•z
≤Ãò¢≈ö ܲÓ˚ˆÏÓ– ~•z ܲyÓ˚ˆÏî !ܲå%È«˛î xhsˇÓ˚ §%•zã˛!ê˛ xö ܲˆÏÓ˚ !Óû˛ˆÏÓÓ˚ öü%öy §Ç@ˇÃ• ܲÓ˚ˆÏï˛ •Î˚– ï˛y ܲÓ˚ˆÏ°
xyí˛zê˛˛õ%ê˛ !Óû˛Ó !ܲå%È«˛î xhsˇÓ˚ •zö˛õ%ê˛ !Óû˛ˆÏÓÓ˚ §üyö%˛õyï˛# ~ܲ ~ܲê˛y !fliÓ˚ üyö !öˆÏò≈¢ ܲˆÏÓ˚ xÌã˛ •zö˛õ%ˆÏê˛Ó˚
xöÓÓ˚ï˛ Äë˛y öyüyÓ˚ !Ó¢,C° xÓfliyê˛yÄ ~í˛¸yˆÏöy ÎyÎ˚–
!Óû˛ˆÏÓÓ˚ öü%öy §Ç@ˇÃˆÏ•Ó˚ •yÓ˚ •zö˛õ%ê˛ !Óû˛ˆÏÓÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ܲ¡õyB˛ Îï˛ ˆÓ!¢ •ˆÏÓñ ï˛ï˛ o&ï˛ öü%öy
§Ç@ˇÃ• ܲÓ˚ˆÏï˛ •ˆÏÓ– !Óû˛ˆÏÓÓ˚ ˛õ!Ó˚Óï≈˛ö Î!ò §y•zö (sinusoidal) ï˛Ó˚D •Î˚ñ ï˛ˆÏÓ Ü˛¡õyˆÏB˛Ó˚ !m=î •yˆÏÓ˚
Sܲ¡õyB˛ 5KHz •ˆÏ° 10KHz •yˆÏÓ˚ñ Óy ≤Ã!ï˛ 10–4 ˆ§ˆÏܲu˛ ˛õˆÏÓ˚ÈÙÈ˛õˆÏÓ˚V öü%öy §Ç@ˇÃ• •ÄÎ˚y í˛z!ã˛ï˛– xyÓ˚
Î!ò xöƒ x!öÎ˚!üï˛ xyÜ,˛!ï˛Ó˚ ï˛Ó˚D •Î˚ñ ï˛yÓ˚ í˛z˛õyòyö=!°Ó˚ ˆÎ §y•zö ï˛Ó˚D!ê˛Ó˚ ܲ¡õyB˛ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ñ
ï˛yÓ˚ !m=î •yˆÏÓ˚ öü%öy §Ç@ˇÃ• •ˆÏï˛ •ˆÏÓ–
~áö ~ܲ!ê˛ xö%¢#°ö#Ó˚ §y•yˆÏ΃ Îy ˛õí˛¸ˆÏ°ö ˆÎ!ê˛ ~ܲÓyÓ˚ é˛y!°ˆÏÎ˚ !öö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2 :
(a) Ó˚ƒyˆÏ°Ó˚ ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ Ó°Î%ˆÏ@¬Ó˚ üyö Óyí˛¸yˆÏï˛ ˆÜ˛yö ˆÜ˛yö ÓƒÓfliy ˆöÄÎ˚y •Î˚⁄
(b) ï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ í˛z˛õÓ˚ ¢∑ï˛Ó˚D xy˛õ!ï˛ï˛ •ˆÏ° ï˛yÓ˚ ˆÓ˚yô ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ ˆÜ˛ö⁄
(c) ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y üy˛õyÓ˚ xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛ˆÏï˛ Óƒ!ï˛ã˛yÓ˚ ˛õ!ê˛Ó˚ xyˆÏ®y°ö ˆã˛yˆÏá ˆòáy ÎyÎ˚ öy– ~Ó˚
ܲyÓ˚î ܲ#⁄
277
10.5 ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ
~•z xyˆÏàÓ˚ xö%ˆÏFåȈÏò xy˛õ!ö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ˛õ!Ó˚üyˆÏ˛õÓ˚ ܲˆÏÎ˚ܲ!ê˛ í˛z˛õyÎ˚ ˛õˆÏí˛¸ˆÏåÈö– ï˛#Ó ï˛yÓ˚ üï˛ ¢ˆÏ∑Ó˚
ܲ¡õyB˛Ä ¢∑ï˛Ó˚ˆÏDÓ˚ ~ܲ!ê˛ =Ó˚&c˛õ)î≈ ôü≈– xÓ¢ƒ ˆÎ ¢∑ˆÏܲ xyüÓ˚y x˛õfl∫Ó˚ Óy ˆày°üy° (noise) ӈϰ
x!û˛!•ï˛ ܲ!Ó˚ ï˛yˆÏï˛ üyôƒˆÏüÓ˚ ӛܲîyÓ˚ ˆÜ˛yöÄ ˆòy°à!ï˛ ÌyˆÏܲ öyñ ~ çyï˛#Î˚ ¢ˆÏ∑Ó˚ ˆÜ˛yöÄ !ö!ò≈‹T
ܲ¡õyB˛Ä ÌyˆÏܲ öy– §%fl∫Ó˚ Óy §%ˆÏÓ˚°y ¢ˆÏ∑ (musical sound) ~ܲ Óy ~ܲy!ôܲ ܲ¡õyˆÏB˛Ó˚ !ü◊î ÌyˆÏܲ–
xyüÓ˚y ~•z çyï˛#Î˚ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ !ܲû˛yˆÏÓ !öî≈Î˚ ܲÓ˚y ÎyÎ˚ñ ˆ§!ê˛•z xyˆÏ°yã˛öy ܲÓ˚Ó–
xy˛õ!ö !öÿ˛Î˚•z xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ ˆÜ˛yöÄ ¢∑ ܲï˛ê˛y áyˆÏò ÌyˆÏܲ Óy ã˛í˛¸y ˆ¢yöyÎ˚ ï˛y ¢ˆÏ∑Ó˚
ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ §Ó˚y§!Ó˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ü!•°y Óy !¢÷Ó˚ ܲt˛fl∫ˆÏÓ˚Ó˚ ܲ¡õyB˛ ˛õ%Ó˚&ˆÏ£ÏÓ˚ ï%˛°öyÎ˚ ˆÓ!¢ •Î˚– ï˛y•z
ü!•°y Óy !¢÷Ó˚ ܲt˛fl∫Ó˚ ˛õ%Ó˚&ˆÏ£ÏÓ˚ ܲt˛fl∫ˆÏÓ˚Ó˚ ï%˛°öyÎ˚ ã˛í˛¸y ˆ¢yöyÎ˚– ˆ◊yï˛yÓ˚ ܲyˆÏö !ӈϢ£Ï ~ܲ!ê˛ ¢∑ ܲï˛ê˛y
ã˛í˛¸y ˆ¢yöyÎ˚ñ ï˛yˆÏܲ xyüÓ˚y ¢ˆÏ∑Ó˚ ï˛#-ï˛y (pitch) Ó!°– ¢ˆÏ∑Ó˚ ܲ¡õyB˛ xyÓ˚ ï˛#-ï˛yÓ˚ â!ö‹T §¡õÜ≈˛
ÌyܲˆÏ°Ä ò%!ê˛ !ܲv ~ܲ öÎ˚– ܲ¡õyB˛ ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ ôü≈ñ !ܲv ï˛#«¯˛ï˛y ˛õ%ˆÏÓ˚y˛õ%!Ó˚ ˆ◊yï˛yÓ˚ xö%û)˛!ï˛Ó˚ í˛z˛õÓ˚
!öû≈˛Ó˚¢#°– ï˛ˆÏÓ ò%•z!ê˛ !Ó÷k˛ ܲ¡õyˆÏB˛Ó˚ §%Ó˚ Î!ò ~ܲ•z ܲ¡õyˆÏB˛Ó˚ •Î˚ ï˛ˆÏÓ ï˛yˆÏòÓ˚ ï˛#«¯˛ï˛yÄ §üyö •Î˚–
ܲyˆÏç•z ˆ◊yï˛y ܲyˆÏö ÷ˆÏö ~ܲ!ê˛ çyöy ܲ¡õyˆÏB˛Ó˚ Ä ~ܲ!ê˛ xçyöy ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛yÓ˚ §üï˛y
§%!ö!ÿ˛ï˛ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏ° !ï˛!ö ò%•z ܲ¡õyB˛ §üyö ӈϰ ôÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ~•zû˛yˆÏÓ xçyöy ܲ¡õyB˛!ê˛
!öî#≈ï˛ •ˆÏÓ– ï˛ˆÏÓ ~•z ˛õk˛!ï˛!ê˛ ˆÜ˛Ó°üye ò%•z ܲ¡õyˆÏB˛Ó˚ §üï˛y !öî≈ˆÏÎ˚Ó˚ çöƒ•z ≤ÈÏÎy烖 xyüÓ˚y ~áyˆÏö
ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üyˆÏ˛õÓ˚ çöƒ ~çyï˛#Î˚ ï%˛ö°y!û˛!_ܲ ˛õk˛!ï˛ åÈyí˛¸yÄ ˆ◊yï˛yÓ˚ xö%û)˛!ï˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° öÎ˚–
~üö ܲˆÏÎ˚ܲ!ê˛ ˛õk˛!ï˛Ó˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
10.5.1 fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´ (Stroboscopic wheel) ˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛
fiê˛ΔˆÏÓyˆÏflÒy˛õ ~üö ~ܲ Îy!sfܲ ÓƒÓfliyñ ÎyˆÏï˛ ˛õÎ≈yÓ,_ à!ï˛§¡õߨ Óy â)î≈yÎ˚üyö Ó›ˆÏܲ !fliÓ˚ xÓfliyÎ˚ ˆòáy
§Ω˛Ó– ~Ó˚ ü)°ö#!ï˛ •° Ó›Ó˚ â)î≈öñ ܲ¡õö Óy xöƒ ôÓ˚ˆÏöÓ˚ ˛õÎ≈yÓ,_ à!ï˛Ó˚ ˛õÎ≈yÎ˚ܲyˆÏ°Ó˚ §üyö §üÎ˚ ˛õˆÏÓ˚
˛õˆÏÓ˚ Ó›!ê˛ˆÏܲ ˆòáyÓ˚ ÓƒÓfliy ܲÓ˚yñ ÎyˆÏï˛ Ó›!ê˛ ~ܲ•z ò¢yÎ˚ !fliÓ˚ xyˆÏåÈ ÓˆÏ° üˆÏö •Î˚– fiê˛ΔˆÏÓyˆÏflÒyˆÏ˛õÓ˚ §y•yˆÏ΃
xy˛õ!ö §Ó˚y§!Ó˚ ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö– ôÓ˚&öñ ¢ˆÏ∑Ó˚ í˛zͧ!ê˛ ~ܲ!ê˛ Ü˛¡õüyö
§%Ó˚¢°yܲy (tuning fork), ÎyÓ˚ ܲ¡õyB˛ !öô≈yÓ˚î ܲÓ˚ˆÏï˛ •ˆÏÓ– xyÓ˚ fiê˛ΔˆÏÓyˆÏflÒy˛õ ~ܲ!ê˛ ã˛yܲyñ ÎyÓ˚ àyˆÏÎ˚ §üyö
ÓƒÓôyˆÏö !ö!ò≈‹T §ÇáƒÜ˛ xÓ˚#Î˚ (radial) ˆÓ˚áy!åÈo xyˆÏåÈ ~ÓÇ ÎyˆÏܲ !ö!ò≈‹T à!ï˛ˆÏï˛ ˆâyÓ˚yˆÏöy ÎyÎ˚– §%Ó˚¢°yܲy
~ÓÇ fiê˛ΔˆÏÓyˆÏflÒy˛õ ã˛e´ˆÏܲ öyöyû˛yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚– ï˛yÓ˚ üˆÏôƒ ~ܲ!ê˛ §%!Óôyçöܲ ÓƒÓfliy 10.5 !ã˛ˆÏe
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
§%Ó˚¢°yܲyÓ˚ àyˆÏÎ˚ ~ܲ!ê˛ !Ó®% Óy òyà (P) !òˆÏÎ˚ ¢°yܲy!ê˛ˆÏܲ ~üöû˛yˆÏÓ Ó˚yáy •Î˚ ÎyˆÏï˛ ï˛yÓ˚ ܲ¡õˆÏöÓ˚
ï˛° xyÓ˚ ã˛e´!ê˛Ó˚ ï˛° §üyhsˇÓ˚y° •Î˚– ¢°yܲyÓ˚ àyˆÏÎ˚Ó˚ òyà!ê˛ˆÏܲ ˆçyÓ˚yˆÏ°y xyˆÏ°yÎ˚ xyˆÏ°y!Ü˛ï˛ Ü˛Ó˚y •Î˚–
278
§%Ó˚¢°yܲy!ê˛ !fliÓ˚ ÌyܲˆÏ° ~ÓÇ ã˛yܲyÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ ˆÓ˚á!åÈo òyà!ê˛Ó˚ ˆ§yçy§%!ç §yüˆÏö ÌyܲˆÏ° ã˛yܲyÓ˚
xöƒ ˛õyˆÏ¢ Ó§yˆÏöy ˆê˛!°ˆÏflÒyˆÏ˛õ òyà!ê˛
flõ‹T ˆòáy ÎyÎ˚– ¢°yܲy ܲ!¡õï˛ •ˆÏ°
òyà!ê˛ ˆÓ˚áy!åȈÏoÓ˚ ú˛yÑܲ !òˆÏÎ˚ xyˆÏ®y!°ï˛
xÓfliyÎ˚ òáy ÎyÎ˚– ~Óy ÏÓ˚ Î!ò §%Ó˚¢°yܲy Ïܲ
ܲ!¡õï˛ Ü˛Ó˚y •Î˚ ~ÓÇ §•z §ˆÏD ã˛yܲy!ê˛ˆÏܲÄ
˜Óò%ƒ!ï˛Ü˛ ˆüyê˛ˆÏÓ˚Ó˚ §y•yˆÏ΃ ˆâyÓ˚yˆÏöy •Î˚ñ
ï˛y •ˆÏ° â)î≈öà!ï˛ Óyí˛¸ˆÏï˛ Óyí˛¸ˆÏï˛ ~üö
~ܲê˛y xÓfliy xy§ˆÏÓ Îáö §%Ó˚¢°yܲyÓ˚
˛õÎ≈yÎ˚ܲy° xyÓ˚ ã˛yܲyÓ˚ ~ܲê˛y ˆÓ˚áy!åȈÏoÓ˚
çyÎ˚àyÎ˚ ˛õˆÏÓ˚Ó˚ê˛y xy§yÓ˚ §üÎ˚ܲy° §üyö
•ˆÏÎ˚ ÎyˆÏÓ– ˆ§•z xÓfliyÎ˚ ˆê˛!°ˆÏflÒyˆÏ˛õ
òyà!ê˛ !fliÓ˚ xÓfliyÎ˚ ˆòáy ÎyˆÏÓ–
Î!ò ã˛yܲyÎ˚ ˛ú˛yшÏܲÓ˚ §Çáƒy m ~ÓÇ ≤Ã!ï˛ ˆ§ˆÏܲu˛ â)î≈ˆÏöÓ˚ §Çáƒy n •Î˚ñ ï˛ˆÏÓ ~ܲ!ê˛ úÑ˛yܲ xyˆÏàÓ˚!ê˛Ó˚
çyÎ˚ày !öˆÏï˛ §üÎ˚ ˆöˆÏÓ
1mn
ˆ§ˆÏܲu˛– ~•z §üÎ˚ •ˆÏÓ ¢°yܲyÓ˚ ˛õÎ≈yÎ˚ ܲyˆÏ°Ó˚ §üyö– xï˛~Óñ ¢°yܲyÓ˚
˛õÎ≈yÎ˚ ܲy° 1mnˆ§ˆÏܲu˛ ~ÓÇ Ü˛¡õyB˛ mm Hz– !flõˆÏí˛y!üê˛yˆÏÓ˚Ó˚ §y•yˆÏ΃ n çyöˆÏï˛ ˛õyÓ˚ˆÏ° ~•z ˛õk˛!ï˛ˆÏï˛
§%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚– xÓ¢ƒ ã˛e´!ê˛ Î!ò ˆ§ˆÏܲˆÏu˛
2n
ÓyÓ˚ â%Ó˚ï˛ñ ï˛y•ˆÏ°Ä §%Ó˚¢°yܲyÓ˚ òyà!ê˛
!fliÓ˚ xÓfliyÎ˚ ˆòáy ˆÎï˛ñ ˆÜ˛ööy ï˛áö ~ܲ!ê˛ ˆÓ˚áy!åȈÏoÓ˚ çyÎ˚àyÎ˚ ˛õˆÏÓ˚Ó˚!ê˛ xy§ˆÏï˛ ˆÎ §üÎ˚ °yàï˛ñ ï˛yÓ˚
üˆÏôƒ ¢°yܲy!ê˛ ò%!ê˛ ˛õ)î≈ ܲ¡õö ˛õÎ≈yÎ˚ §üyÆ Ü˛Ó˚ï˛– ~çöƒ ã˛ˆÏe´Ó˚ §ˆÏÓ≈yFã˛ ˆÎ ˆÜ˛Ô!îܲ ˆÓˆÏà òyà!ê˛ !fliÓ˚
ˆòáyÎ˚ñ ˆ§!ê˛•z ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ çöƒ @ˇÃ•î ܲÓ˚ˆÏï˛ •Î˚– §%Ó˚¢°yܲy!ê˛ˆÏܲ §yôyÓ˚îï˛ ˜Óò%ƒ!ï˛Ü˛û˛yˆÏÓ Ü˛!¡õï˛
ܲÓ˚y •Î˚ñ ÎyˆÏï˛ Ü˛¡õö !fliÓ˚ Ä ò#â≈fliyÎ˚# •Î˚–
10.5.2 Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´Ó˚ƒyˆÏ°Ó˚ ¢∑ ã˛e´ (Rayleigh’s phonic wheel)
~!ê˛ !Óò%ƒÍÈÙÈã˛y!°ï˛ §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ !öî≈ˆÏÎ˚Ó˚ ~ܲ!ê˛ Îsfñ ˆÎ!ê˛ xy§ˆÏ° ~ܲ!ê˛ §ü°Î˚ ˆüyê˛Ó˚
(synchronous motor).– 10.6 !ã˛ˆÏe Îsf!ê˛Ó˚ àë˛ö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö ˆÎ §%Ó˚¢°yܲyÓ˚ (T) ܲ¡õyB˛
!öî≈Î˚ •ˆÏÓ ˆ§!ê˛ˆÏܲ ~ܲ!ê˛ ï˛!í˛¸ÍÓï≈˛ö#Ó˚ xhs≈ˇû%˛_´ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– B Óƒyê˛y!Ó˚!ê˛ M1, M2 Ä M ~•z !ï˛ö!ê˛ ï˛!í˛¸Í
ã%˛¡∫ܲñ §%Ó˚¢°yܲy ~ÓÇ §%- §ÇˆÏÎyà!Ó®% P ~Ó˚ üôƒ !òˆÏÎ˚ ï˛!í˛¸Í ≤ÃÓy• ã˛y!°ï˛ ܲˆÏÓ˚– ~•z ï˛!í˛¸Í ≤ÃÓy• M
ï˛!í˛¸Í ã%˛¡∫ܲˆÏܲ ܲyÎ≈ܲÓ˚# ܲˆÏÓ˚ ~ÓÇ ˆ§!ê˛ §%Ó˚¢°yܲyÓ˚ Óy‡ ò%!ê˛ˆÏܲ xyܲ£Ï≈î ܲˆÏÓ˚– xÓ¢ƒ ~çöƒ •Î˚ §%Ó˚¢°yܲy!ê˛
279
ˆ°y•yñ !fiê˛°ñ ≤Ãû,˛!ï˛ ˆÜ˛yö xÎ˚ÿ%˛¡∫ܲ ˛õòyˆÏÌ≈ ˜ï˛!Ó˚ •ˆÏï˛ •Î˚ñ öï%˛Óy ˆ§!ê˛Ó˚ Óy‡ ò%!ê˛Ó˚ §ˆÏD M ã%˛¡∫ˆÏܲÓ˚ ˆüÓ˚&
ò%!ê˛Ó˚ §yüˆÏö ò%!ê˛ ˆ°y•yÓ˚ ê%˛Ü˛ˆÏÓ˚y Î%_´ ܲÓ˚ˆÏï˛ •Î˚– §%Ó˚¢°yܲyÓ˚ Óy‡ xyÜ,˛‹T •ˆÏ°•z P §ÇˆÏÎyà !Ó!FåÈߨ •Î˚ñ
ú˛ˆÏ° ï˛!í˛¸Í ≤ÃÓy• ÓyÓ˚ ÓyÓ˚ ã˛y°% Ä Ó¶˛ •ˆÏï˛ ÌyˆÏܲ ~ÓÇ §%Ó˚¢°yܲy!ê˛Ó˚ ï˛yÓ˚ !öçfl∫ ܲ¡õyˆÏB˛ §%!fliï˛û˛yˆÏÓ
ܲ!¡õï˛ •ˆÏï˛ ÌyˆÏܲ–
xy˛õ!ö !öÿ˛Î˚•z °«˛ƒ ܲˆÏÓ˚ˆÏåÈöñ 10.6 !ã˛ˆÏe ~ܲ!ê˛ ã˛e´ (
ω
) ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~!ê˛ˆÏܲ•z xyüÓ˚y ¢∑ ã˛e´
Ó!°– ~!ê˛ ˆ°y•yñ Óy !fiê˛ˆÏ°Ó˚ ˜ï˛!Ó˚
~ÓÇ í˛zÕ‘¡∫ xˆÏ«˛Ó˚ í˛z˛õÓ˚ á%Ó ü§,îû˛yˆÏÓ
â%Ó˚ˆÏï˛ õyˆÏÓ˚– M1 Ä M2 Îáö ã%˛¡∫!ܲï˛
•Î˚ñ ï˛áö ˆ§=!° ã˛ˆÏe´Ó˚ !Ó˛õÓ˚#ï˛
!òˆÏܲÓ˚ ò%!ê˛ òyÑï˛ˆÏܲ xyܲ£Ï≈î ܲˆÏÓ˚–
ï˛!í˛¸Í ≤ÃÓy• !Ó!FåÈߨ •ˆÏ° ~•z xyܲ£Ï≈î
ˆÌˆÏü ÎyÎ˚ ~ÓÇ ã˛e´!ê˛ !öçfl∫ çyˆÏí˛ƒÓ˚
çöƒ â%Ó˚ˆÏï˛ ÌyˆÏܲ– M1 Ä M2 Îáö
xyÓyÓ˚ ã%˛¡∫!Ü˛ï˛ •Î˚ ï˛áö Î!ò !ë˛Ü˛
˛õˆÏÓ˚Ó˚ ò%!ê˛ !Ó˛õÓ˚#ï˛ òyÑï˛ xyˆÏàÓ˚ ò%!ê˛Ó˚
fliyˆÏö xyˆÏ§ñ ï˛y•ˆÏ° ˆ§ ò%!ê˛Ä ~ܲ•zû˛yˆÏÓ xyÜ,˛‹T •Î˚ ~ÓÇ ~•zû˛yˆÏÓ ã˛e´!ê˛ §üà!ï˛ˆÏï˛ §%!fliï˛û˛yˆÏÓ â%ˆÏÓ˚ ã˛ˆÏ°–
xy§ˆÏ° â£Ï≈ˆÏîÓ˚ ú˛ˆÏ° ã˛e´!ê˛Ó˚ â)î≈ö à!ï˛¢!_´Ó˚ ˆÎê%˛Ü%˛ «˛Î˚ •Î˚ñ ã%˛¡∫ܲ M1 Ä M2 ÈÙÈ ~Ó˚ xyܲ£Ï≈î ˆ§ê%˛Ü%˛ ˛õ)Ó˚î
ܲˆÏÓ˚– ~•z xÓfliyÎ˚ ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ §%Ó˚¢°yܲyÓ˚ Îï˛=!° ܲ¡õö §¡õ)î≈ •Î˚ñ !ë˛Ü˛ ï˛ï˛=!° òyÑï˛ M1 Óy M2
ã%˛¡∫ܲˆÏܲ x!ï˛e´ü ܲˆÏÓ˚ ÎyÎ˚– Î!ò ¢∑ ã˛ˆÏe´Ó˚ òyшÏï˛Ó˚ §Çáƒy n ~ÓÇ §%!fliï˛ â)î≈ˆÏöÓ˚ §üÎ˚ ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ â)î≈ˆÏöÓ˚
§Çáƒy N •Î˚ñ ï˛ˆÏÓ ~•z §Çáƒy!ê˛ •° n N ~ÓÇ ~!ê˛•z §%Ó˚¢°yܲyÓ˚ !öˆÏî≈Î˚ ܲ¡õyB˛–
10.5.3 ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ö (Cagniard de la Tour’s siren)
~ÓyÓ˚ xyüÓ˚y ~üö ~ܲ!ê˛ ÎˆÏsfÓ˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚Ó ˆÎ!ê˛Ó˚ §y•yˆÏ΃ ˆÎ ˆÜ˛yöÄ !öî≈Î˚ˆÏÎyàƒ Ü˛¡õyˆÏB˛Ó˚
¢∑ í˛zͲõߨ ܲÓ˚y ÎyÎ˚– ~•z Îsf!ê˛ ~ܲ ôÓ˚ˆÏöÓ˚ §y•zˆÏÓ˚ö– xy˛õ!ö •Î˚ï˛ x§yü!Ó˚ܲ ≤Ã!ï˛Ó˚«˛y !Óû˛yˆÏàÓ˚ §y•zˆÏÓ˚ˆÏöÓ˚
¢∑ ÷ˆÏöˆÏåÈö– ˆ§ˆÏ«˛ˆÏe xy˛õ!ö •Î˚ï˛ °«˛ƒ ܲˆÏÓ˚ˆÏåÈö ˆÎñ ˙ §y•zˆÏÓ˚ˆÏöÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ï˛#«¯˛ï˛y í˛zû˛ˆÏÎ˚•z
≤Ã̈Ïü ÓyˆÏí˛¸ !ܲå%È«˛î !fliÓ˚ ÌyˆÏܲ Óy Äë˛yÈÙÈöyüy ܲˆÏÓ˚ ~ÓÇ ï˛yÓ˚˛õÓ˚ ܲˆÏü ÎyÎ˚– ~•z ΈÏsfÓ˚ §y•yˆÏ΃ ܲ# û˛yˆÏÓ
~ܲ!ê˛ xK˛yï˛ Ü˛¡õyB˛ !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚ñ ~ÓyÓ˚ xyüÓ˚y ˆ§!ê˛ ˆòáÓ–
ܲƒy!öÎ˚yÓ˚ òƒ °yï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ˆÏöÓ˚ àë˛ö !ã˛e 10.7 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~ܲ!ê˛ ÓyÎ˚%≤ÈÏܲy¤˛ A-ˆï˛
˛õyˆÏ¡õÓ˚ §y•yˆÏ΃ ÓyÎ˚% ≤ÈÏÓ¢ ܲÓ˚yˆÏöy •Î˚– ≤ÈÏܲyˆÏ¤˛Ó˚ í˛z˛õˆÏÓ˚Ó˚ ï˛° B-ˆï˛ xˆÏöܲ=!° !åÈo §üyö ò)Ó˚ˆÏc
280
Ó,_yܲyˆÏÓ˚ §yçyˆÏöy ÌyˆÏܲ– S!ã˛e Îsf!ê˛Ó˚ xö%˜Ïòâ≈ƒˆÏFåÈò ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ˛õyˆÏ¢•z xyÑܲy xyˆÏåÈ !åÈo=!°Ó˚
xÓfliyöV– B-~Ó˚ í˛z˛õˆÏÓ˚ ~ܲ!ê˛ ã˛yܲ!ï˛ (C) ÌyˆÏܲñ ÎyˆÏܲ ~ܲ!ê˛ ˆÜ˛w#Î˚ °¡∫ xˆÏ«˛Ó˚ §yˆÏ˛õˆÏ«˛ §•ˆÏç•z
ˆâyÓ˚yˆÏöy ÎyÎ˚– C ã˛yܲ!ï˛ˆÏï˛ B-~Ó˚ !åÈo=!°Ó˚ ˆ§yçy§%!ç §ü§Çáƒyܲ !åÈo ÌyˆÏܲñ ÎyˆÏï˛ !åÈo=!° ü%ˆÏáyü%!á
˛õí˛¸ˆÏ° ÓyÎ˚%≤ÈÏܲy¤˛ ˆÌˆÏܲ Óyï˛y§ !öà≈ï˛ •ˆÏï˛ ˛õyˆÏÓ˚ñ !ܲv !åÈo=!° §ˆÏÓ˚ ˆàˆÏ° ÓyÎ˚%≤ÃÓy• Ó¶˛ •ˆÏÎ˚ ÎyÎ˚–
~•z §!ÓÓ˚yü ÓyÎ˚%≤ÃÓy• ¢ˆÏ∑Ó˚ §,!‹T ܲˆÏÓ˚ ÎyÓ˚ ü)° §%ˆÏÓ˚Ó˚ ܲ¡õyB˛ ÓyÎ˚%≤ÃÓy• ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ Îï˛ÓyÓ˚ Óyôy≤ÃyÆ
•Î˚ ï˛yÓ˚ §üyö– C ã˛yܲ!ï˛Ó˚ â)î≈ö x«˛ (rotation shaft) S-~Ó˚ §ˆÏD à!ï˛üy˛õܲ Îsf °yàyˆÏöy ÌyˆÏܲñ ÎyˆÏï˛
C ã˛yܲ!ï˛!ê˛ ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ ܲï˛ÓyÓ˚ â%Ó˚ˆÏåÈñ ï˛y çyöy ÎyÎ˚–
ˆÜ˛yöÄ ~ܲ!ê˛ fl∫öˆÏܲÓ˚ xK˛yï˛ Ü˛¡õyB˛ !öî≈ˆÏÎ˚Ó˚ çöƒ §y•zˆÏÓ˚ˆÏöÓ˚ C ã˛yܲ!ï˛!ê˛ ˜Óò%ƒ!ï˛Ü˛ ˆüyê˛ˆÏÓ˚Ó˚ §•yˆÏ΃
e´ü¢ o&ï˛ï˛Ó˚ ˆÓˆÏà ˆâyÓ˚yˆÏöy •Î˚ñ Îï˛«˛î öy xK˛yï˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ Ä §y•zˆÏÓ˚ˆÏöÓ˚ ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛y ܲyˆÏö
÷ˆÏö §üyö üˆÏö •Î˚– §yôyÓ˚îï˛ñ ò%!ê˛ Ü˛¡õyB˛ ܲyåÈyܲy!åÈ ~ˆÏ° fl∫Ó˚ܲ¡õ ˆ¢yöy ÎyÎ˚– ˆ§ xÓfliyÎ˚ §y•zˆÏÓ˚ˆÏöÓ˚
â)î≈ö ˆÓà §yüyöƒ ܲüˆÏÓ!¢ ܲˆÏÓ˚ fl∫Ó˚ܲ¡õ ò)Ó˚ ܲÓ˚ˆÏï˛ •Î˚– ôÓ˚y Îyܲñ ò%•z ï˛#«¯˛ï˛y Îáö §üyö ï˛áö ã˛yܲ!ï˛!ê˛
ˆ§ˆÏܲˆÏu˛ n ÓyÓ˚ â%Ó˚ˆÏåÈ– !åȈÏoÓ˚ §Çáƒy Î!ò m •Î˚ ï˛ˆÏÓ ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ m, n §ÇáƒÜ˛ •yÄÎ˚yÓ˚ 鲰ܲ
A ˆÌˆÏܲ ˆÓ!Ó˚ˆÏÎ˚ xy§ˆÏåÈ– §y•zˆÏÓ˚ˆÏöÓ˚ í˛zͲõy!òï˛ §%ˆÏÓ˚Ó˚ ܲ¡õyB˛ ï˛áö m, n ~ÓÇ ~!ê˛•z xK˛yï˛ Ü˛¡õyˆÏB˛Ó˚
!öˆÏî≈Î˚ üyö–
281
10.5.3 ˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲˆ•°ü‰ˆÏ•y°‰Í§‰ xö%öyòܲ (Helmholtz’s Resonator)
ˆ•°ü‰ˆÏ•y°‰ÍˆÏ§Ó˚ xö%öyòܲ ÓyhflψÏÓ ôyï%˛ Óy ܲyˆÏã˛Ó˚ ˜ï˛!Ó˚ ~ܲ!ê˛ ú˛yÑ˛õy ˛õye– ~Ó˚ !û˛ï˛ˆÏÓ˚Ó˚ Óyï˛yˆÏ§Ó˚
˛õÎ≈yÎ˚Ó,_ ≤çyÓ˚î Ä §ÇöüˆÏöÓ˚ ú˛ˆÏ° ~Ó˚ ~ܲ!ê˛ !öçfl∫ ܲ¡õyB˛ ÌyˆÏܲ– xö%öyòܲ!ê˛Ó˚ í˛z˛õÓ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚
¢∑ xy˛õ!ï˛ï˛ •ˆÏ° xyˆÏÓ˚y!˛õï˛ Ü˛¡õyˆÏB˛Ó˚ ˆÜ˛Ó° ~ܲ!ê˛ !ö!ò≈‹T üyˆÏöÓ˚ §ˆÏD•z ~Ó˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !üˆÏ°
ÎyÎ˚ ~ÓÇ ˙ !ö!ò≈‹T ܲ¡õyˆÏB˛ ~!ê˛Ó˚ xö%öyò âˆÏê˛– ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòˆÏܲÓ˚ ~•z !ö!ò≈‹T ܲ¡õyˆÏB˛ §yí˛¸y
ˆòÄÎ˚yÓ˚ «˛üï˛y!ê˛ˆÏܲ öyöy ܲyˆÏç ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚– ˆÎüöÈÙÙÙÈ¢ˆÏ∑Ó˚ xK˛yï˛ Ü˛¡õyB˛ !öî≈Î˚ñ ¢∑ !ӈϟ’£Ïî
Ä ¢∑ !ÓÓô≈ö– ~áyˆÏö xyüÓ˚y ≤ÃÌü ÓƒÓ•yÓ˚!ê˛•z xyˆÏ°yã˛öy ܲÓ˚Ó–
!ã˛e 10.8 (a)-ˆï˛ ˆ•°üˆÏ•y°‰Í§‰ xö%öyòˆÏܲÓ˚ àë˛ö °«˛ƒ ܲÓ˚&ö– ~!ê˛ ~ܲ!ê˛ ú˛yÑ˛õy ôyï˛Ó ˆày°Ü˛–
~ܲ!òˆÏܲ Ü%Ñ˛ˆÏçyÓ˚ ü%ˆÏáÓ˚ üˆÏï˛y ˆÓ°öyܲyÓ˚ ü%á xyˆÏåÈñ ÎyˆÏܲ @ˇÃ#Óy (neck) Ó°y •Î˚– xöƒ!òˆÏܲ §Ó˚& ˆáy°y
ö° (B) xyˆÏåÈ– ¢∑ï˛Ó˚D A-ˆï˛ xy˛õ!ï˛ï˛ •Î˚ ~ÓÇ Î!ò ˙ ¢ˆÏ∑Ó˚ ü)° §%Ó˚ Óy ˆÜ˛yöÄ í˛z˛õ§%Ó˚ xö%öyòˆÏܲÓ˚
xö%Óyò# ܲ¡õyˆÏB˛Ó˚ §üyö •Î˚ñ ï˛ˆÏÓ B öˆÏ°Ó˚ §yüˆÏö ܲyö ˛õyï˛ˆÏ° ˆçyÓ˚yˆÏ°y ¢∑ ˆ¢yöy ÎyÎ˚ xÌÓy ˛õyï˛°y
ܲyàç Óy xˆÏºÓ˚˛ ˛õyï˛ Ó˚yáˆÏ° ˆÜÑ˛ˆÏ˛õ ĈÏ벖 ï˛ˆÏÓ ~•z ΈÏsf xö%öyò ˆÜ˛Ó° ~ܲ!ê˛ Ü˛¡õyˆÏB˛Ó˚ çöƒ•z •Î˚–
˛õÓ˚#«˛yÓ˚ ܲyˆÏç ˆåÈyê˛ ˆÌˆÏܲ Óí˛¸ !Ó!û˛ß¨ xyܲyˆÏÓ˚Ó˚ ~ÓÇ !Ó!û˛ß¨ ܲ¡õyˆÏB˛ xö%öyò âˆÏê˛ ~üö xˆÏöܲ=!°
ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòˆÏܲÓ˚ ~ܲ!ê˛ ˆ§ê˛ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ~ܲ•z xö%öyòˆÏܲÓ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛ xö%öyò ˛õyÄÎ˚yÄ
§Ω˛Óñ ï˛ˆÏÓ ï˛yÓ˚ çöƒ xö%öyòܲ!ê˛Ó˚ xyܲyÓ˚ ˛õ!Ó˚Óï≈˛ö ܲÓ˚ˆÏï˛ •Î˚– ˆ§Ó˚ܲü ~ܲ ÓƒÓfliy ˆÜ˛y!öà (Koenig)
ܲˆÏÓ˚!åȈϰöñ Îy !ã˛ˆÏe 10.8(b)-ˆï˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~ˆÏï˛ ò%!ê˛ ˆÓ°öyܲyÓ˚ ˛õyˆÏeÓ˚ ~ܲ!ê˛ xöƒ!ê˛Ó˚ üˆÏôƒ
áyˆÏ˛õÈÙÈáyˆÏ˛õ ~шÏê˛ ÎyÎ˚– ˆë˛ˆÏ° ì%˛!ܲˆÏÎ˚ Óy ˆê˛ˆÏö ÓyÓ˚ ܲˆÏÓ˚ñ xö%öyòˆÏܲÓ˚ xyÎ˚ï˛ö ܲüˆÏÓ!¢ ܲÓ˚y ÎyÎ˚– ~û˛yˆÏÓ
ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòܲ!ê˛ˆÏܲ ~üö xÓfliyÎ˚ xyöy •Î˚ñ ÎyˆÏï˛ xçyöy ܲ¡õyˆÏB˛Ó˚ §ˆÏD xö%öyò §,!‹T •Î˚– ˆ§•z
xÓfliyÎ˚ xö%öyòˆÏܲÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ Îyñ xçyöy ܲ¡õyB˛ ï˛y•z– §%ï˛Ó˚yÇñ ≤ÃÌü!ê˛ çyöy ÌyܲˆÏ° !mï˛#Î˚!ê˛
çyöy ÎyˆÏÓ– ~ˆÏ«˛ˆÏe ܲ#û˛yˆÏÓ xö%öyò âˆÏê˛ ~ÓÇ xö%öyò ܲ¡õyB˛ ˆÜ˛yö ˆÜ˛yö !ӣψÏÎ˚Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ ˆ§!ê˛
xy˛õ!ö !öÿ˛Î˚•z çyöˆÏï˛ ã˛y•zˆÏÓö– ~ÓyÓ˚ ~ !ӣψÏÎ˚ !ܲå%È xyˆÏ°yã˛öy ܲÓ˚y Îyܲ–
xçyöy ܲ¡õyˆÏB˛Ó˚ ¢∑ï˛Ó˚D ~ˆÏ§ xö%öyòˆÏܲÓ˚ @ˇÃ#ÓyÓ˚ ü%ˆÏá ˛õí˛¸ˆÏ° ï˛yÓ˚ üˆÏôƒÜ˛yÓ˚ ÓyÎ˚% ܲ!¡õï˛ •ˆÏÎ˚ o&ï˛
!˛õfiê˛ˆÏöÓ˚ üˆÏï˛y Äë˛yöyüy ܲÓ˚ˆÏÓ– ú˛ˆÏ° ˆày°ˆÏܲÓ˚ üˆÏôƒ Óyï˛yˆÏ§Ó˚ e´üyß∫ˆÏÎ˚ Ó˚&k˛ï˛y˛õ §ÇˆÏܲyã˛ö Ä ≤çyÓ˚î
(Adiabatic compression and rarefaction) •ˆÏÓ–
Î!ò @ˇÃ#ÓyÓ˚ ≤ÃfliˆÏFåÈò ˆ«˛eú˛° a Ä ˜òâ≈ƒ l •Î˚ ~ÓÇ ÓyÎ˚%Ó˚ âöc
ρ
•Î˚ñ ï˛ˆÏÓ @ˇÃ#ÓyÓ˚ üˆÏôƒ ÓyÎ˚%Ó˚ û˛Ó˚
al
ρ
•ˆÏÓ– üˆÏö ܲÓ˚y Îyܲñ ≤ÃyÌ!üܲ §yüƒyÓfliyÎ˚ xö%öyòˆÏܲÓ˚ ˆày°Ü˛!ê˛ˆÏï˛ ÓyÎ˚%Ó˚ xyÎ˚ï˛ö !åÈ°
ν
~ÓÇ ã˛y˛õ
!åÈ° ÓyÎ˚%ü[˛ˆÏ°Ó˚ ã˛y˛õ P– §yüƒyÓfliyö ˆÌˆÏܲ @ˇÃ#ÓyÓ˚ ÓyÎ˚%Ó˚ x ˛õ!Ó˚üyî §Ó˚î âê˛ˆÏ° ˆày°Ü˛!ê˛Ó˚ ÓyÎ˚%Ó˚ xyÎ˚ï˛ö
ˆÓˆÏí˛¸ •Î˚ V + xa– ôÓ˚y Îyܲñ ˙ ≤çyÓ˚ˆÏîÓ˚ ú˛ˆÏ° ÓyÎ˚%Ó˚ ã˛y˛õ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏÎ˚ P1 •Î˚– ~áö !öí˛zê˛ˆÏöÓ˚ !mï˛#Î˚
§)e xö%ÎyÎ˚# @ˇÃ#ÓyÓ˚ üôƒfli ÓyÎ˚%Ó˚ à!ï˛ §ü#ܲÓ˚î •ˆÏÓ≠
282
al
ρ2
2dxdt
= (P1 – P)a
!ã˛e≠10.8(b)
ˆÎˆÏ•ï%˛ ÓyÎ˚%Ó˚ ≤çyÓ˚î Óy §ÇˆÏܲyã˛ö ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛ xï˛ƒhsˇ o&ï˛ âˆÏê˛ñ xï˛~Ó ~=!° Ó˚&k˛ï˛y˛õ ≤Ã!e´Î˚y
~ÓÇ ~Ó˚˛ çöƒ Ó˚&k˛ï˛y˛õ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ §)e ˛≤ÈÏÎyçƒ •ˆÏÓ– Ó˚&k˛ï˛y˛õ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ !öÎ˚ü xö%ÎyÎ˚#
P1(V+ xa)
γ
= PV
γ
– ~áyˆÏö
γ
= !fliÓ˚ ã˛y˛õ Ä !fliÓ˚ xyÎ˚ï˛ö ÓyÎ˚%Ó˚ ò%•z xyˆÏ˛õ!«˛Ü˛ ï˛yˆÏ˛õÓ˚ xö%˛õyï˛–
∴
P1
1xaV
⎛⎞ + ⎜⎟ ⎝⎠
= P
Óyñ P1
1xaVγ ⎛⎞ + ⎜⎟ ⎝⎠
= P, ˆÎˆÏ•ï%˛ xa <<
ν
–
Óyñ P1 – P =
xaVγ−
P1 ...10.9
§ü#ܲÓ˚î 10.8-ˆÜ˛ 10.9 §ü#ܲÓ˚ˆÏî ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° xy˛õ!ö ˛õyˆÏÓöñ
l
ρ21
2xaP dxV dtγ−
=
Óyñ
21
2−
=xaPd x x
Vlpdtγ
(a)
(b)
283
Óyñ
22
2 0d x xdt
ω+ = ˆÎáyˆÏö
21 aPVlγ ω
ρ=
~•z ˛õ!Ó˚!ã˛!ï˛ §Ó˚° ˆòy°à!ï˛Ó˚ §ü#ܲÓ˚î– §%ï˛Ó˚yÇñ xö%öyòˆÏܲÓ˚ üˆÏôƒ ÓyÎ˚% §Ó˚° ˆòy°à!ï˛ˆÏï˛ xyˆÏ®y!°ï˛
•Î˚ ~ÓÇ ï˛yÓ˚ ܲ¡õyB˛ •Î˚ñ
n = 112 2
aPVlγω
π π ρ=
Óyñ n = 2c a
lVπ ...10.10
ˆÎáyˆÏö c = 1Pγρ = xö%öyòˆÏܲÓ˚ üˆÏôƒ ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà–
§ü#ܲÓ˚î (10.10) ˆÌˆÏܲ ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòˆÏܲÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚ñ ܲyÓ˚î c, a,
l ~ÓÇ V §Ó•z xyüyˆÏòÓ˚ çyöy– xöƒ §ühflÏ Ó˚y!¢=!° !fliÓ˚ Ó˚yáˆÏ°
n =
∞1V
xÌ≈yÍñ xö%öyòˆÏܲÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ ï˛yÓ˚ xyÎ˚ï˛ˆÏöÓ˚ Óà≈ü)ˆÏ°Ó˚ ÓƒyhflÏyö%˛õyï˛# •Î˚–
10.5.5 !í˛!çê˛y° ܲ¡õyB˛üy˛õܲ!í˛!çê˛y° ܲ¡õyB˛üy˛õܲ!í˛!çê˛y° ܲ¡õyB˛üy˛õܲ!í˛!çê˛y° ܲ¡õyB˛üy˛õܲ!í˛!çê˛y° ܲ¡õyB˛üy˛õܲ (Digital Frequencymeter)
~!ê˛ §¡õ)î≈ •zˆÏ°Ü˛ê˛Δ!öܲ Óï≈˛ö#– ~!ê˛Ó˚ ç!ê˛° ܲyÎ≈˛õk˛!ï˛ ~áyˆÏö Óƒyáƒy ܲÓ˚y §Ω˛Ó öÎ˚– ~Ó˚ ˆÜ˛yö ÎsfyLjϢÓ˚
ܲ# ܲyçñ ï˛y xyˆÏ°yã˛öy ܲÓ˚y Îyܲ– ˆÎ ¢∑ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ !öô≈yÓ˚î ܲÓ˚y •ˆÏÓñ ï˛yˆÏܲ ê˛Δy™!í˛í˛z§yÓ˚S§yôyÓ˚îï˛
üy•zˆÏe´yˆÏú˛yöVÈÙÈ~Ó˚ §y•yˆÏ΃ §üyö%˛õyï˛# ï˛!í˛¸Í ï˛Ó˚ˆÏD Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛ˆÏÓ˚ Ú•zö˛õ%ê˛Û !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚–
~ÓyˆÏÓ˚ !ã˛e 10.9 ˆòá%ö– ¢)öƒ x!ï˛e´üî !öˆÏò¢≈ܲ (Zero crossing detector) ~üö ~ܲ!ê˛ Óï≈˛ö#ñ ÎyˆÏï˛
˛õÎ≈yÎ˚à!ï˛Ó˚ ï˛!í˛¸Í ï˛Ó˚D ≤ÃÎ%_´ •ˆÏ° Îï˛ÓyÓ˚ ï˛y ôöydܲ ˆÌˆÏܲ îydܲ ~ÓÇ îydܲ ˆÌˆÏܲ ôöydܲ •Î˚ñ
xÌ≈yÍ ¢)öƒ xÓfliy x!ï˛e´ü ܲˆÏÓ˚ñ ï˛ï˛ÓyÓ˚ ï˛yÓ˚ xyí˛zê˛˛õ%ê˛ !Óû˛Ó (A) !ö!ò≈‹T !ÓhflÏyˆÏÓ˚Ó˚ ôöydܲ Óy îydܲ
xÓfliy ôyÓ˚î ܲˆÏÓ˚– ~Ó˚ ú˛ˆÏ° ˛õÎ≈yÎ˚à!ï˛Ó˚ ï˛Ó˚D!ê˛ ï˛yÓ˚ §üyö ܲ¡õyˆÏB˛Ó˚ Óà≈ï˛Ó˚ˆÏD (square wave)
Ó˚*˛õyhsˇ!Ó˚ï˛ •Î˚– x˛õyˆÏÓ˚¢öy° !ÓÓô≈ܲ (operational amplifier, §ÇˆÏ«˛ˆÏ˛õ OP-AMP) öyüܲ ~ܲ !ӈϢ£Ï
•zˆÏ°Ü˛ê˛Δ!öܲ ï˛ˆÏsfÓ˚ (electric device) §y•yˆÏ΃ ~•z Óï≈˛ö#!ê˛ !ö!ü≈ï˛ •Î˚– xyÓ˚ ~ܲ!ê˛ Óï≈˛ö# !öá%Ñï˛ 1 ˆ§ˆÏܲu˛
fliy!Î˚ˆÏcÓ˚ ôöydܲ !Óû˛Óflõ®ö (B) í˛zͲõyòö ܲˆÏÓ˚– ~Ó˚ çöƒ T-flip-flop öyüܲ ~ܲ!ê˛ •zˆÏ°Ü˛ê˛Δ!öܲ ï˛sf
~ÓÇ ˆÜ˛°y§ flõ®Ü˛ (crystal oscillator) ÓƒÓ•*ï˛ •Î˚–
284
A ~ÓÇ B í˛zû˛Î˚ !Óû˛Ó flõ®ö (voltage pulse)-ˆÜ˛ ~ܲˆÏe AND ˆàê˛ öyüܲ ~ܲ!ê˛ •zˆÏ°Ü˛!ê˛Δܲ ï˛ˆÏsf
˛≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~•z ï˛sf!ê˛Ó˚ ˜Ó!¢‹Tƒ ~•z ˆÎñ A ~ÓÇ B í˛zû˛ˆÏÎ˚•z ôöydܲ •ˆÏ° ï˛ˆÏÓ C !Ó®%ˆÏï˛ ôöydܲ
!Óû˛Ó ˛õyÄÎ˚y ÎyˆÏÓ– ~áö B-~Ó˚ !Óû˛Ó ~ܲ ˆ§ˆÏܲˆÏu˛ ôˆÏÓ˚ ôöydܲ– ~•z §üˆÏÎ˚ A Îï˛ÓyÓ˚ ôöydܲ •ˆÏÓ
SxÌ≈yÍ ˆÎ ܲê˛y ï˛Ó˚D˜Ïòâ≈ƒ §¡õ)î≈ •ˆÏÓVñ C ï˛ï˛ÓyÓ˚ ôöydܲ •ˆÏÓ– §%ï˛Ó˚yÇñ C •ˆÏÓ §Ó˚y§!Ó˚ •zö˛õ%ˆÏê˛Ó˚
ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ– ≤Ã!ï˛ ˆ§ˆÏܲˆÏu˛ •zö˛õ%ê˛ ï˛Ó˚D Îï˛ÓyÓ˚ ܲ¡õö §¡õ)î≈ ܲÓ˚ˆÏÓñ C-ˆï˛ ï˛ï˛=!° !Óû˛Óflõ®ö
í˛zͲõy!òï˛ •ˆÏÓ– ~•z §Çáƒy!ê˛ àîܲ (Counter) öyüܲ ~ܲ!ê˛ •zˆÏ°Ü˛!ê˛Δܲ Óï≈˛ö#Ó˚ §y•yˆÏ΃ ˆÓ˚ܲí≈˛ ܲÓ˚y •Î˚–
10.5.6 x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛x!§ˆÏ°yˆÏflÒy˛õ ˛õk˛!ï˛
~•z ˛õk˛!ï˛Ó˚ §y•yˆÏ΃ xy˛õ!ö ~ܲ!ê˛ x!í˛ÄÈÙÈ!ú ˛ˆÏܲyˆÏÎ˚™ x!§ˆÏ°ê˛ˆÏÓ˚Ó˚ í˛z˛õˆÏÎyçöˆÏÎyàƒ (adjustable)
ܲ¡õyˆÏB˛Ó˚ ˛õÎ≈yÓ,_ xyí˛zê˛˛õ%ˆÏê˛Ó˚ §ˆÏD xK˛yï˛ Ü˛¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ §Ó˚y§!Ó˚ ï%˛°öy ܲˆÏÓ˚ ˛õˆÏÓ˚Ó˚ ܲ¡õyB˛!ê˛ ÓyÓ˚
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– 10.10 !ã˛e ˆÌˆÏܲ xy˛õ!ö ˛õk˛!ï˛!ê˛ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!öˆÏî≈Î˚ ܲ¡õyˆÏB˛Ó˚ ¢∑!ê˛ ≤Ã̈Ïü ~ܲ!ê˛ üy•zˆÏe´yˆÏú˛yˆÏö xy˛õ!ï˛ï˛ ܲÓ˚y •Î˚ ~ÓÇ üy•zˆÏe´yˆÏú˛yö ˆÌˆÏܲ ˆÎ
!Óû˛Ó ˛õyÄÎ˚y ÎyÎ˚ ˆ§!ê˛ˆÏܲ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚Y-xˆÏ«˛ ≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~Ó˚ ú˛ˆÏ° x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ •zˆÏ°Ü˛ê˛Δö
Ó˚!Ÿ¬ ˛õò≈yÎ˚ ˆÎ í˛zIμ° xyˆÏ°yܲ!Ó®%!ê˛ Ó˚ã˛öy ܲˆÏÓ˚ñ ˆ§!ê˛ Y Óy í˛zÕ‘¡∫ !òˆÏܲ üy•zˆÏe´yˆÏú˛yˆÏö xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚
ܲ¡õyB˛ xö%ÎyÎ˚# xyˆÏ®y!°ï˛ •ˆÏï˛ ÌyܲˆÏÓ–
~ÓyÓ˚ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ X xˆÏ«˛ ~ܲ!ê˛ x!í˛ÄÈÙÈ!ú ˛ˆÏܲyˆÏÎ˚!™ Óy ◊yÓƒ ܲ¡õyˆÏB˛Ó˚ x!§ˆÏ°ê˛ˆÏÓ˚Ó˚ ˛õÎ≈yÓ,_
xyí˛zê˛ÈÙÈ˛õ%ê˛ !Óû˛Ó ≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~•z ôÓ˚ˆÏöÓ˚ x!§ˆÏ°ê˛Ó˚ ≤ÃyÎ˚ 20Hz ˆÌˆÏܲ 20000Hz ܲ¡õyˆÏB˛Ó˚ ˛õÎ≈yÓ,_
B
C
A
285
!Óû˛Ó í˛zͲõyòö ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ ~ÓÇ ~ܲ!ê˛ í˛yÎ˚y°Î%_´ öÓ(knob)-~Ó˚ §y•yˆÏ΃ ~•z ܲ¡õyB˛!ê˛ §¡õ)î≈ ˛õyÕ‘yÓ˚
üˆÏôƒ •zFåÈyüˆÏï˛y ˛õ!Ó˚Ó!ï≈˛ï˛ ܲÓ˚y ÎyÎ˚– x!§ˆÏ°ê˛Ó˚!ê˛ ã˛y°% ܲÓ˚ˆÏ° x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ ˛õò≈yÎ˚ xyˆÏ°yܲ!Ó®%!ê˛ X
Óy xö%û)˛!üܲ !òˆÏܲ x!§ˆÏ°ê˛ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛ xyˆÏ®y!°ï˛ •ˆÏÓ–
xy˛õ!ö !öÿ˛Î˚ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ~áö xyˆÏ°yܲ!Ó®%!ê˛Ó˚ X Ä Y x«˛ ÓÓ˚yÓÓ˚ ò%•z §Ó˚° ˆòy°à!ï˛Ó˚
§üy˛õï˛ö âê˛ˆÏÓ ~ÓÇ ò%•z ˆòy°à!ï˛Ó˚ ܲ¡õyˆÏB˛ Î!ò §Ó˚° xö%˛õyï˛ ÌyˆÏܲ ï˛ˆÏÓ xy°yܲ!Ó®%!ê˛ ˛õò≈yÎ˚ !°§yç%Ó˚
!ã˛e ˜ï˛!Ó˚ ܲÓ˚ˆÏÓ– !ӈϢ£Ïï˛ñ ò%•z ܲ¡õyB˛ Î!ò §üyö •Î˚ ï˛ˆÏÓ ˛õò≈yÎ˚ §Ó˚°ˆÏÓ˚áy xÌÓy Ó,_ xÌÓy ~ܲ!ê˛ í˛z˛õÓ,_
ˆòáy ÎyˆÏÓ– ~•z xÓfliyÎ˚ x!§ˆÏ°ê˛ˆÏÓ˚Ó˚ í˛yÎ˚y° ˆÌˆÏܲ ˆ§!ê˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õyë˛ !öˆÏ° ˆ§!ê˛•z •ˆÏÓ !öˆÏî≈Î˚ ܲ¡õyB˛–
~ÓyÓ˚ ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ §¡∫¶˛#Î˚ ò%!ê˛ ≤Èϟ¿Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 3 :
fiê˛ΔˆÏÓyˆÏflÒy!˛õܲ ã˛e´ ˛õk˛!ï˛ˆÏï˛ ~ܲ!ê˛ §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚ˆÏï˛ !àˆÏÎ˚ ˆòáy ˆà° ˆÎñ 80 !ê˛
ˆÓ˚áy!åÈoÎ%_´ ã˛e´!ê˛ Îáö !ü!öˆÏê §ˆÏÓ≈yFã˛ 90 ÓyÓ˚ ˆâyˆÏÓ˚ñ ï˛áö §%Ó˚¢°yܲy!ê˛ !fliÓ˚ ˆòáyÎ˚– §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛
ܲï˛⁄ ã˛e´!ê˛ !ü!öˆÏê˛ 45 ÓyÓ˚ â%Ó˚ˆÏ° ܲ# ˆòáy ˆÎï˛⁄
10.6 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
ˆÎ ˆÜ˛yöÄ ¢ˆÏ∑Ó˚ ~ܲ!ê˛ ≤Ãôyö ˆÓ!¢‹Tƒ ï˛yÓ˚ ï˛#Ó ï˛y– ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó°ˆÏï˛ ˆÓyé˛yÎ˚ üyôƒˆÏüÓ˚ ~ܲܲ °¡∫
ˆ«˛eú˛ˆÏ°Ó˚ üˆÏôƒ !òˆÏÎ˚ ~ܲܲ §üˆÏÎ˚ Óy!•ï˛ ¢!_´– ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ xö%˛õyï˛ˆÏܲ ï˛yÓ˚ °ày!Ó˚ò‰ü‰ !•§yˆÏÓ
≤Ãܲy¢ ܲÓ˚y •Î˚ ~ÓÇ ï˛yÓ˚ ò¢üyÇ¢ˆÏܲ ˆí˛!§ˆÏÓ° ~ܲܲ Ó°y •Î˚– ï˛#Ó ï˛yÓ˚ §ˆÏD §¡∫!¶˛ï˛ ~ܲ!ê˛ Ó˚y!¢ •°
¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒ– ¢∑ ˆçyˆÏÓ˚ •ˆÏFåÈñ öy xyˆÏhflÏ •ˆÏFåÈ ï˛yÓ˚ xö%û)˛!ï˛ˆÏܲ•z xyüÓ˚y ≤ÃyÓ°ƒ Ó!°– ĈÏÎ˚ÓyÓ˚ÈÙȈú˛Ü˛öyˆÏÓ˚Ó˚
§)e xö%ÎyÎ˚# ≤ÃyÓ°ƒ Ó,!k˛ ï˛#Ó ï˛yÓ˚ Ó,!k˛Ó˚ °ày!Ó˚ò‰ˆÏüÓ˚ §üyö%˛õyï˛# •Î˚– ï˛#Ó ï˛y üy˛õyÓ˚ !Ó!û˛ß¨ ˛õk˛!ï˛ÈÙÙÙÈÓ˚ƒyˆÏ°Ó˚
ã˛yܲ!ï˛ñ ï˛Æ ï˛yÓ˚ üy•zˆÏe´yˆÏú˛yöñ xyˆÏ°yܲ Óƒ!ï˛ã˛yÓ˚ ˛õk˛!ï˛ •zï˛ƒy!ò ~áyˆÏö Óî≈öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– !í˛!çê˛y°
ï˛#Ó ï˛yüy˛õ# ΈÏsfÓ˚ ܲyÎ≈˛õk˛!ï˛Ä xy˛õ!ö çyöˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈö–
¢ˆÏ∑Ó˚ ܲ¡õyB˛ üy˛õyÓ˚ ˆÓ¢ ܲˆÏÎ˚ܲ!ê˛ ˛õk˛!ï˛ xy˛õ!ö ~áyˆÏö ˛õˆÏí˛¸ˆÏåÈö– ~•z üˆÏôƒ ܲï˛Ü˛=!° §¡õ)î≈ Îy!sfܲñ
ˆÎüö ‹T…ˆÏÓyˆÏflÒy!˛õܲ ˛õk˛!ï˛ Óy Ó˚ƒyˆÏ°Ó˚ ¢∑ã˛e´ ˛õk˛!ï˛– xyÓyÓ˚ ܲï˛Ü˛=!° ˛õk˛!ï˛ˆÏï˛ ˆ◊yï˛yˆÏܲ ܲyˆÏö ÷ˆÏö
ò%!ê˛ ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛yˆÏܲ §üyö ܲÓ˚ˆÏï˛ •Î˚ Óy xö%öyò âê˛ˆÏåÈ !ܲöyñ ˆ§ Óƒy˛õyˆÏÓ˚ §%!ö!ÿ˛ï˛ •ˆÏï˛ •Î˚–
ˆ¢y£Ïy_´ ˛õk˛!ï˛ˆÏï˛ ˆÎ ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòܲ ÓƒÓ•*ï˛ •Î˚ ˆ§!ê˛ñ ¢∑!ÓK˛yˆÏöÓ˚ •z!ï˛•yˆÏ§ ~ܲ!ê˛
=Ó˚&c˛õ)î≈ fliyö x!ôܲyÓ˚ ܲˆÏÓ˚ˆÏåÈ– ~•z §Ó˚° ÓƒÓfliy!ê˛ ¢∑ !ӈϟ’£ÏˆÏî öyöyû˛yˆÏÓ ÓƒÓ•*ï˛ •Î˚– ~áyˆÏö xyüÓ˚y
~•z ÓƒÓfliy!ê˛Ó˚ ܲyÎ≈˛õk˛!ï˛Ó˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ܲˆÏÓ˚!åÈ ~ÓÇ ~!ê˛Ó˚ xö%öyò ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ Ó˚y!¢üy°yÄ
≤Ã!ï˛!¤˛ï˛ ܲˆÏÓ˚!åÈ–
10.7 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. Ó˚ƒyˆÏ° ã˛yܲ!ï˛Ó˚ §y•yˆÏÎσ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲ#û˛yˆÏÓ üy˛õy •Î˚⁄
2. ôù!ö ˆÓ˚!í˛Ä!üê˛yˆÏÓ˚Ó˚ ü)°ö#!ï˛ Óƒyáƒy ܲÓ˚&ö–
286
3. ܲƒy!öÎ˚yÓ˚ òƒ °y ï%˛ˆÏÓ˚Ó˚ §y•zˆÏÓ˚ˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈˛õk˛!ï˛Ó˚ !ÓÓÓ˚î !òö– ~•z Îsf!ê˛ Ü˛¡õyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ
åÈyí˛¸y xyÓ˚ ܲ#ÈÙÈܲ# ܲyˆÏç °yàˆÏï˛ ˛õyˆÏÓ˚ ӈϰ xy˛õöyÓ˚ üˆÏö •Î˚⁄
4. ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòˆÏܲÓ˚ Óƒyáƒy ܲÓ˚&ö– ≤Ãüyî ܲÓ˚&ö ˆÎñ ~Ó˚ xö%öyò# ܲ¡õyB˛ xyÎ˚ï˛ˆÏöÓ˚ Óà≈ü)ˆÏ°Ó˚
ÓƒhflÏyö%˛õyï˛#–
5. ܲƒyˆÏÌyí˛ ˆÓ˚ x!§ˆÏ°yˆÏflÒyˆÏ˛õÓ˚ §y•yˆÏ΃ ܲ# ܲˆÏÓ˚ ~ܲ!ê˛ !Ó÷k˛ §%ˆÏÓ˚Ó˚ ܲ¡õyB˛ !öî≈Î˚ ܲÓ˚y ÎyÎ˚ ï˛y Óî≈öy
ܲÓ˚&ö–
6. !í˛!çê˛y° !ú ˛ˆÏܲyˆÏÎ˚!™!üê˛yˆÏÓ˚Ó˚ ܲyÎ≈˛õk˛!ï˛ Óƒyáƒy ܲÓ˚&ö–
10.8 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠
1. (a) üôƒÓ˚y!eˆÏï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y hflÏÓ˚ 10 log 3200 Óy 45 ˆí˛!§ˆÏÓ° ܲüñ xÌ≈yÍ 80 – 45= 35dB–
(b) ÓyˆÏí˛¸ñ ã˛yÓ˚=îñ Wm–2, ≤ÃÓ°ï˛yÓ˚ xö%û)˛!ï˛–
2. (a) ÓƒÓfliy=!° •° (i) ã˛yܲ!ï˛!ê˛ ¢∑ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓˆÏàÓ˚ §ˆÏD 45° ˆÜ˛yˆÏî ˆé˛y°yˆÏöy •Î˚ó (ii)
ã˛yܲ!ï˛!ê˛ˆÏܲ öˆÏ°Ó˚ fliyî%ï˛Ó˚ˆÏDÓ˚ §%flõ® !Ó®%ˆÏï˛ Ó˚yáy •Î˚–
(b) ¢∑ï˛Ó˚D ˆÎ ÓyÎ˚%≤ÃÓy• í˛zͲõߨ ܲˆÏÓ˚ ï˛y ï˛y˛õ ˛õ!Ó˚ã˛°ö myÓ˚y ï˛Æ ï˛yÓ˚!ê˛Ó˚ í˛z£èï˛y ܲ!üˆÏÎ˚ ˆòÎ˚–
~Ó˚ ú˛ˆÏ° ôyï˛Ó ï˛yˆÏÓ˚Ó˚ ˆÓ˚yô ܲˆÏü ÎyÎ˚–
(c) ÓyÎ˚%Ó˚ ã˛y˛õñ âöc ~ÓÇ ≤Ã!ï˛§Ó˚yˆÏB˛Ó˚ ˛õ!Ó˚Óï≈˛ö ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛ âˆÏê˛– §%ï˛Ó˚yÇñ Óƒ!ï˛ã˛yÓ˚ ˛õ!ê˛Ó˚
xyˆÏ®y°öÄ ~ܲ•z ܲ¡õyˆÏB˛ âˆÏê˛– !ܲv xyüyˆÏòÓ˚ ˆã˛yˆÏá ˆÎ ˆÜ˛yöÄ ò,¢ƒ ≤ÃyÎ˚ 1
10 ˆ§ˆÏܲu˛ fliyÎ˚# •Î˚–
ÎyÓ˚ ú˛ˆÏ° o&ï˛ xyˆÏ®y!°ï˛ ˛õ!ê˛ xyüyˆÏòÓ˚ ˆã˛yˆÏᲠflõ‹Tû˛yˆÏÓ ôÓ˚y ˛õˆÏí˛¸ öy–
3. §%Ó˚¢°yܲyÓ˚ ܲ¡õyB˛
809012Hz60×=
–
ã˛e´!ê˛ !ü!öˆÏê˛ 45 ÓyÓ˚ â%Ó˚ˆÏ° ~ܲ!ê˛ ˆÓ˚áy!åȈÏoÓ˚ çyÎ˚àyÎ˚ ˛õˆÏÓ˚Ó˚!ê˛ xy§ˆÏï˛ ˆÎ §üÎ˚ °yàï˛ñ xÌ≈yÍ
60 180 45 60
=×
ˆ§ˆÏܲˆÏu˛ §%Ó˚¢°yܲyÓ˚ ò%!ê˛ Ü˛¡õö §¡õ)î≈ •ï˛– §%ï˛Ó˚yÇñ ~ˆÏ«˛ˆÏeÄ §%Ó˚¢°yܲy!ê˛ !fliÓ˚
ӈϰ üˆÏö •ˆÏï˛y–
287
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠
1. 10.4.1 xLjϢ ~!ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
2. 10.4.4 xLjϢ ôù!öÈÙȈÓ˚!í˛Ä!üê˛yˆÏÓ˚Ó˚ ü)° ö#!ï˛ Ó!î≈ï˛ •ˆÏÎ˚ˆÏåÈ– ~!ê˛ ˆòˆÏá !öö–
3. ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ 10.5.3 xLjϢ xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ– ˛õ!Ó˚Óï≈˛öˆÏÎyàƒ Ü˛¡õyˆÏB˛Ó˚ ¢∑ í˛zͲõyòö
åÈyí˛¸yÄ ~•z §y•zˆÏÓ˚ö !Ó˛õò§)ã˛Ü˛ §ÇˆÏܲï˛ñ ܲyÓ˚áyöyÓ˚ ÓyÑ!¢ñ xy¡∫%°ƒy™ ≤Ãû,˛!ï˛ ày!í˛¸Ó˚ ‡ê˛yÓ˚ !•§yˆÏÓ
ÓƒÓÓyÓ˚ ܲÓ˚y ÎyÎ˚–
4. 10.5.4 xLjϢ ˆ•°ü‰ˆÏ•y°Í§‰ xö%öyòˆÏܲÓ˚ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5. 10.5.6 xLjϢ ~ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
6. 10.5.5 xLjϢ ~•z ܲyÎ≈˛õk˛!ï˛ Óƒyáƒy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
288
~ܲܲ ~ܲܲ ~ܲܲ ~ܲܲ ~ܲܲ 11 ≠ ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliy≠ ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliy≠ ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliy≠ ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliy≠ ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliy (Auditorium Acoustics)
àë˛ö
11.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
11.2 xö%Ó˚îö (Reverberation)
11.3 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛
11.4 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e
11.5 !ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç (Eyring)-~Ó˚ §)e
11.6 ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ
11.7 ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢y (Acoustic design of auditorium)
11.8 §yÓ˚yÇ¢
11.9 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
11.10 í˛z_Ó˚üy°y
11.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
xy˛õ!ö !öÿ˛Î˚•z !§ˆÏöüy ÈÙÈ !̈ÏÎ˚ê˛yÓ˚ ˆòáˆÏï˛ Óy Ó_,´ï˛yñ àyö Óyçöy ÷öˆÏï˛ !Ó!û˛ß¨ ˆ≤ë˛yà,ˆÏ• !àˆÏÎ˚ˆÏåÈö–
xy˛õ!ö •Î˚ï˛ °«˛ƒ ܲˆÏÓ˚ˆÏåÈöñ ˆÜ˛yöÄ ˆÜ˛yöÄ ˆ≤ë˛yà,ˆÏ• ¢∑ flõ‹Tû˛yˆÏÓ ˆ¢yöy ÎyÎ˚ ~ÓÇ ˆ≤ë˛yà,ˆÏ•Ó˚ §Ó
xLjϢ•z ¢∑ §üyöû˛yˆÏÓ ˆ˛õÔÑåÈÎ˚– xyÓyÓ˚ xyüÓ˚y ~Ä °«˛ƒ ܲ!Ó˚ ˆÎñ ˆÜ˛yöÄ ˆÜ˛yöÄ •ˆÏ° !flõܲyˆÏÓ˚Ó˚ §y•y΃
åÈyí˛¸y §Ó çyÎ˚ày ˆÌˆÏܲ ¢∑ ˆ¢yöy ÎyÎ˚ öy Óy ˆÓ!¢ àü‰àü‰ ܲÓ˚yÎ˚ flõ‹T ˆÓyé˛y ÎyÎ˚ öy– ˆ≤ë˛yà,ˆÏ•Ó˚ ~•z
˜Ó!¢‹Tƒ=!° ï˛yÓ˚ àë˛ˆÏöÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚–
§û˛yÈÙÈÓ_,´ï˛y xÌÓy §yÇfl,Ò!ï˛Ü˛ xö%¤˛yˆÏöÓ˚ í˛z˛õÎ%_´ ˆÜ˛yöÄ Óí˛¸ ˆ≤ë˛yà,• Óy •°âÓ˚ ~üö •ÄÎ˚y í˛z!ã˛ï˛ñ
ÎyˆÏï˛ ï˛yÓ˚ ≤Ã!ï˛!ê˛ xLjϢ í˛z˛õ!Ó‹T ˆ◊yï˛yÓ˚ ܲyˆÏåÈ ¢∑ flõ‹Tû˛yˆÏÓ ˆ˛õÔÑåÈÎ˚– ˆ§ ܲyÓ˚ˆÏî ¢∑!ÓK˛yˆÏöÓ˚ !ܲå%È ü)°
!öÎ˚ü ˆüˆÏö ~Ó˚ àë˛ö !ӈϢ£Ï ôÓ˚ˆÏö ܲÓ˚y •ˆÏÎ˚ ÌyˆÏܲ– xyÓyÓ˚ !Ó˛õÓ˚#ï˛ !òܲê˛yÄ û˛yÓ%ö– ¢∑ ˆÓ˚ܲ!í≈˛ÇÈÙÈ~Ó˚
fiê%˛!í˛Äñ ˆê˛!°ˆÏú˛yö Ó%Ì !ܲÇÓy •y§˛õyï˛yˆÏ°Ó˚ x˛õyˆÏÓ˚¢ö ܲ«˛ •ÄÎ˚y í˛z!ã˛ï˛ ¢∑ˆÏÓ˚yô# (sound proof), ÎyˆÏï˛
Óy•zˆÏÓ˚Ó˚ ˆày°üy° öy xy§ˆÏï˛ ˛õyˆÏÓ˚– ˆ§ çöƒ ï˛yÓ˚ àë˛ö ≤Ãîy°#Ä ˛õ,Ìܲ •ÄÎ˚y òÓ˚ܲyÓ˚– ~Ó˚ çöƒ
˛õòyÌ≈!ÓK˛yˆÏöÓ˚ !ܲå%È ü)° !öÎ˚ü üyöˆÏï˛ •Î˚– ¢∑ï˛Ó˚D §MÈ˛y°ˆÏöÓ˚ ˆ§•z !üÎ˚ü §ü!‹TˆÏܲ Ó°y •Î˚ ˆ≤ë˛yà,ˆÏ•Ó˚
¢∑ÓƒÓfliy–
§yôyÓ˚î Ó§ÓyˆÏ§Ó˚ âˆÏÓ˚Ó˚ §ˆÏD ˆ≤ë˛yà,ˆÏ•Ó˚ ü)° ˛õyÌ≈ܲƒ ˆ§!ê˛Ó˚ Ó,•Í xyܲyÓ˚ ~ÓÇ ¢∑!ÓöƒyˆÏ§Ó˚ ÓƒÓfliy–
~ܲ!ê˛ û˛y° ˆ≤ë˛yà,ˆÏ•Ó˚ ܲ# ܲ# =î Ìyܲy ≤ÈÏÎ˚yçö ӈϰ xy˛õöyÓ˚ üˆÏö •Î˚⁄ xy˛õ!ö !öÿ˛Î˚•z !ö¡¨!°!áï˛
˜Ó!¢‹Tƒ=!°Ó˚ í˛z˛õÓ˚ ˆçyÓ˚ ˆòˆÏÓö ≠
289
(i) ˆ≤ë˛yà,ˆÏ•Ó˚ §ühflÏ xLjϢ ˆÎ ≤Ã!ï˛!ê˛ ¢∑yÇ¢ ◊&!ï˛ˆÏàyã˛Ó˚ •ÄÎ˚yÓ˚ üï˛ ÎˆÏÌ‹T ˆçyÓ˚yˆÏ°y ¢∑ ˆ¢yöy
ÎyÎ˚–
(ii) ~ܲ!ê˛ ¢∑yLjϢÓ˚ ≤Ã!ï˛ôù!ö ˆÎö ˛õÓ˚Óï˛#≈ ¢∑yLjϢˆÏܲ ì˛yܲy öy ˆòÎ˚ñ ÎyˆÏï˛ Ó_´yÓ˚ í˛zFã˛y!Ó˚ï˛
¢∑yÇ¢=!° ˛õ,Ìܲû˛yˆÏÓ flõ‹T ˆÓyé˛y ÎyÎ˚– xÓ¢ƒ !ܲå%Èê˛y ≤Ã!ï˛ôù!ö Ìyܲy ÓyN˛ö#Î˚ ˆÜ˛ööy ï˛y ¢ˆÏ∑Ó˚
ï˛#Ó ï˛y Óyí˛¸yÎ˚ ~ÓÇ ˆ§!ê˛ˆÏܲ ◊&!ï˛§%áܲÓ˚ ܲˆÏÓ˚–
(iii) ü)° ¢∑ ˆÎö !ÓÜ,˛ï˛ •ˆÏÎ˚ öy ÎyÎ˚ñ xÌ≈yÍ ü)° ¢ˆÏ∑ ü)°§%Ó˚ Ä í˛z˛õ§%Ó˚=!° ˆÎ xö%˛õyˆÏï˛ ÌyˆÏܲñ ˆ◊yï˛yÓ˚
ܲˆÏî≈ ˆÎö ˆ§•z xö%˛õyˆÏï˛•z ˆ˛õÔÑåÈyÎ˚–
~•z ˆÓ!¢‹Tƒ=!° ÓçyÎ˚ Ó˚yáyÓ˚ çöƒ ˆÎ !Ó£ÏÎ˚!ê˛ !ÓˆÏÓã˛öy ܲÓ˚y §ÓˆÏã˛ˆÏÎ˚ =Ó˚&c˛õ)î≈ñ ï˛y •° xö%Ó˚îö– ï˛y
åÈyí˛¸yÄ Ü˛ˆÏÎ˚ܲ!ܲ!ê˛ !Óã˛yÎ≈ !Ó£ÏÎ˚ xyˆÏåÈ Îy ˛õˆÏÓ˚ í˛zˆÏÕ‘á ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ˆ≤ë˛yà,ˆÏ•Ó˚ Óy›!Óòƒyàï˛ !öü≈yî ˆÜ˛Ô¢°
¢∑!ÓK˛yˆÏöÓ˚ ˆÎ §Ó !öÎ˚ü Ä §)ˆÏeÓ˚ í˛z˛õÓ˚ xyôy!Ó˚ï˛ñ ~•z ~ܲˆÏܲ ˆ§=!° xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠
~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° xy˛õ!ö
ˆÜ˛yöÄ ˆÜ˛yöÄ •ˆÏ° ¢∑ ˆÜ˛ö û˛y° ˆ§yöy ÎyÎ˚ öy ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ˆÓyé˛yˆÏï˛ ˛õyÓ˚ˆÏÓö xö%Ó˚îö ܲyˆÏܲ ӈϰ ~ÓÇ ç#!Óï˛ Ä ü,ï˛ Ü˛«˛ ܲ#–
Ók˛ ˆ≤ë˛yà,ˆÏ• ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛ Ä •…yˆÏ§Ó˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
~ܲ!ê˛ í˛zÍÜ,˛‹T ˆ≤ë˛yà,• !öü≈yˆÏîÓ˚ çöƒ ܲ# ܲ# !Ó£ÏÎ˚ !ÓˆÏÓã˛öy ܲÓ˚y ≤ÈÏÎ˚yçö ˆ§ !ӣψÏÎ˚ ˛õÓ˚yü¢≈ !òˆÏï˛
˛õyÓ˚ˆÏÓö–
11.2 xö%Ó˚îöxö%Ó˚îöxö%Ó˚îöxö%Ó˚îöxö%Ó˚îö (Reverberation)
§yôyÓ˚î x!û˛K˛ï˛y ˆÌˆÏܲ xyüÓ˚y çy!ö ˆÎñ ˆÜ˛yöÄ Ók˛ âˆÏÓ˚ ˛Ó_,´ï˛yñ §D#ï˛ Óy ÓyˆÏòƒÓ˚ ôù!ö ˆÌˆÏü ˆàˆÏ°Ä
!ܲå%È«˛î ï˛yÓ˚ ˆÓ˚¢ ˆÌˆÏܲ ÎyÎ˚– ˆ◊yï˛yÓ˚ ܲyˆÏö ¢∑ =O!Ó˚ï˛ •ˆÏï˛ •ˆÏï˛ !ü!°ˆÏÎ˚•z ÎyÎ˚– ˆÜ˛yöÄ «˛îfliyÎ˚# ¢∑
í˛zͲõy!òï˛ •ÄÎ˚yÓ˚ ˛õÓ˚ ܲˆÏ«˛Ó˚ ã˛y!Ó˚!òˆÏܲÓ˚ ˆòÄÎ˚y°ñ åÈyò Ä xöƒyöƒ ï˛° ˆÌˆÏܲ ÓyÓ˚ÓyÓ˚ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚
ˆ◊yï˛yÓ˚ ܲyˆÏö ˆ˛õÔÑåÈyÎ˚– ˆÎ ˆÜ˛yöÄ «˛îfliyÎ˚# ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö ˆÎ ¢∑yö%û)˛!ï˛ÈÙȧ,!‹T ܲˆÏÓ˚ ï˛y 0.1s fliyÎ˚#
•Î˚– ˙ §üˆÏÎ˚Ó˚ üˆÏôƒ xöƒ ˆÜ˛yö ¢∑ ܲyˆÏö ~ˆÏ°Ä xyüÓ˚y ˆ§!ê˛ xy°yòy ܲˆÏÓ˚ Ó%é˛ˆÏï˛ ˛õy!Ó˚ öy– ~Ó˚ ú˛ˆÏ°
ü)°¢∑ Ä ≤Ã!ï˛ôù!ö=!° !üˆÏ° ~ܲ!ê˛ ≤ð!¡∫ï˛ ¢∑yö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚ñ ÎyÓ˚ ï˛#Ó ï˛y e´ü¢ •…y§ ˛õyÎ˚ Ä xӈϢˆÏ£Ï
!ü!°ˆÏÎ˚ ÎyÎ˚–
ôù!öÓ˚ ~•z ≤ð!¡∫ï˛ •ÄÎ˚yˆÏܲ xö%Ó˚îö Ó°y •Î˚– ò#â≈fliyÎ˚# xö%Ó˚îö ˆÜ˛yöÄ ˆÜ˛yöÄ ˆ«˛ˆÏe ÷öˆÏï˛ û˛y°
°yàˆÏ°Ä Ó_,ï˛yñ öyê˛Ü˛ ≤Ãû,˛!ï˛ ˆÎ §Ó ˆ«˛ˆÏe í˛zFã˛y!Ó˚ï˛ ¢∑=!° flõ‹Tû˛yˆÏÓ ˆ¢yöyÓ˚ ≤ÈÏÎ˚yçö •Î˚ñ ˆ§áyˆÏö
xö%Ó˚îö ~ܲ!ê˛ ÓyܲƒyÇ¢ xyÓ˚ ~ܲ!ê˛ˆÏܲ ˆì˛ˆÏܲ ˆòÄÎ˚yÓ˚ ú˛ˆÏ° ◊ÓˆÏî x§%!ÓôyÓ˚ §,!‹T ܲˆÏÓ˚– xyÓyÓ˚ Î!ò xö%Ó˚îö
~ˆÏܲÓyˆÏÓ˚•z öy ÌyˆÏܲñ §D#ï˛ÈÙÈÓyòƒ ˆ◊yï˛yÓ˚ ܲyˆÏö §%áܲÓ˚ •ˆÏÓ öy– ˆ§ çöƒ ˆ≤ë˛yà,• xö%Ó˚îˆÏöÓ˚ §üÎ˚ •ÄÎ˚y
í˛z!ã˛ï˛ á%Ó ˆÓ!¢Ä öÎ˚ñ ~ˆÏܲÓyˆÏÓ˚ ܲü öÎ˚ñ ò%•zˆÏÎ˚Ó˚ üyé˛yüy!鲖 ~•z ≤çˆÏD ܲˆÏÎ˚ܲ!ê˛ §ÇK˛y ˆçˆÏö ˆöÄÎ˚y Îyܲ–
290
(i) xö%Ó˚îö ܲy° xö%Ó˚îö ܲy° xö%Ó˚îö ܲy° xö%Ó˚îö ܲy° xö%Ó˚îö ܲy° (Re-verberation time) : ¢∑ í˛zͧy!•ï˛ •ˆÏÎ˚ ˆÌˆÏü ÎyÄÎ˚yÓ˚ ˛õˆÏÓ˚ ≤Ã!ï˛ôù!öï˛
•ˆÏï˛ •ˆÏï˛ Îï˛ê˛y §üÎ˚ ôˆÏÓ˚ ≤Ã!ï˛ôù!öï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ◊Óî§#üyÓ˚ í˛z˛õˆÏÓ˚ ÌyˆÏܲñ §yôyÓ˚îû˛yˆÏÓ ï˛yˆÏܲ xö%Ó˚îö
ܲy° ӈϰ– ï˛ˆÏÓ ~•z §ÇK˛y ˆÌˆÏܲ ˛õ!Ó˚üyîàï˛û˛yˆÏÓ xö%Ó˚îö ܲy° !öî≈Î˚ ܲÓ˚y ÎyÎ˚ öy– xyˆÏü!Ó˚ܲyÓ˚ !ÓK˛yö#
ˆ§!ÓˆÏöÓ˚ à,•#ï˛ §ÇK˛y xö%ÎyÎ˚#ñ ˆÎ §üˆÏÎ˚ ≤Ã!ï˛ôù!öï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ü)° ¢ˆÏ∑Ó˚ 10–6 =î •ˆÏÎ˚ ö%ƒöï˛ü
◊yÓƒ§#üyÎ˚ ˆ˛õÔшÏåÈ˛ ÎyÎ˚ñ ï˛yˆÏܲ•z xö%Ó˚îö ܲy° Ó°y •Î˚ xÌ≈yÍ Îáö ü)° ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ö)ƒöï˛ü ◊yÓƒ§#üyÓ˚
106 =î xÌ≈yÍñ ï˛#Ó ï˛y hflÏÓ˚ 60db !åÈ°ñ ˆ§•z ü%•)ï≈˛ ˆÌˆÏܲ ï˛#Ó ï˛y hflÏÓ˚ 0db ~ öyüyÓ˚ çöƒ Îï˛ê˛y §üÎ˚ °yˆÏà
ˆ§!ê˛•z •° xö%Ó˚îö ܲy°–
(ii) ¢∑ˆÏ¢y£Ïî =îyB˛ ¢∑ˆÏ¢y£Ïî =îyB˛ ¢∑ˆÏ¢y£Ïî =îyB˛ ¢∑ˆÏ¢y£Ïî =îyB˛ ¢∑ˆÏ¢y£Ïî =îyB˛ (Sound absorption coefficient) : ¢∑ï˛Ó˚D Îáö ˆòÄÎ˚y°ñ ˛õò≈yñ ܲyˆÏë˛Ó˚
˛õyê˛yï˛ö ≤Ãû,˛!ï˛Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •Î˚ñ ï˛áö ï˛Ó˚D!ê˛ §¡õ)î≈ ≤Ã!ï˛ú˛!°ï˛ •Î˚ öy– ≤Ã!ï˛ú˛°Ü˛ ï˛° ¢∑ï˛Ó˚ˆÏDÓ˚
!ܲå%Èê˛y ¢!_´ ˆ¢y£Ïî ܲˆÏÓ˚– ¢∑ï˛Ó˚D ˆÜ˛yöÄ ï˛ˆÏ°Ó˚ í˛z˛õÓ˚ °¡∫û˛yˆÏÓ xy˛õ!ï˛ï˛ •ˆÏ° xy˛õ!ï˛ï˛ ¢!_´Ó˚ ˆÎ xÇ¢
˙ ï˛ˆÏ° ˆ¢y!£Ïï˛ •Î˚ ï˛yˆÏܲ ˙ ï˛ˆÏ°Ó˚ ¢∑ˆÏ¢y£Ïî =îyB˛ ӈϰ– xy˛õ!ö !öÿ˛Î˚•z xy®yç ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏåÈö
ˆÎñ ˆáy°y çyöy°y Óy òÓ˚çy ˆÎˆÏ•ï%˛ ï˛Ó˚ˆÏDÓ˚ ˆÜ˛yöÄ ¢!_´•z ≤Ã!ï˛ú˛!°ï˛ ܲˆÏÓ˚ öyñ xï˛~Ó ˆ§=!°Ó˚ ¢∑ˆÏ¢y£Ïî
=îyB˛ 1.0 •ˆÏÓ– ~•z =îyˆÏB˛Ó˚ üyö ~ܲ ~ܲ ôÓ˚ˆÏöÓ˚ ï˛ˆÏ°Ó˚ ˆ«˛ˆÏe ~ܲ ~ܲ Ó˚ܲü •Î˚– xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚
ܲ¡õyˆÏDÓ˚ í˛z˛õˆÏÓ˚Ä ~•z üyö !öû≈˛Ó˚ ܲˆÏÓ˚–
ˆÜ˛yöÄ ~ܲ!ê˛ ï˛ˆÏ°Ó˚ ˆ«˛eú˛°ˆÏܲ ¢∑ˆÏ¢y£Ïî =îyB˛ !òˆÏÎ˚ =î ܲÓ˚ ˙ ï˛ˆÏ°Ó˚ ÚÚˆ¢y£ÏîÛÛ ˛õyÄÎ˚y ÎyÎ˚–
(iii) ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ≠ ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ≠ ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ≠ ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ≠ ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ≠ ˆÜ˛yöÄ Ü˛ˆÏ«˛Ó˚ xö%Ó˚îö ܲy° ܲ«˛!ê˛Ó˚ xyÎ˚ï˛ö ~ÓÇ Ü˛ˆÏ«˛Ó˚ §Ó=!°
ï˛ˆÏ°Ó˚ ˆüyê˛ ˆ¢y£ÏˆÏîÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– xy˛õ!ö ~•z ~ܲˆÏܲ ˛õˆÏÓ˚ ˆòáˆÏÓö ˆÎñ xö%Ó˚îˆÏöÓ˚ ܲy° ܲˆÏ«˛Ó˚
xyÎ˚ï˛ˆÏöÓ˚ §üyö%˛õyï˛# ~ÓÇ ˆüyê˛ ˆ¢y£ÏˆÏîÓ˚ ÓƒyhflÏyö%˛õyï˛#– ˆÎ ܲˆÏ«˛Ó˚ ˆüyê˛ ˆ¢y£Ïî xyÎ˚ï˛ˆÏöÓ˚ ï%˛°öyÎ˚ ܲüñ
ˆ§áyˆÏö xö%Ó˚îö ˆÓ!¢ •Î˚ñ ú˛ˆÏ° ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ˆ§yçy§%!ç xy§y ¢ˆÏ∑Ó˚ ï%˛°öyÎ˚ ˆÓ!¢ •Î˚ñ ¢ˆÏ∑Ó˚ ˆÓ˚¢
xˆÏöܲ«˛î ÌyˆÏܲ– ~ çyï˛#Î˚ ܲ«˛ˆÏܲ ≤ÃyîÓhsˇ ܲ«˛ Ó°y •Î˚– x˛õÓ˚!òˆÏܲñ Î!ò xyÎ˚ï˛ˆÏöÓ˚ ï%˛°öyÎ˚ ˆ¢y£Ïî ˆÓ!¢
•Î˚ ï˛ˆÏÓ ¢ˆÏ∑Ó˚ ˆÓ˚¢ o&ï˛ !ü!°ˆÏÎ˚ ÎyÎ˚– ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ú˛yÑܲy çyÎ˚àyÎ˚ Îï˛ê˛y •ï˛ ≤ÃyÎ˚ ï˛ï˛ê˛y•z •Î˚– ~ ôÓ˚ˆÏöÓ˚
ܲ«˛ˆÏܲ xyüÓ˚y !ö‹±yî ܲ«˛ Ó!°–
(iv) xö%Ü)˛° xö%Ó˚îö ܲy° xö%Ü)˛° xö%Ó˚îö ܲy° xö%Ü)˛° xö%Ó˚îö ܲy° xö%Ü)˛° xö%Ó˚îö ܲy° xö%Ü)˛° xö%Ó˚îö ܲy° (Optimum Re-verberation time) : ˆ≤ë˛yà,ˆÏ• í˛zͲõy!òï˛ ôù!öÓ˚
◊&!ï˛üyô%Î≈ Ó˚«˛y ܲÓ˚yñ ≤Ã!ï˛ú˛°ˆÏöÓ˚ §üÎ˚ ˆ¢y£ÏˆÏîÓ˚ ú˛ˆÏ° ¢∑¢!_´Ó˚ x!ï˛!Ó˚_´ «˛Î˚ ˆÓ˚yô ܲˆÏÓ˚ ¢ˆÏ∑Ó˚ ΈÏÌ‹T
ï˛#Ó ï˛y ÓçyÎ˚ Ó˚yáyñ xyÓyÓ˚ ~ܲ•z §ˆÏD í˛zFã˛y!Ó˚ï˛ ¢∑ñ §D#ï˛ ≤Ãû,˛!ï˛Ó˚ flõ‹Tï˛y Ó˚«˛y ܲÓ˚yÓ˚ çöƒ xyüÓ˚y ã˛y•z
xö%Ó˚îˆÏöÓ˚ ~ܲ!ê˛ üyé˛yüy!é˛ xÓfliy– ~•z §Ó≈y!ôܲ xö%Ó˚îˆÏöÓ˚ ܲy°!ê˛ˆÏܲ•z xyüÓ˚y xö%Ü)˛° xö%Ó˚îö ܲy° Ó!°–
~!ê˛Ó˚ üyö ܲˆÏ«˛Ó˚ xyÎ˚ï˛ö ~ÓÇ Ü˛«˛!ê˛ Ü˛#û˛yˆÏÓ ÓƒÓ•yÓ˚ •ˆÏÓ ~•z ò%•zÈÙÈ~Ó˚ í˛z˛õÓ˚•z !öû≈˛Ó˚ ܲˆÏÓ˚– !fiê˛ˆÏú˛ö Ä
ˆÓê˛ !Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ ܲˆÏ«˛Ó˚ xyò¢≈ xö%Ó˚îö ܲy° !öî≈ˆÏÎ˚Ó˚ çöƒ ˆÎ ≤ÃyˆÏÎ˚y!àܲ §)e ÓƒÓ•yÓ˚ ܲˆÏÓ˚ö ˆ§!ê˛ˆÏܲ
ˆü!ê˛Δܲ ~ܲˆÏܲ ˆ°áy ÎyÎ˚–
T = N (0.0036
13 V
+ 0.107) ˆ§ Ïܲu˛–– ...11.1
291
ˆÎáy Ïö V = m3 ~ܲˆÏܲ ܲˆÏ«˛Ó˚ xyÎ˚ï˛öñ N Ó˚y!¢Ó˚ üyö Ó_,´ï˛y Óy öyê˛Ü˛ñ Îsf§D#ï˛ Ä Ó,®àyö Óy xˆÏÜ≈˛‹T…yÓ˚
ˆ«˛ˆÏe ÎÌye´ˆÏü 4, 5 Ä 6 – ~ܲ!ê˛ í˛zòy•Ó˚î ˆòÄÎ˚y Îyܲ– ôÓ˚&öñ 30m × 20m × 10m = 600m3 xyÎ˚ï˛ˆÏöÓ˚
~ܲ!ê˛ Ü˛«˛ xˆÏÜ≈˛‹T…yÓ˚ çöƒ ÓƒÓ•,ï˛ •ˆÏÓ– 11.1 §)e xö%ÎyÎ˚# ~•z ܲˆÏ«˛Ó˚ xö%Ü)˛° xö%Ó˚îö ܲy° •ˆÏÓ≠
6(0·0036 ×
13 6000
+ 0·107) = 1·03 ˆ§ˆÏܲu˛
xy˛õ!ö !•§yÓ Ü˛ˆÏÓ˚ ˆòˆÏáˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ ~•z ܲ«˛!ê˛Ó˚ xö%Ó˚îö ܲy° 1·3 ˆ§ˆÏܲu˛ •ˆÏ° ~!ê˛ Ó_,´ï˛yÓ˚
çöƒ í˛z˛õÎ%_´ •ï˛–
11.3 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ó,!k˛
!öhflÏ∏˛ âˆÏÓ˚ Îáö ˆÜ˛yöÄ ¢∑ xyÓ˚Ω˛ •Î˚ ï˛áö âˆÏÓ˚Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ˆ§•z ü%•)ˆÏï≈˛•z ˆÜ˛yö fliyÎ˚# üyˆÏö
ˆ˛õÔÑåÈyÎ˚ öy– ï˛#Ó ï˛y ¢)öƒüyö ˆÌˆÏܲ ܲ#û˛yˆÏÓ Ó,!k˛ ˛õyÎ˚ñ ~ÓyÓ˚ xyüÓ˚y ˆ§!ê˛ !öô≈yÓ˚î ܲÓ˚ˆÏï˛ ˆã˛‹Ty ܲÓ˚Ó– üˆÏö
ܲÓ˚y Îyܲñ ˆÜ˛yöÄ âˆÏÓ˚ ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ §ü•yˆÏÓ˚ ¢∑¢!_´ í˛zͲõy!òï˛ •ˆÏFåÈ– âˆÏÓ˚Ó˚ ˆòÄÎ˚y° Óy xöƒ
ï˛ˆÏ° xy˛õ!ï˛ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ !ܲå%È ≤Ã!ï˛ú˛!°ï˛ ~ÓÇ !ܲå%È xӈϢy!£Ïï˛ •ˆÏFåÈ– ÓyÓ˚ÓyÓ˚ ≤Ã!ï˛ú˛!°ï˛ •ÄÎ˚yÓ˚
ú˛ˆÏ° §ühflÏ Ü˛«˛ ç%ˆÏí˛¸ ¢∑¢!_´Ó˚ âöc §üyö ~ÓÇ ¢!_´≤ÃÓy•Ä §Ó!òˆÏܲ §üyöû˛yˆÏÓ •ˆÏFåÈ ÓˆÏ° ôˆÏÓ˚ ˆöÄÎ˚y
ÎyÎ˚–
ˆòÄÎ˚yˆÏ°Ó˚ ˆÜ˛yöÄ ~ܲ fliyˆÏö «%˛o ˆ«˛eú˛° ds ~ÓÇ ï˛y ˆÌˆÏܲ r ò)Ó˚ˆÏc ~ܲ «%˛o xyÎ˚ï˛ö dv ܲ“öy
ܲÓ˚y •°– r ~Ó˚˛!ò¢y ~ÓÇ ds ÈÙÈ ~Ó˚ ï˛ˆÏ°Ó˚ °¡∫!ò¢yÓ˚ üˆÏôƒ ˆÜ˛yî
θ
ñ ˆÎüö !ã˛e 11.1-~ ˆòáy ÎyˆÏFåÈ–
Î!ò âˆÏÓ˚Ó˚ üˆÏôƒ ¢∑ ¢!_´Ó˚ âöc u •Î˚ ï˛y •ˆÏ° dv «%˛o xyÎ˚ï˛ÏˆÏöÓ˚ üˆÏôƒ ¢!_´Ó˚ ˛õ!Ó˚üyî udv •ˆÏÓ–
ˆÎˆÏ•ï%˛ ¢!_´ ≤ÃÓy• §ü˜Ïò!¢Ü˛ (isotropic), ds «%˛o ï˛ˆÏ°Ó˚ í˛z˛õÓ˚ dv xyÎ˚ï˛Ïö ˆÌˆÏܲ ~•z ¢!_´Ó˚
dE =
4udvdω
π
xÇ¢ xy˛õ!ï˛ï˛ •ˆÏÓ–
ˆÎáyˆÏö d
ω
•° ds myÓ˚y dv ˆï˛ §,‹T âö ˆÜ˛yî (solid angle)
~áöñ dv = r2 sin
( dddr θθφφ
= !òàÇ¢ ˆÜ˛yîV
~ÓÇñ d
ω
=
2cos dsr
θ
ܲyÓ˚îñ r !ò¢yÎ˚ ds ~Ó˚ °¡∫ ≤Èϫ˛˛õ (perpendicular projection) •ˆÏFåÈ ds cos
θ
–
!ã˛e ≠ 11.1
292
xï˛~Óñ
dE =
4uπ
r2 sin
θ
d
θ
d
φ
dr
2cos dsr
θ
=
4uds
π
sin
θ
cos
θ
d
θ
d
φ
dr ...11.2
í˛z˛õˆÏÓ˚Ó˚ Ó˚y!¢üy°y!ê˛ •° dv xyÎ˚ï˛ö ˆÌˆÏܲ xy§y ¢!_´Ó˚ ˛õ!Ó˚üy˛õ– Î!ò §ühflÏ Ü˛ˆÏ«˛Ó˚ xyÎ˚ï˛ˆÏöÓ˚ !•§yÓ
ܲÓ˚ˆÏï˛ •Î˚ñ ï˛ˆÏÓ
θ
ˆÜ˛yˆÏî 0 ˆÌˆÏܲ
2π
˛õÎ≈hsˇ ~ÓÇ
φ
ˆÜ˛yî 0 ˆÌˆÏܲ 2
π
˛õÎ≈hsˇ ÓƒyÆ ÓˆÏ° ôÓ˚ˆÏï˛ •ˆÏÓ–
ˆû˛ˆÏÓ ˆòá%ö r ~Ó˚ Óƒy!Æ Ü˛# •ˆÏÓ⁄
r •° ò)Ó˚ˆÏcÓ˚ üy˛õ ~ÓÇ ¢ˆÏ∑Ó˚ à!ï˛ c Ó°ˆÏï˛ ˆÓyé˛yÎ˚ ~ܲܲ §üˆÏÎ˚ ¢∑ myÓ˚y x!ï˛e´yhsˇ ò)Ó˚c– ï˛y•z
r
→
0 ˆÌ Ïܲ c ˛õÎ≈hsˇ ˛õ!Ó˚Óï≈˛ö ܲÓ˚ˆÏ° ~ܲܲ §üˆÏÎ˚ ds-~ xy§y ¢∑¢!_´Ó˚ ˛õ!Ó˚üyî ˛õyÄÎ˚y ÎyˆÏÓ–
§%ï˛Ó˚yÇñ 11.2 §ü#ܲÓ˚îˆÏܲ §üyܲ°ö ܲˆÏÓ˚ ~ܲܲ §üˆÏÎ˚ ds ˆ«˛eú˛ˆÏ° ܲˆÏ«˛Ó˚ ˆüyê˛ xyÎ˚ï˛ö ˆÌˆÏܲ
xy˛õ!ï˛ï˛ ¢∑¢!_´ ˛õyÄÎy ÎyˆÏÓ ≠
E = 22
000
sincos4∫∫∫
c udsdddrππ
θθθφπ
= 1· ·2 ·
4 2uds cππ
=
4cuds ∴
~ܲܲ §üˆÏÎ˚ ~ܲܲ ˆ«˛eú˛ˆÏ° xy˛õ!ï˛ï˛ ¢!_´ •ˆÏÓ
I =
4cu
...11.3
§ÇK˛y xö%ÎyÎ˚# ~!ê˛•z ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y–
ôÓ˚y Îyܲñ ܲˆÏ«˛ !Ó!û˛ß¨ í˛z˛õyòyˆÏö !ö!ü≈ï˛ ï˛° xyˆÏåÈñ ˆÎ=!°Ó˚ ¢∑ ˆ¢y£Ïî =îyB˛ xy°yòy–
Î!ò i ï˛ü ï˛ˆÏ°Ó˚ ¢∑ˆÏ¢y£Ïî =îyB˛
α
i ~ÓÇ ˆ«˛eú˛°
i s δ
•Î˚ñ ï˛ˆÏÓ ˙ ï˛ˆÏ° ~ܲܲ §üˆÏÎ˚
i Is δ
˛õ!Ó˚üyî
¢!_´ xy˛õ!ï˛ï˛ •ˆÏÓ ~ÓÇ ï˛yÓ˚ üˆÏôƒ
ii Is αδ
˛õ!Ó˚üyî ¢!_´ ˆ¢y!£Ïï˛ •ˆÏÓ– §%ï˛Ó˚yÇñ ~ܲܲ §üˆÏÎ˚ ܲˆÏ«˛ ˆüyê˛
ˆ¢y!£Ïï˛ ¢!_´Ó˚ ˛õ!Ó˚üyî ≠
293
Ea =
44 iiiicucua Iss αδαδ ∑=∑=
...11.4
ˆÎáyˆÏö ܲˆÏ«˛Ó˚ §Ó=!° ï˛ˆÏ°Ó˚ ˆüyê˛ ˆ¢y£Ïî ≠
a = i ii
sα δ∑
§¡õ)î≈ âÓ˚!ê˛Ó˚ xyÎ˚ï˛ö Î!ò V •Î˚ ~ÓÇ ï˛yÓ˚ üˆÏôƒ í˛zͧ ˆÌˆÏܲ ≤Ã!ï˛ ~ܲܲ §üˆÏÎ˚ í˛zͲõy!òï˛ ¢!_´ P
•Î˚ñ ï˛ˆÏÓ ¢!_´§ÇÓ˚«˛ˆÏîÓ˚ !öÎ˚ü xö%ÎyÎ˚# âˆÏÓ˚Ó˚ üˆÏôƒ ¢∑¢!_´ Ó,!k˛Ó˚ •yÓ˚ •ˆÏÓñ
¢!_´ í˛zͲõyòˆÏöÓ˚ •yÓ˚ ÈÙÈ ¢!_´ ˆ¢y£ÏˆÏîÓ˚ •yÓ˚˛
xÌ≈yÍñ
()4
dduacu uVVPdtdt
==−
...11.5
Óyñ / 4
du dtP acu V
=−
¢∑ xyÓ˚Ω˛ •ÄÎ˚yÓ˚ ü%•)ˆÏï≈˛ñ xÌ≈≈yÍ Îáö t = 0, ï˛áö u = 0 !åÈ°– t §üÎ˚ ˛õˆÏÓ˚ ¢∑¢!_´Ó˚ âöc u •°–
ï˛y•ˆÏ°
00/4
tuIdu dtVPacu
=− ∫∫
Î!ò P – acu/4 = x ôÓ˚y •Î˚ñ ï˛ˆÏÓ í˛zû˛Î˚!òˆÏܲ §üyܲ°ö ܲˆÏÓ˚ ˛õyÄÎy ÎyÎ˚ñ
/ 4
0
4P acu
t dxv ac x
−
= − ∫
=
4/4 logPacuacP−− ∴
log
/44
PacuactPV
−− =
Óyñ/ 4/ 4 act VP acu e
P−− =
Óyñ u = ( )/ 44 1 act VP eac
−− ...11.6
¢!_´ âöc Óyí˛¸ˆÏï˛ Óyí˛¸ˆÏï˛ ~ܲê˛y í˛zFã˛ï˛ü üyˆÏö (um) ˆ˛õÔÑåȈÏÓ– §%ï˛Ó˚yÇñ t = ∞ •ˆÏ° u = um=
4Pac
294
∴
u = um(1– e–act/4V) ...11.7
§ü#ܲÓ˚î 11.7 ˆÌˆÏܲ âˆÏÓ˚ ¢∑¢!_´Ó˚ âöc Ó,!k˛Ó˚ ≤ÃÜ,˛!ï˛ ˆÓyé˛y ÎyˆÏFåÈ– §ü#ܲÓ˚î 11.3 ˆÌˆÏܲ xyüÓ˚y
˛õy•zñ I =
4cu
, xÌ≈yÍñ ¢!_´ âöc xyÓ˚ ï˛#Ó ï˛y ˛õÓ˚flõÓ˚ §üyö%˛õyï˛#– xï˛~Óñ §ü#ܲÓ˚î 11.7 ˆÌˆÏܲ ˛õyÄÎ˚y
ÎyÎ˚ñ I = Im(1 – e–act/4V) ...11.8
ˆÎáyˆÏö Im =
4m cu
= ï˛#Ó ï˛yÓ˚ §ˆÏÓ≈yFã˛ üyö–
§ü#ܲÓ˚î 11.8 âˆÏÓ˚ ¢∑¢!_´Ó˚ ï˛#Ó ï˛y Ó,!k˛Ó˚ ≤ÃÜ,˛!ï˛
!öˆÏò≈¢ ܲÓ˚ˆÏåÈ– §üˆÏÎ˚Ó˚ §ˆÏD ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲ#û˛yˆÏÓ Óyí˛¸ˆÏÓñ
ï˛y !ã˛e 11.2 ÈÙÈ ~ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ï˛#Ó ï˛y Óyí˛¸ˆÏï˛ Óyí˛¸ˆÏï˛
~ܲê˛y üyˆÏö !àˆÏÎ˚ !fliÓ˚ •ˆÏÎ˚ ÎyÎ˚– ~•z Ó,!k˛Ó˚ •yÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚
âˆÏÓ˚Ó˚ xyÎ˚ï˛ö ~ÓÇ û˛ï˛Ó˚ܲyÓ˚ òÄÎ˚y° •zï˛ƒy!òÓ˚ xӈϢy£Ïî
«˛üï˛yÓ˚ í˛z˛õˆÏÓ˚–
11.4 ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…y§ Ä ˆ§!ÓˆÏöÓ˚ §)e
Î!ò ¢!_´ âöc í˛zFã˛ï˛ü üyˆÏö ˆ˛õÔшÏåÈ ÎyÓyÓ˚ ˛õÓ˚ í˛zͧ!ê˛ˆÏܲ Ó¶˛ ܲˆÏÓ˚ ˆòÄÎ˚y •Î˚ñ ï˛áöÄ Ü˛ˆÏ«˛Ó˚ !Ó!û˛ß¨
ï˛ˆÏ° ¢∑ ¢!_´ ˆ¢y!£Ïï˛ •ˆÏï˛ ÌyܲˆÏÓ– ~Ó˚ ú˛ˆÏ° ï˛áö ¢!_´ âöc ܲüˆÏï˛ ÌyܲˆÏÓ– 11.5 xÓܲ° §ü#ܲÓ˚ˆÏî
P = 0 Ó§y Ï°
V
4duacudt
=−
Óyñ 4du dtacu V
= −
í˛zû˛Î˚ !òܲ §üyܲ°ö ܲÓ˚ˆÏ° ˛õy•zñ
logu = ,4act AV
− + ˆÎáyˆÏö A = §üyܲ°ö ô &Óܲ–
~ˆÏ«˛ˆÏe Îáö t = 0, u = um, §%ï˛Ó˚yÇñ A = logum
295
∴
logu =
4act
V−
+ logum
Óyñ u = ume–act/4V ...11.9
˛õ)ˆÏÓ≈Ó˚ üï˛ñ ˆÎˆÏ•ï%˛ I = ,4cu
Im =
4m cu
I = Ime–act/4V ...11.10
§ü#ܲÓ˚î 11.10 ˆÌˆÏܲ ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ˆÜ˛yöÄ Ók˛ âˆÏÓ˚ ¢∑¢!_´ !ܲå%È«˛î í zͲõy!òï˛ • ÏÎ˚ Ì Ïü ÎyÄÎ˚yÓ˚
˛õÓ˚ ï˛#Ó ï˛y í˛zFã˛ï˛ü üyö Im ̈Ïܲ §)ã˛Ü˛ (exponential) •yˆÏÓ˚
ܲüˆÏï˛ ÌyܲˆÏÓ– ܲ#û˛yˆÏÓ §üˆÏÎ˚Ó˚ §ˆÏD ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲüˆÏÓñ
ï˛y !ã˛e 11.3-~ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
í˛zͧ Ó¶˛ ܲˆÏÓ˚ !òˆÏ° ï˛Í«˛îyÍ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ¢)öƒ •ˆÏÎ˚
ÎyÎ˚ öy– Ü˛ï˛ o&ï˛ Óy ô#ˆÏÓ˚ ï˛y ܲüˆÏÓñ ï˛y !öû≈˛Ó˚ ܲˆÏÓ˚ âˆÏÓ˚Ó˚
xyÎ˚ï˛ö ~ÓÇ ˆû˛ï˛Ó˚ܲyÓ˚ ˆòÄÎ˚y° ≤Ãû,˛!ï˛Ó˚ ˆ¢y£ÏˆÏîÓ˚ í˛z˛õˆÏÓ˚–
~ÓyÓ˚ ôÓ˚&öñ !öhflÏ∏˛ ܲˆÏ«˛ ¢ˆÏ∑Ó˚ í˛zͧ ã˛y°% •ÄÎ˚yÓ˚
!ܲå%È«˛î ˛õÓ˚ Îáö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Óy ¢!_´ âöc §ˆÏÓ≈yFã˛ üyˆÏö
ˆ˛õÔшÏåȈÏåÈñ ï˛áö í˛zͧ!ê˛ Ó¶˛ ܲˆÏÓ˚ ˆòÄÎ˚y •°– ~áö ¢ˆÏ∑Ó˚
ï˛#Ó ï˛yÓ˚ ˆÎ ˛õ!Ó˚Óï≈˛ö •ˆÏÓ ï˛y 11.4 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚ÓˆÏï≈˛ ¢!_´ âöˆÏcÓ˚ ˆ°á!ã˛ˆÏe
xÑyܲˆÏ° ˆ§!ê˛ ~ܲ•zÓ˚ܲü ˆòáyï˛–
11.9 §ü#ܲÓ˚î ˆÌˆÏܲ xyüÓ˚y xö%Ó˚îö
ܲyˆÏ°Ó˚ ~ܲ!ê˛ Ó˚y!¢üy°y ˆ˛õˆÏï˛ ˛õy!Ó˚– ˙
§ü#ܲÓ˚î ˆÌˆÏܲ um/u = eact/4V xy˛õ!ö
xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ ¢!_´ âöc 106 =î
˛õ!Ó˚üyˆÏî •…y§ ˆ˛õˆÏÎ˚ ö)ƒöï˛ü ◊yÓƒ§#üyÎ˚
ˆ˛õÔÑåȈÏöyÓ˚ ˆÎ §üÎ˚ñ ˆ§ê˛y•z xö%Ó˚îö ܲy°
TR–
∴
eacTR/4V= 106
Óyñ acTR/4V = In(106) = 6 × 2·303
296
Óyñ TR=
2·30364Vca××
Î!ò ¢ˆÏ∑Ó˚ à!ï˛ 343ms–1 ôÓ˚y •Î˚ ~ÓÇ V Ä a ÎÌye´ˆÏü m3 Ä m2 ~ܲˆÏܲ üy˛õy •Î˚ ï˛ˆÏÓ
TR=
0·161Va
...11.11
~!ê˛•z ˆ§!ÓˆÏöÓ˚ §)e–
ˆ¢y£ÏˆÏîÓ˚ ~ú˛ !˛õ ~§ ~ܲˆÏܲÓ˚ öyü ˆ§!ÓˆÏöÓ˚ öyüyö%§yˆÏÓ˚ Sabin-•z ˆòÄÎ˚y •ˆÏÎ˚!åÈ°– ˆáy°y çyöy°y
Óy Ä•z çyï˛#Î˚ xyò¢≈ ˆ¢y£ÏˆÏܲÓ˚ ~ܲ Óà≈ú%˛ê˛ ˆ«˛eú˛ˆÏ°Ó˚ ˆ¢y£ÏîˆÏܲ Ú~ܲ ˆ§!ÓöÛ (Sabin) Ó°y •Î˚– ï˛ˆÏÓ
xyüÓ˚y xyò¢≈ ˆ¢y£ÏˆÏܲÓ˚ ~ܲ Óà≈!üê˛yÓ˚ ˆ«˛eú˛ˆÏ°Ó˚ ˆ¢y£ÏîˆÏܲ•z ~ܲܲ !•§yˆÏÓ ôÓ˚Ó–
ˆ§!ÓˆÏöÓ˚ §)e!ê˛Ó˚ §y•yˆÏ΃ ˆÜ˛yöÄ Ü˛ˆÏ«˛Ó˚ xyÎ˚ï˛ö ~ÓÇ §Ó=!° ï˛ˆÏ°Ó˚ ˆ«˛eú˛° Ä ¢∑ˆÏ¢y£Ïî =îyB˛
çyöy ÌyܲˆÏ° ï˛yÓ˚ ˆÌˆÏܲ xy˛õ!ö xö%Ó˚îö ܲy° !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~•z §)e!ê˛ !ܲv e&!ê˛ü%_´ öÎ˚– ~!ê˛
≤Ã!ï˛¤˛y ܲÓ˚ˆÏï˛ !àˆÏÎ˚ xyüÓ˚y ܲˆÏÎ˚ܲ!ê˛ !Ó£ÏÎ˚ ôˆÏÓ˚ !öˆÏÎ˚!åÈ Îy !ë˛Ü˛ öyÄ •ˆÏï˛ ˛õyˆÏÓ˚– ˆòáy Îyܲ ~=!° ܲ#
ܲ#–
(i) í˛z͈ϧÓ˚ ¢∑¢!_´ í˛zͲõyòˆÏöÓ˚ •yÓ˚ §¡õ)î≈ !fliÓ˚ ~ÓÇ Ü˛ˆÏ«˛Ó˚ ï˛#Ó ï˛yÓ˚ §ˆÏD §¡õÜ≈˛•#ö–
(ii) ܲˆÏ«˛Ó˚ §Ó xLjϢ ¢∑¢!_´ §üû˛yˆÏÓ Ó!rê˛ï˛ ~ÓÇ ¢∑¢!_´Ó˚ ≤ÃÓy• §¡õ)î≈ §ü˜Ïò!¢Ü˛–
(iii) ¢∑ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ˆÜ˛yö Ä ≤Ãû˛yÓ ÌyܲˆÏÓ öy–
(iv) ¢ˆÏ∑Ó˚ ˆ¢y£Ïî ˆÜ˛Ó°üye˛ ≤Ã!ï˛ú˛°ˆÏöÓ˚ §üÎ˚•z âê˛ˆÏÓñ üyôƒˆÏü xÌ≈yÍ ÓyÎ˚%ˆÏï˛ ˆÜ˛yöÄ ˆ¢y£Ïî•z âê˛ˆÏÓ
öy–
(v) ï˛°=!°Ó˚ ˆ¢y£Ïî =îyB˛ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ Ä ï˛#Ó ï˛yÓ˚ í˛z˛õÓ˚ ~ˆÏܲÓyˆÏÓ˚•z !öû≈˛Ó˚¢#° öÎ˚–
ÓyhflψÏÓ §D#ï˛ñ öyê˛Ü˛ Óy Ó_,´ï˛y ˆÜ˛yöÄ!ê˛ˆÏï˛•z ¢∑¢!_´ §ü•yˆÏÓ˚ í˛zͲõߨ •Î˚ öy– §yüƒyÓfliyÎ˚ ¢ˆÏ∑Ó˚
í˛z˛õ!Ó˚˛õyï˛ö âê˛ˆÏÓ ~ÓÇ fliyî%ï˛Ó˚D ˜ï˛!Ó˚ •ˆÏÓ– ÎyÓ˚ ú˛ˆÏ° ¢∑¢!_´Ó˚ §%£Ïü Órê˛ö âê˛ˆÏÓ öy– ÓyÎ˚%ˆÏï˛ ¢∑ï˛Ó˚ˆÏDÓ˚
ˆ¢y£Ïî âˆÏê˛ ~ÓÇ ˆ¢y£ÏˆÏîÓ˚ üyey¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ §ˆÏD Ó,!k˛ ˛õyÎ˚– !ӈϢ£Ïï˛ñ Óí˛¸ xyܲyˆÏÓ˚Ó˚ ˆ≤ë˛yà,ˆÏ•Ó˚
ˆ«˛ˆÏe ÓyÎ˚%üyôƒˆÏü ¢ˆÏ∑Ó˚ ˆ¢y£Ïî ˆüyˆÏê˛•z í˛zˆÏ˛õ«˛y ܲÓ˚y ÎyÎ˚ öy– ~åÈyí˛¸y Î!ò ˆÜ˛yö ܲˆÏ«˛Ó˚ ≤Ã!ï˛!ê˛ ï˛ˆÏ°Ó˚
¢∑ˆÏ¢y£Ïî =îyB˛
α
= 1 •Î˚ñ ï˛áö a Ó˚y!¢!ê˛ §Ó=!° ï˛ˆÏ°Ó˚ ˆüyê˛ ˆ«˛eú˛ˆÏ°Ó˚ §üyö •ˆÏÓ ~ÓÇ ˆ§ˆÏ«˛ˆÏeÄ
!ܲå%Èê˛y xö%Ó˚îö ܲy° ˛õyÄÎ˚y ÎyˆÏÓ– !ܲv ÓyhflψÏÓ Ü˛# •ˆÏÓ ï˛y ˆû˛ˆÏÓ ˆòá%ö– ~ xÓfliyÎ˚ ¢ˆÏ∑Ó˚ ˆÜ˛yöÄ
≤Ã!ï˛ú˛°ö•z âê˛ˆÏÓ öy ~ÓÇ xö%Ó˚îö ~ˆÏܲÓyˆÏÓ˚•z ÌyܲˆÏÓ öy–
xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ˆ§!ÓˆÏöÓ˚ §)e!ê˛ §Ó xÓfliyÎ˚ áyê˛ˆÏÓ öy– xy§ˆÏ° ܲˆÏ«˛Ó˚ üy˛õ ¢ˆÏ∑Ó˚
ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ ï%˛ö°yÎ˚ xˆÏöܲ Óí˛¸ ~ÓÇ ˆ¢y£ÏˆÏîÓ˚ ˛õ!Ó˚üyî ܲü •ˆÏ° (
α
i < 0·2), ï˛ˆÏÓ•z ˆ§!ÓˆÏöÓ˚ §)e ÓƒÓ•yÓ˚
ܲÓ˚y ÎyÎ˚–
~ÓyˆÏÓ˚ ˆòáy Îyܲ !ö‹±yî ܲˆÏ«˛ñ ˆÎáyˆÏö ˆ¢y£ÏˆÏîÓ˚ üyö á%Ó•z ˆÓ!¢ñ ˆ§áyˆÏö ܲ#û˛yˆÏÓ xö%Ó˚îö ܲy° !öî≈Î˚
ܲÓ˚y ÎyÎ˚–
297
11.5 !ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç!ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç!ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç!ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç!ö‹±yî ܲˆÏ«˛ xö%Ó˚îö Ä xy•z!Ó˚Ç (Eyring)-~Ó˚ §)e~Ó˚ §)e~Ó˚ §)e~Ó˚ §)e~Ó˚ §)e
xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈöñ ˆÎ §Ó ܲˆÏ«˛ ˆ¢y£ÏˆÏîÓ˚ üyö ˆÓ!¢ñ ˆ§=!°Ó˚ ˆ«˛ˆÏe ˆ§!ÓˆÏöÓ˚ §)e ÓƒÓ•yÓ˚
ܲÓ˚y ÎyÎ˚ öy– ~ ôÓ˚ˆÏöÓ˚ ܲˆÏ«˛Ó˚ çöƒ xy•z!Ó˚Ç !ܲå%Èê˛y !Ó!û˛ß¨ ˛õk˛!ï˛ˆÏï˛ xö%Ó˚îö ܲyˆÏ°Ó˚ §ˆÏD ܲˆÏ«˛Ó˚ xyÎ˚ï˛öñ
ï˛°=!°Ó˚ ˆüyê˛ ˆ«˛eú˛° ~ÓÇ ˆ§=!°Ó˚ àí˛¸ ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ §¡õÜ≈˛ !öî≈Î˚ ܲˆÏÓ˚ö–
!ö‹±yî ܲˆÏ«˛ ï˛#Ó ï˛yÓ˚ !•§yÓ Ü˛Ó˚ˆÏï˛ ˆàˆÏ° ÓyÓ˚ÓyÓ˚ ≤Ã!ï˛ú˛°ˆÏö ¢∑ï˛Ó˚ˆÏDÓ˚ ˆ¢y£Ïî !ÓˆÏÓã˛öy ܲÓ˚ˆÏï˛ •ˆÏÓ–
üˆÏö ܲÓ˚y Îyܲñ ܲˆÏ«˛Ó˚ ï˛°=!°Ó˚ àí˛¸ ¢∑ˆÏ¢y£Ïî =îyB˛
α
– ï˛y •ˆÏ° ≤Ã!ï˛ÓyÓ˚ ¢∑ xy˛õ!ï˛ï˛ •ÄÎ˚yÓ˚ ˛õÓ˚
ï˛#Ó ï˛yÓ˚
α
xÇ¢ ˆ¢y!£Ïï˛ •ˆÏÓ ~ÓÇ (1 –
α
) xÇ¢ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÓ–
xï˛~Óñ ≤ÃyÓ˚!Ω˛Ü˛ ï˛#Ó ï˛y Î!ò I0 •Î˚ñ ≤ÃÌüñ !mï˛#Î˚ñ ï,˛ï˛#Î˚.......•zï˛ƒy!ò ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õˆÏÓ˚ ï˛#Ó ï˛y •ˆÏÓ
I0 (1 –
α
), I0(1 –
α
)2, I0(1 –
α
)3......•zï˛ƒy!ò–
ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ ¢ˆÏ∑Ó˚ í˛zͧ Ó¶˛ ܲˆÏÓ˚ !òˆÏ° n §ÇáƒÜ˛ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •ˆÏÓ
I = I0(1 –
α
)n ...11.12
Î!ò ~!ê˛ ö)ƒöï˛ü ◊yÓƒ§#üy •Î˚ñ ï˛ˆÏÓ xö%Ó˚îö ܲyˆÏ°Ó˚ §ÇK˛y xö%ÎyÎ˚# ≠
I/I0= 10–6
§ü#ܲÓ˚î 11.12 ˆÌˆÏܲ (1 – α
)n,= 10–6
Óyñ nlog(l –
α
) = –2·303 × 6
Óyñ n =
()2·3036
log1α−×
−
...11.13
§%!ÓôyÓ˚ çöƒ ˆÜ˛Ó° ã˛yÓ˚ ˆòÄÎ˚y° ˆÌˆÏܲ ≤Ã!ï˛ú˛°ö•z !ÓˆÏÓã˛öy ܲÓ˚y •ˆÏFåÈ– ò%•z ≤Ã!ï˛ú˛°ˆÏöÓ˚ üˆÏôƒ ¢∑
myÓ˚y x!ï˛e´yhsˇ àí˛¸ ò)Ó˚c •ˆÏÓ≠
/ 4Vb
S= ...11.14
ˆÎáyˆÏö V = âˆÏÓ˚Ó˚ xyÎ˚ï˛ö ~ÓÇ S = ã˛yÓ˚ ˆòÄÎ˚yˆÏ°Ó˚ ˆüyê˛ ˆ«˛eú˛°– n §ÇáƒÜ˛ ≤Ã!ï˛ú˛°ˆÏö x!ï˛e´yhsˇ
ò)Ó˚c = ¢ˆÏ∑Ó˚ à!ï˛ X xö%Ó˚îö ܲy°
Óyñ nb = cTR ...11.15
ܲyÓ˚îñ n §ÇáƒÜ˛ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ö)ƒöï˛ü ◊yÓƒ§#üyÎ˚ ˆ˛õÔÑåÈyˆÏFåÈ–
298
§ü#ܲÓ˚î 11.13, 11.14 ~ÓÇ 11.15 xö%§yˆÏÓ˚
n = cTR · ( )2·303 6
4 log 1SV α
− ×=−
¢ˆÏ∑Ó˚ à!ï˛ 343ms–1 ôÓ˚ˆÏ° TR = ( )
0·161V
log 1⎡ ⎤− −⎣ ⎦S α...11.16
§ü#ܲÓ˚î 11.26 ˆÜ˛ xy•z!Ó˚ÇÈÙÈ~Ó˚ §)e ӈϰ– ~!ê˛ !ö‹±yî ܲˆÏ«˛ xö%Ó˚îö ܲyˆÏ°Ó˚ àîöy ܲÓ˚ˆÏï˛ §y•y΃
ܲˆÏÓ˚– ò%!ê˛ xÓfliy ܲ“öy ܲÓ˚y Îyܲ–
(i) Îáö ˆ¢y£Ïî á%Ó ˆÓ!¢ xÌ≈yÍ
α
~Ó˚ üyö 1-~Ó˚ ܲyåÈyܲy!åÈñ ï˛áö ≠ –log(1 –
α
) Óy log
11α−
á%Ó Óí˛¸ñ ú˛ˆÏ° TR á%Ó ˆåÈyê˛– ˆ§ ˆ«˛ˆÏe xö%Ó˚îö öàîƒ Óy ܲ«˛!ê˛ !ö‹±yî–
(ii) Îáö ˆ¢y£Ïî á%Ó Ü˛ü ≠
log(1 –
α
) = –
α23
.....23αα −−−
– α( α- ~Ó˚ í˛zFã˛ï˛Ó˚ âyï˛=!° öàîƒ ôˆÏÓ˚V
ˆ§ ˆ«˛ˆÏe TR=
0·161VSα
S
α
= ˆüyê˛ xӈϢy£Ïîñ §%ï˛Ó˚yÇ í˛z˛õˆÏÓ˚Ó˚ §)e!ê˛ ˆ§!ÓˆÏöÓ˚ §)ˆÏeÓ˚ xö%Ó˚*˛õ– ~•z xÓfliyÎ˚ ܲ«˛!ê˛ ≤ÃyîÓhsˇ
ӈϰ !ÓˆÏÓ!ã˛ï˛ •ˆÏÓ–
§%ï˛Ó˚yÇñ xyüÓ˚y ˆòá°yü xy•z!Ó˚Ç ~Ó˚ §)e ˆÌˆÏܲ ≤ÃyîÓhsˇ Ä !ö‹±yî ܲ«˛ ÈÙÙÙÈ ò%•zÈÙÈ~Ó˚•z xö%Ó˚îö ܲy°
!öôy≈Ó˚î ܲÓ˚y ÎyÎ˚–
!ü!°Çê˛ö xy•z!Ó˚Ç ÈÙÈ ~Ó˚ §)ˆÏeÓ˚ §ÇˆÏ¢yôö ܲˆÏÓ˚ ˆÎ §)e!ê˛Ó˚ ≤ÃhflÏyÓ Ü˛ˆÏÓ˚ˆÏåÈö ˆ§!ê˛ •° ≠
TR =
()0·05Vlog1
−∑− ii Sα
~áyˆÏö Si Ä
α
i •° iÈÙÈï˛ü ï˛ˆÏ°Ó˚ ˆ«˛eú˛° Ä ¢∑ˆÏ¢y£Ïî =îyB˛–
Îáö §Ó
α
i «%˛o üyˆÏöÓ˚ñ ï˛áö ~•z §)e!ê˛ ˆ§!ÓˆÏöÓ˚ §)ˆÏe ˛õ!Ó˚îï˛ •Î˚– !ܲv Îáö
α
iÈÙÈ ~Ó˚ üyö=!°
Óí˛¸ •Î˚ñ ï˛áö ~•z §)e!ê˛Ó˚ §ˆÏD ˛õÓ˚#«˛y°∏˛ üyˆÏöÓ˚ û˛y° §D!ï˛ ˆòáy ÎyÎ˚–
299
~ÓyÓ˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ üyôƒˆÏü xö%Ó˚îö ܲy° §¡∫ˆÏ¶˛ Îy ˛õí˛¸ˆÏ°ö ï˛y ~ܲÓyÓ˚ é˛y!°ˆÏÎ˚ !öˆÏï˛ ˛õyˆÏÓ˚ö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 1 :
(i) §!ë˛Ü ¢∑ ˆÓˆÏåÈ !öˆÏÎ˚ ¢)öƒfliyö ˛õ)î≈ ܲÓ˚&ö ≠
(a) ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliyÎ˚ §ÓˆÏã˛ˆÏÎ˚ =Ó˚&c˛õ)î≈ !Ó£ÏÎ˚ •°––––––––– Sܲ¡õyB˛† xö%Ó˚îö†ï˛#Ó ï˛yV
(b) ˆ≤ë˛yà,ˆÏ•Ó˚ àë˛ö ~üö •ÄÎ˚y í˛z!ã˛ï˛ ÎyˆÏï˛ xö%Ó˚îö ܲy° xö%Ü)˛° xö%Ó˚îö ܲyˆÏ°Ó˚ ———
—•Î˚– Sܲü†ˆÓ!¢†§üyöV
(c) ˆ≤ë˛yà,ˆÏ•Ó˚ ˆüˆÏé˛ˆÏï˛ ˆüyê˛y ܲyˆÏ˛õ≈ê˛ ˛õyï˛y •ˆÏ° xö%Ó˚îö ܲy°————– SÓyí˛¸ˆÏÓ†§üyö
ÌyܲˆÏӆܲüˆÏÓV
(d) Îáö ˆ¢y£ÏˆÏîÓ˚ üyey á%Ó Ü˛ü ÌyˆÏܲñ ï˛áö ˆ§!ÓˆÏöÓ˚ §)e Ä xy•z!Ó˚Ç ÈÙÈ ~Ó˚ §)ˆÏeÓ˚ üˆÏôƒ §!ë˛Ü˛
§)e •° ————– S÷ô% ≤ÃÌü!ê˛†÷ô% !mï˛#Î˚!ê†í˛zû˛Î˚•zV
(e) xöƒ !Ó£ÏÎ˚=!° x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyܲˆÏ° xö%Ó˚îö ܲy° ˆ≤ë˛yà,ˆÏ•Ó˚ xyÎ˚ï˛ˆÏöÓ˚———— •Î˚–
S§üyö%˛õyï˛#†ÓƒyhflÏyö%˛õyï˛#†ÓˆÏà≈Ó˚ §üyö%˛õyï˛#V
(ii) ~ܲ!ê˛ •°âˆÏÓ˚Ó˚ xyÎ˚ï˛ö 10000m3 ~ÓÇ ï˛yÓ˚ §Ó=!° ï˛ˆÏ°Ó˚ ˆüyê˛ ˆ¢y£Ïî 300 ~ܲܲ S~ܲܲ =
Im2 ˛õ)î≈ ˆ¢y£ÏîܲyÓ˚# ï˛°V– ˙ âˆÏÓ˚Ó˚ xö%Ó˚îö ܲy° Ü˛ï˛ •ˆÏÓ⁄ ~áö ˙ âˆÏÓ˚ Ü˛ï˛ çö ò¢≈ܲ ≤ÈÏÓ¢ ܲÓ˚ˆÏ°
xö%Ó˚îö ܲy° xˆÏô≈ܲ •ˆÏÓ⁄ ≤ÈÏï˛ƒÜ˛ ò¢≈ˆÏܲÓ˚ ˆ¢y£Ïî àˆÏí˛¸ 0·4 ~ܲܲ ôˆÏÓ˚ !öö–
11.6 ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ¢∑ˆÏ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ
xö%Ó˚îö ܲy°ˆÏܲ ˆ≤ë˛yà,ˆÏ•Ó˚ xyÎ˚ï˛ö ~ÓÇ ï˛yÓ˚ §Ω˛yÓƒ ÓƒÓ•yÓ˚ xö%ÎyÎ˚# §ÓˆÏã˛ˆÏÎ˚ í˛z˛õˆÏÎyà# üyˆÏö Ó˚yáyÓ˚
≤ÈÏÎ˚yçö#Î˚ï˛y xy˛õ!ö !öÿ˛Î˚•z í˛z˛õ°!∏˛ ܲˆÏÓ˚ˆÏåÈö– xö%Ó˚îöˆÏܲ ΈÏÌy˛õÎ%_´ xÓfliyö Ó˚yáyÓ˚ çöƒ åÈyò Ä
ˆòÄÎ˚yˆÏ°Ó˚ ï˛°=!°Ó˚ ¢∑ˆÏ¢y£Ïî =îyB˛ !öô≈yÓ˚î ܲÓ˚yÓ˚ ≤ÈÏÎ˚yçö •Î˚– !Ó!û˛ß¨ ˛õk˛!ï˛ˆÏï˛ ï˛y ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
~ܲ!ê˛ ≤Ãã˛!°ï˛ ˛õk˛!ï˛ •° ~ܲ!ê˛ Ü˛ˆÏ«˛ ò%!ê˛ í˛zͧ ˆÌˆÏܲ í˛zͲõy!òï˛ ¢ˆÏ∑Ó˚ xö%Ó˚îö ܲyˆÏ°Ó˚ ï%˛°öy–
üˆÏö ܲÓ˚y Îyܲñ P1 Ä P2 ò%!ê˛ ¢∑ í˛z͈ϧÓ˚ «˛üï˛y ~ÓÇ ~ˆÏòÓ˚ xö%˛õyï˛ çyöy xyˆÏåÈ– ≤ÃÌü!ê˛ ˆÌˆÏܲ
ˆÜ˛yöÄ xö%Ó˚îö ܲˆÏ«˛ ¢∑ í˛zͲõyòö ܲÓ˚y •°– ˆÓ¢ !ܲå%È«˛î ܲˆÏÓ˚ñ Îáö ¢∑ ¢!_´Ó˚ âöc !ö!ÿ˛ï˛ û˛yˆÏÓ
í˛zFã˛ï˛ü üyˆÏö ˆ˛õÔшÏåÈ ˆàˆÏåÈñ ï˛áö í˛zͧ Ó¶˛ ܲÓ˚y •° ~ÓÇ ¢∑ ö)ƒöï˛ü ◊yÓƒ§#üyÎ˚ ˆ˛õÔÑåȈÏöyÓ˚ §üÎ˚ T1
üy˛õy •°– 11.7 §ü#ܲÓ˚î xö%ÎyÎ˚# í˛zFã˛ï˛ü ¢!_´ âöc 4P1/ac ˆÎáyˆÏö a xû˛ƒhsˇÓ˚#î åÈyò Ä ˆòÄÎ˚yˆÏ°Ó˚
300
ˆüyê˛ ˆ¢y£Ïî ~ÓÇ c ¢ˆÏ∑Ó˚ ˆÓà– ~ܲ•z ˛õÓ˚#«˛y !mï˛#Î˚ í˛zͧ!ê˛ !öˆÏÎ˚Ä Ü˛Ó˚y •° ~ÓÇ §üÎ˚ T2 üy˛õy •°–
~áö 11.7 §ü#ܲÓ˚î xö%§yˆÏÓ˚ñ
12 /4/4 12 44 acTVacTV PP eeacac
−− =
ܲyÓ˚î T1 §üÎ˚ ÓyˆÏò ≤ÃÌü!ê˛ ~ÓÇ T2 §üÎ˚ ÓyˆÏò !mï˛#Î˚!ê˛ ö)ƒöï˛ü ◊yÓƒ§#üyÎ˚ ˆ˛õÔшÏåȈÏåÈ– í˛zû˛ˆÏÎ˚Ó˚ ˆ«˛ˆÏe•z
ï˛áö ï˛#Ó ï˛y §üyö– í˛zû˛ˆÏÎ˚Ó˚ ˆ«˛ˆÏe•z ܲ«˛!ê˛ ~ܲ•zñ ÎyÓ˚ xyÎ˚ï˛ö V–
∴
() 12 14
2
acTTV Pe
P−
=
Óyñ In ( )11 24
P ac T TP V
⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠
∴ a =
()12
12
4(/)−
VInPPcTT
...11.17
í˛z˛õˆÏÓ˚Ó˚ §ü#ܲÓ˚î ˆÌˆÏܲ ܲˆÏ«˛Ó˚ ˆüyê˛ ˆ¢y£Ïî ÓyÓ˚ ܲÓ˚y ÎyÎ˚– !ܲv ܲˆÏ«˛Ó˚ xû˛ƒhsˇÓ˚#î ˆòÄÎ˚y° Óy åÈyˆÏòÓ˚
í˛z˛õyòyˆÏöÓ˚ ˆ¢y£Ïî =îyB˛ ÓyÓ˚ ܲÓ˚y ÎyˆÏÓ öy– ï˛yÓ˚ çöƒ 11.4 §ü#ܲÓ˚ˆÏîÓ˚ §y•y΃ !öˆÏï˛ •ˆÏÓ– àí˛¸ ˆ¢y£Ïî
=îyB˛ ≠
i i
i
ds ads S
αα ∑= =∑
Óyñ a = α S ...11.18
ˆÎáyˆÏö S ˆüyê˛ xû˛ƒhsˇÓ˚#î ï˛°=!°Ó˚ ˆ«˛eú˛°– §ü#ܲÓ˚î 11.17 Ä 11.18 ˆÌˆÏܲ ˛õy•zñ
()12
12
4(/ VInPP acSTT
=−
...11.19
§ü#ܲÓ˚î 11.19 ˆÌˆÏܲ àí˛¸ ˆ¢y£Ïî =îyB˛ ÓyÓ˚ ܲÓ˚y ÎyˆÏÓ– §yÓ˚!î 11.2-~ ܲˆÏÎ˚ܲ!ê˛ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî
=îyB˛ ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ–
301
§yÓ˚!î §yÓ˚!î §yÓ˚!î §yÓ˚!î §yÓ˚!î 11.2 !Ó!û˛ß¨ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî =îyB˛!Ó!û˛ß¨ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî =îyB˛!Ó!û˛ß¨ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî =îyB˛!Ó!û˛ß¨ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî =îyB˛!Ó!û˛ß¨ í˛z˛õyòyˆÏöÓ˚ ¢∑ˆÏ¢y£Ïî =îyB˛
ï˛ˆÏ°Ó˚ í˛z˛õyòyöï˛ˆÏ°Ó˚ í˛z˛õyòyöï˛ˆÏ°Ó˚ í˛z˛õyòyöï˛ˆÏ°Ó˚ í˛z˛õyòyöï˛ˆÏ°Ó˚ í˛z˛õyòyö ˆ¢y£Ïî =îyB˛ˆ¢y£Ïî =îyB˛ˆ¢y£Ïî =îyB˛ˆ¢y£Ïî =îyB˛ˆ¢y£Ïî =îyB˛
(500 Hz ܲ¡õyˆÏB˛V
ˆáy°y òÓ˚çy†çyöy°y 1.00
âö ò¢≈ܲ §ü!‹T 0.96
û˛yÓ˚# ˛õò≈y 0.50
ܲyˆÏ˛õ≈ê˛ S˛õyï˛°y ˆÌˆÏܲ û˛yÓ˚#V 0.1 - 0.4
§yôyÓ˚î •zˆÏê˛Ó˚ ˆòÄÎ˚y° 0.03
˛õy•zö ܲyˆÏë˛Ó˚ ˛õƒyˆÏö° 0.10
!ӈϢ£Ï ü§,î ≤’yfiê˛yÓ˚ 0.2 - 0.4
çyöy°yÓ˚ ܲyã˛ 0.05
ܲÇ!e´ê˛†ˆüyçy•zܲ ˆüˆÏé˛ 0.02
¢∑ˆÏ¢y£Ïî !ӈϢ£Ï ˛õƒyˆÏö° 0.50
âˆÏÓ˚Ó˚ ˆòÄÎ˚y°ñ ˆüˆÏé˛ Ä åÈyò ˆÌˆÏܲ ¢ˆÏ∑Ó˚ ≤Ã!ï˛ú˛°ö ܲüyˆÏï˛ ≤ÃyÎ˚•z û˛yÓ˚# ˛õò≈yñ ܲyˆÏ˛õ≈ê˛ñ §!åÈo ≤’y•zí˛zˆÏí˛Ó˚
˛õƒyˆÏö° ≤Ãû,˛!ï˛Ó˚ ÓƒÓ•yÓ˚ •Î˚– ~Ó˚ ܲyÓ˚î xy˛õ!ö 11.2 §yÓ˚î#!ê˛ °«˛ƒ ܲÓ˚ˆÏ°•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
11.7 ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢yˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢yˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢yˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢yˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑!ÓK˛yö§ß√ï˛ öܲ¢y (Acoustic design of auditorium)
ˆ≤ë˛yà,ˆÏ•Ó˚ àë˛ö ˛õ!Ó˚ܲ“öyÎ˚ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ ≤Ãyôyöƒ ˛õyÎ˚ xö%Ó˚îö– ~ ˛õÎ≈hsˇ ï˛yÓ˚ !ÓhflÏy!Ó˚ï˛ !ÓÓÓ˚î ˆòÄÎ˚y
•°– ï˛ˆÏÓ xö%Ó˚îö åÈyí˛¸yÄ xyÓ˚Ä Ü˛ˆÏÎ˚ܲ!ê˛ !Ó£ÏÎ˚ !ÓˆÏÓã˛öy ܲÓ˚ˆÏï˛ •Î˚– ~ÓyÓ˚ ˆòáy Îyܲ ˆ§=!° ܲ#–
(i) §%£Ïü ï˛#Ó ï˛y §%£Ïü ï˛#Ó ï˛y §%£Ïü ï˛#Ó ï˛y §%£Ïü ï˛#Ó ï˛y §%£Ïü ï˛#Ó ï˛y (uniform intensity) : ¢∑ ÎyˆÏï˛ ˆ≤ë˛yà,ˆÏ•Ó˚ §Ó çyÎ˚àyˆÏï˛ §üyö ˆçyÓ˚yˆÏ°y •Î˚ñ
ˆ§ çöƒ !Óò%ƒÍã˛y!°ï˛ °yí˛zí˛!flõܲyÓ˚ Ó˚yáy •Î˚– §yôyÓ˚îï˛ ~=!° ˆ◊yï˛y Óy Ó_´yÓ˚ üyÌyÓ˚ ˆã˛ˆÏÎ˚ ˆÓ!¢ í˛zFã˛ï˛yÎ˚
Ó˚yáy •Î˚– Ü,˛!eü åÈyˆÏòÓ˚ (false ceiling) §y•yˆÏ΃ âˆÏÓ˚Ó˚ í˛zFã˛ï˛y ܲü ܲˆÏÓ˚ !òˆÏ°Ä ◊yÓƒï˛y û˛y° •Î˚– ˆòÄÎ˚y°
Óy åÈyˆÏò ˆày° xÌÓy ˆÓ°öyܲyÓ˚ xÇ¢ xÓöï˛û˛yˆÏÓ ÌyܲˆÏ° ï˛y ˆÌˆÏܲ ¢∑ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ x§üyöû˛yˆÏÓ
ˆú˛yܲy!§ï˛ •ˆÏÎ˚ ÎyÄÎ˚yÓ˚ §Ω˛yÓöy ÌyˆÏܲ– ï˛yˆÏï˛ !Ó!û˛ß¨ çyÎ˚àyÎ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲü ˆÓ!¢ •ˆÏÎ˚ ÎyˆÏÓ– ï˛y
åÈyí˛¸y !flõܲyˆÏÓ˚Ó˚ !˛õåȈÏöÓ˚ xÇ¢ ˆÌˆÏܲ ≤Ã!ï˛ú˛°öÄ Óyôy §,!‹T ܲˆÏÓ˚– ï˛y ò)Ó˚ ܲÓ˚yÓ˚ çöƒ ˆ§ xLjϢ ܲyˆÏ˛õ≈ê˛
çyï˛#Î˚ xӈϢy£Ïܲ ˛õòyˆÏÌ≈Ó˚ xyhflÏÓ˚î Ìyܲy û˛y°–
302
(ii) xö%öyò xö%öyò xö%öyò xö%öyò xö%öyò (Resonance) x˛õ§yÓ˚î ≠ x˛õ§yÓ˚î ≠ x˛õ§yÓ˚î ≠ x˛õ§yÓ˚î ≠ x˛õ§yÓ˚î ≠ Î!ò í˛zͲõy!òï˛ ôù!ö§ü!‹TÓ˚ ˆÜ˛yöÄ !ӈϢ£Ï ܲ¡õyˆÏB˛Ó˚ çöƒ
xö%öyò §,!‹T •Î˚ñ ˆ§•z ܲ¡õyB˛Î%_´ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y xöƒ=!°Ó˚ ˆã˛ˆÏÎ˚ ˆÓ!¢ •ˆÏÎ˚ ~ܲ ˆÜ˛y°y•° §,!‹T ܲÓ˚ˆÏÓ–
xö%öyò# ܲ¡õyB˛ ≤ÃyÎ˚¢•z •ˆÏ°Ó˚ xyÎ˚ï˛ˆÏöÓ˚ Óà≈ü)ˆÏ°Ó˚ ÓƒyhflÏö%˛õyï˛# •Î˚– ï˛y•z •ˆÏ°Ó˚ xyÎ˚ï˛ö á%Ó Óí˛¸ •ˆÏ°
xö%öyò# ܲ¡õyB˛ ◊yÓƒ§#üyÓ˚ !öˆÏã˛ ÌyˆÏܲ ~ÓÇ xö%öyˆÏòÓ˚ ≤Ãû˛yÓ ~í˛¸yˆÏöy ÎyÎ˚–
(iii) ˆÜ˛y°y•° ˆÜ˛y°y•° ˆÜ˛y°y•° ˆÜ˛y°y•° ˆÜ˛y°y•° (Noise) ˆÌˆÏܲ ü%!_´ ≠ ˆÌˆÏܲ ü%!_´ ≠ ˆÌˆÏܲ ü%!_´ ≠ ˆÌˆÏܲ ü%!_´ ≠ ˆÌˆÏܲ ü%!_´ ≠ ˆ≤ë˛yà,ˆÏ•Ó˚ Óy•zˆÏÓ˚Ó˚ ˆÌˆÏܲ xy§y ˆày°üy° ~ÓÇ ˆû˛ï˛ˆÏÓ˚
˜Ïï˛!Ó˚ •ÄÎ˚y ˆày°üy°ñ SˆÎüö ˆ◊yï˛yˆÏòÓ˚ ˛õòôù!öñ ˜Óò%ƒ!ï˛Ü˛ ˛õyáyÓ˚ ¢∑V ò%•zÈÙÈ•z ˆÜ˛y°y•° §,!‹T ܲÓ˚ˆÏï˛
˛õyˆÏÓ˚–
Óy•zˆÏÓ˚ ˆÌˆÏܲ xy§y ¢∑ Óy!í˛¸Ó˚ ܲyë˛yˆÏüyÓ˚ üôƒ !òˆÏÎ˚ñ òÓ˚çyÈÙÈçyöy°yñ ˆòÄÎ˚y° ˆû˛ò ܲˆÏÓ˚ ~ÓÇ ÓyÎ˚%Óy!•ï˛
•ˆÏÎ˚ xy§ˆÏï˛ ˛õyˆÏÓ˚– Óy!í˛¸Ó˚ ˆüˆÏé˛ Óy åÈyˆÏòÓ˚ üôƒ !òˆÏÎ˚ xy§y ¢∑ ˆÓ˚yô ܲÓ˚ˆÏï˛ •ˆÏ° ܲyë˛yˆÏüy ˆÌˆÏܲ !Ó!FåÈߨ
ܲyë˛ Óy ¢∑ xhsˇÓ˚ܲ ˆÓyˆÏí≈˛Ó˚ Óyí˛¸!ï˛ ˆüˆÏé˛ Óy åÈyò àë˛ö ܲÓ˚ˆÏï˛ •Î˚– òÓ˚çyÓ˚ !ܲöyÓ˚yÓ˚ ú˛yÑܲ Ó˚ÓyÓ˚ Óy ˆú˛Œê˛
!ö!ü≈ï˛ ˆüyê˛y xyhflÏÓ˚î !òˆÏÎ˚ Ó¶˛ Ó˚yáˆÏï˛ •ÏÎ˚– ˛õÓ˚˛õÓ˚ ò%!ê˛ òÓ˚çy Ó§yˆÏ° ˆ§!ê˛ ¢∑ xhsˇÓ˚ܲ !•§yˆÏÓ û˛y°
ܲyç ܲˆÏÓ˚– ÓyÎ˚%Ó˚ üˆÏôƒ Ó˚yáy ˆòÄÎ˚y° Óy ˆÜ˛yöÄ ˛õyê˛yï˛ö ˆû˛ò ܲÓ˚yÓ˚ §üˆÏÎ˚ °¡∫û˛yˆÏÓ xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚
20log nc
πσρ
db «˛Î˚ •Î˚– ~áyˆÏö
σ
= kgm–2 ~ܲˆÏܲ ÓyôyÓ˚ ï˛° âöcñ n = ¢ˆÏ∑Ó˚ ܲ¡õyB˛ñ
ρ
= ÓyÎ˚%Ó˚
âöc Ä c = ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚ ˆÓà– ~Ó˚ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎñ x!ôܲ âöˆÏcÓ˚ ˛õ%Ó˚& ˆòÄÎ˚y° ¢∑ xyê˛Ü˛yˆÏï˛
ˆÓ!¢ ܲyÎ≈ܲÓ˚# ~ÓÇ x!ôܲ ܲ¡õyˆÏB˛Ó˚ ¢∑ ˆÓ˚yô ܲÓ˚y xˆÏ˛õ«˛yÜ,˛ï˛ §•ç–
ˆ≤ë˛yà,ˆÏ•Ó˚ ˆüˆÏé˛Ó˚ ܲyˆÏ˛õ≈ê˛ñ x!ú˛§ âˆÏÓ˚ ê˛y•z˛õÓ˚y•zê˛yˆÏÓ˚Ó˚ ï˛°yÎ˚ Ó˚ÓyÓ˚ Óy ˆú˛fiê˛ÈÙÈ~Ó˚ ˛õƒyí˛ Ü˛ˆÏ«˛Ó˚
xû˛ƒhsˇˆÏÓ˚ í˛zͲõߨ ¢∑ ܲüyˆÏï˛ §y•y΃ ܲˆÏÓ˚– ï˛ˆÏÓ ˜Óò%ƒ!ï˛Ü˛ ˛õyáyñ ¢#ï˛ï˛y˛õ !öÎ˚sfî Îsfñ Ü%˛°yÓ˚ñ °yí˛zí˛!flõܲyÓ˚
§ü)• e&!ê˛ü%_´ •ÄÎ˚y ˛≤ÈÏÎ˚yçö ÎyˆÏï˛ ~=!° xÓy!N˛ï˛ ¢ˆÏ∑Ó˚ í˛zͧ öy •Î˚–
(v) ˆ§y˛õyö ≤Ãû˛yÓ ˆ§y˛õyö ≤Ãû˛yÓ ˆ§y˛õyö ≤Ãû˛yÓ ˆ§y˛õyö ≤Ãû˛yÓ ˆ§y˛õyö ≤Ãû˛yÓ (Echelon effect) : Î!ò •°âˆÏÓ˚ !§Ö!í˛¸ñ ˆÓ˚!°Ç ≤Ãû,˛!ï˛ ò#â≈yÜ,˛!ï˛ !ö!ò≈‹T xyܲyÓ˚
ÌyˆÏܲñ ï˛yˆÏï˛ ¢∑ ÓyˆÏÓ˚ÓyˆÏÓ˚ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ ï˛yÓ˚ ܲyåÈyܲy!ë˛ xLjϢ ~ܲ ôÓ˚ˆÏöÓ˚ àüàˆÏü xyÄÎ˚yç ˆ¢yöy
ÎyÎ˚– ~ˆÏܲ ˆ§y˛õyö ≤Ãû˛yÓ Ó°y •Î˚– ~ˆÏܲ ~í˛¸yˆÏöyÓ˚ çöƒ !§Ö!í˛¸Ó˚ ôyˆÏ˛õ ܲyˆÏ˛õ≈ê˛ •zï˛ƒy!ò xӈϢy£Ïܲ
˛õòyÌ≈ Ó˚yáy ÎyÎ˚– xÌÓy !§Ö!í˛¸Ó˚ ܲyë˛yˆÏüy ~ܲê˛yöy öy ܲˆÏÓ˚ üyˆÏé˛ üyˆÏé˛ xyÜ,˛!ï˛Ó˚ ~ܲ!ê˛ ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ˆòÄÎ˚y
ÎyÎ˚–
(vi) ˆ◊yï˛yˆÏòÓ˚ í˛z˛õ!fli!ï˛ÈÙÈxö%˛õ!fli!ï˛ ≠ ˆ◊yï˛yˆÏòÓ˚ í˛z˛õ!fli!ï˛ÈÙÈxö%˛õ!fli!ï˛ ≠ ˆ◊yï˛yˆÏòÓ˚ í˛z˛õ!fli!ï˛ÈÙÈxö%˛õ!fli!ï˛ ≠ ˆ◊yï˛yˆÏòÓ˚ í˛z˛õ!fli!ï˛ÈÙÈxö%˛õ!fli!ï˛ ≠ ˆ◊yï˛yˆÏòÓ˚ í˛z˛õ!fli!ï˛ÈÙÈxö%˛õ!fli!ï˛ ≠ §yÓ˚!î 11.2 ˆÌˆÏܲ ˆÓyé˛y ÎyˆÏFåÈ ˆÎñ fl∫Î˚Ç ˆ◊yï˛y Óy ò¢≈ܲ
!•§yˆÏÓ í˛z˛õ!fliï˛ Óƒ!_´ ¢∑ˆÏ¢y£Ïܲ •ˆÏï˛ ˛õyˆÏÓ˚ö– à!òÎ%_´ ˆã˛Î˚yÓ˚Ä üö%£ÏƒˆÏòˆÏ•Ó˚ üˆÏï˛y•z ¢∑ˆÏ¢y£Ïܲ í˛z˛õyòyö
!•§yˆÏÓ Ü˛yç ܲˆÏÓ˚– ï˛y•z ú˛yÑܲy •°âˆÏÓ˚ áy!° xy§ö=!° !ܲå%Èê˛y ¢∑ ˆ¢y£Ïî ܲˆÏÓ˚– xÓ¢ƒ ~ˆÏï˛Ä ò¢≈ˆÏܲ
˛õ!Ó˚˛õ)î≈ xÓfliyÓ˚ §ˆÏD !ܲå%Èê˛y ˛õyÌ≈ܲƒ ˆÌˆÏܲ ÎyÎ˚– ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ ˛õ!Ó˚ܲ“öy ~üöû˛yˆÏÓ Ü˛Ó˚y •Î˚ ˆÎö
≤Ãï˛ƒy!¢ï˛ §ÇáƒÜ˛ ò¢≈ܲ í˛z˛õ!fliï˛ ÌyܲˆÏ° xö%Ó˚îö ܲy° xyò¢≈ xö%Ó˚îö ܲyˆÏ°Ó˚ §üyö •Î˚–
303
~ÓyÓ˚ ˆ¢y£Ïî =îyˆÏB˛Ó˚ ˛õ!Ó˚üy˛õ §¡∫¶˛#Î˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2 É ~ܲ!ê˛ âˆÏÓ˚Ó˚ xyÎ˚ï˛ö 1500m3 ~ÓÇ §Ó ï˛ˆÏ°Ó˚ ˆüyê˛ ˆ«˛eú˛° 1200m2– 0.5W
«˛üï˛yÓ˚ ¢∑ í˛zͧ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆòáy ˆà° í˛zͧ!ê˛ˆÏܲ ÌyüyˆÏöyÓ˚ 2s ˛õˆÏÓ˚ âˆÏÓ˚Ó˚ ¢∑
◊yÓƒ§#üyÎ˚ ˆ˛õÔÑåÈÎ˚– 100w «˛üï˛yÓ˚ í˛zͧ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° ˙ §üÎ˚!ê˛ 3s •Î˚– âˆÏÓ˚Ó˚ ˆüyê˛
ˆ¢y£Ïî ܲï˛⁄ âˆÏÓ˚Ó˚ §Ó ï˛ˆÏ°Ó˚ àí˛¸ ˆ¢y£Ïî =îyB˛ ܲï˛⁄
11.8 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~ܲ!ê˛ Ü˛ˆÏ«˛ í˛zͲõy!òï˛ ¢∑ ˆòÄÎ˚y° ≤Ãû,˛!ï˛ ˆÌˆÏܲ !ܲå%Èê˛y ≤Ã!ï˛ú˛!°ï˛ •Î˚ñ xyÓyÓ˚ ˆòÄÎ˚y°ñ xy§ÓyÓ
•zï˛ƒy!ò myÓ˚y !ܲå%Èê˛y ˆ¢y!£Ïï˛ •Î˚– ~•z ≤Ã!ï˛ú˛°ö Ä ˆ¢y£ÏˆÏîÓ˚ üyey ܲˆÏ«˛Ó˚ üôƒÜ˛yÓ˚ Ó› §ü!‹TÓ˚ í˛z˛õyòyö
Ä âˆÏÓ˚Ó˚ üyˆÏ˛õÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ~ !ӣψÏÎ˚ ¢∑!ÓK˛yˆÏöÓ˚ !öÎ˚ü Ä §)e=!°Ó˚ myÓ˚y !öô≈y!Ó˚ï˛ •Î˚ ܲˆÏ«˛Ó˚
üˆÏôƒ ܲï˛ê˛y xö%Ó˚îö •ˆÏÓ ~ÓÇ âˆÏÓ˚Ó˚ §Ó≈e ¢∑ ˆçyÓ˚yˆÏ°y Ä flõ‹T ˆ¢yöy ÎyˆÏÓ !ܲöy– Ó§ÓyˆÏ§Ó˚ §yôyÓ˚î
xyܲyˆÏÓ˚Ó˚ âˆÏÓ˚Ó˚ ï%˛°öyÎ˚ Ó,•Í xyܲyˆÏÓ˚Ó˚ ˆ≤ë˛yà,ˆÏ•Ó˚ ˆ«˛ˆÏe•z ~=!° =Ó˚&c˛õ)î≈ •ˆÏÎ˚ òyÑí˛¸yÎ˚– ˆ§ ܲyÓ˚ˆÏî öyê˛Ü˛ñ
Ó_,´ï˛yñ !§ˆÏöüy ≤Ãû,˛!ï˛Ó˚ çöƒ ˆ≤ë˛yà,ˆÏ•Ó˚ !öü≈yîܲyˆÏÎ≈ Ä•z !öÎ˚ü=!° xö%§Ó˚î ܲÓ˚y •Î˚– ¢∑!ÓK˛yˆÏöÓ˚ ˆ§•z
!öÎ˚ü=!° Ä ˆ§=!°Ó˚ ˛õ!Ó˚ˆÏ≤Ã!«˛ˆÏï˛ ˆ≤ë˛yà,• àë˛ˆÏöÓ˚ §)e§ü)• ÈÙÙÙÈ ~•z §Ó !üˆÏ°•z ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ ÓƒÓfliyÓ˚
xyˆÏ°yã˛öy–
xö%Ó˚îöñ xÌ≈yÍ ÓyˆÏÓ˚ÓyˆÏÓ˚ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ú˛ˆÏ° ¢ˆÏ∑Ó˚ ò#â≈fliyÎ˚# =OÓ˚î í˛zFã˛y!Ó˚ï˛ ¢ˆÏ∑Ó˚ flõ‹Tï˛yˆÏܲ
§ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ ≤Ãû˛y!Óï˛ Ü˛ˆÏÓ˚– ˆÜ˛yöÄ Ü˛ˆÏ«˛ ~ܲ!ê˛ í˛zͧ ˆÌˆÏܲ e´üyàï˛ ¢∑ í˛zͲõy!òï˛ •ˆÏ° ≤Ã!ï˛ú˛°ö
~ÓÇ ˆ¢y£ÏˆÏîÓ˚ üôƒ !òˆÏÎ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Óyí˛¸ˆÏï˛ Óyí˛¸ˆÏï˛ ~ܲê˛y üyˆÏö !àˆÏÎ˚ !fliÓ˚ •ˆÏÎ˚ ÎyÎ˚– ˆ§•z xÓfliyÎ˚
Î!ò í˛zͧ!ê˛ Ó¶˛ ܲˆÏÓ˚ ˆòÄÎ˚y ÎyÎ˚ñ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ï˛Í«˛îyÍ ¢)öƒ •ˆÏÎ˚ ÎyÎ˚ öy– ô#ˆÏÓ˚ ô#ˆÏÓ˚ ܲüˆÏï˛ Ü˛üˆÏï˛
ö)ƒöï˛ü ◊yÓƒ§#üyÓ˚ !öˆÏã˛ ã˛ˆÏ° ÎyÎ˚– ~•z §üÎ˚ܲy°ˆÏܲ Ó°y •Î˚ xö%Ó˚îö ܲy°xö%Ó˚îö ܲy°xö%Ó˚îö ܲy°xö%Ó˚îö ܲy°xö%Ó˚îö ܲy°– xö%Ó˚îö ܲyˆÏ°Ó˚ §ÇK˛y åÈyí˛¸yÄ
~ܲ!ê˛ Ü˛ˆÏ«˛Ó˚ xö%Ó˚îö ܲy° Ü˛ï˛ •ˆÏÓ ˆ§!ê˛ Ü˛#û˛yˆÏÓ !•§yÓ Ü˛Ó˚y ÎyÎ˚ ~ÓÇ ˛õÓ˚#«˛y myÓ˚y ܲ#û˛yˆÏÓ xö%Ó˚îö
ܲy° !öî≈ˆÏÎ˚Ó˚ myÓ˚y ܲˆÏ«˛Ó˚ ˆ¢y£ÏˆÏîÓ˚ ˛õ!Ó˚üyî !öî≈Î˚ ܲÓ˚y ÎyÎ˚ñ ˆ§à=!° xy˛õ!ö çyöˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈö– xyò¢≈
xö%Ó˚îö ܲy° Ó°ˆÏï˛ Ü˛# ˆÓyé˛yÎ˚ ~ÓÇ ~ܲ!ê˛ Ü˛ˆÏ«˛ ܲ#û˛yˆÏÓ xö%Ó˚îö ܲy° ܲüyˆÏöy Óyí˛¸yˆÏöy ÎyÎ˚ñ ï˛yÄ ~•z
~ܲˆÏܲ xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ– §ÓˆÏ¢ˆÏ£Ï ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ÓƒÓfliyÓ˚ ܲˆÏÎ˚ܲ!ê˛ !òˆÏܲÓ˚ í˛z˛õÓ˚ xyˆÏ°yܲ˛õyï˛ Ü˛Ó˚y
•ˆÏÎ˚ˆÏåÈñ ˆÎ=!° •Î˚ï˛ xy˛õöyÓ˚ ܲyˆÏåÈ ˆÜ˛Ôï%˛•ˆÏ°yj#˛õܲ ӈϰ üˆÏö •ˆÏÓ–
11.9 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ~ܲ!ê˛ âˆÏÓ˚Ó˚ ˜òâ≈ƒ 10m, ≤Ãfli 6m Ä í˛zFã˛ï˛y 5m – åÈyò Ä Ü˛yˆÏ˛õ≈ˆÏê˛ ì˛yܲy ˆüˆÏé˛Ó˚ ¢∑ˆÏ¢y£Ïî =îyB˛
ÎÌye´ˆÏü 0·05 Ä 0·2– ˆòÄÎ˚y°=!°Ó˚ 10 ¢ï˛yÇ¢ ˆáy°y òÓ˚çyÈÙÈçyöy°y ~ÓÇ Óy!ܲ xLjϢÓ˚
¢∑ˆÏ¢y£Ïî =îyB˛ 0·02– ˙ âˆÏÓ˚Ó˚ xö%Ó˚îö ܲy° Ü˛ï˛ •ˆÏÓ⁄
304
2. ~ܲ!ê˛ Ók˛ âˆÏÓ˚Ó˚ üy˛õ ˜òâ≈ƒ 8m × ≤Ãfli 8m × í˛zFã˛ï˛y 4m– ~!ê˛Ó˚ ˆòÄÎ˚y° Ä åÈyˆÏò ¢∑ˆÏ¢y£Ïܲ
˛õƒyˆÏö° °yàyˆÏöy xyˆÏåÈ ÎyÓ˚ ˆ¢y£Ïî =îyB˛ 0·5 – ˆüˆÏé˛Ó˚ xˆÏô≈ܲ xLjϢ ˆüyê˛y ܲyˆÏ˛õ≈ˆÏê˛ ì˛yܲyñ ÎyÓ˚
ˆ¢y£Ïî =îyB˛ 0·4 – ˆüˆÏé˛Ó˚ Óy!ܲ xLjϢ 40!ê˛ à!òˆÏüyí˛¸y xy§ö xyˆÏåÈñ ˆÎ=!°Ó˚ ≤Ã!ï˛!ê˛Ó˚ ï˛°
1·25m2 ~ÓÇ ˆ¢y£Ïî =îyB˛ 0·6– ˆ§!Óö Ä xy•z!Ó˚ÇÈÙÈ~Ó˚ §)ˆÏeÓ˚ §y•yˆÏ΃ ܲ«˛!ê˛Ó˚ xö%Ó˚îö ܲy°
!öî≈Î˚ ܲÓ˚&ö–
3. ~ܲ!ê˛ Ü˛ˆÏ«˛ ~ܲ!ê˛ ¢ˆÏ∑Ó˚ í˛zͧ !ö!ò≈‹T «˛üï˛yÎ˚ ¢∑ í˛zͲõyòö ܲÓ˚ˆÏï˛ ÷Ó˚& ܲÓ˚°– ܲˆÏ«˛ ï˛#Ó ï˛y ܲ#û˛yˆÏÓ
Ó,!k˛ ˛õyˆÏÓ ï˛yÓ˚ ày!î!ï˛Ü˛ §)e !öî≈Î˚ ܲÓ˚&ö– ˙ ï˛#Ó ï˛y Ü˛ï˛ §üˆÏÎ˚ §Ó≈y!ôܲ ï˛#Ó ï˛yÓ˚ xˆÏô≈ܲ üyˆÏö
ˆ˛õÔÑåÈyˆÏÓ⁄
4. ~ܲ!ê˛ ˆ≤ë˛yà,ˆÏ• ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y fliyÎ˚# üyˆÏö ˆ˛õÔÑåÈÓyÓ˚ ˛õÓ˚ ¢ˆÏ∑Ó˚ í˛zͧ=!° ~ܲ§ˆÏD Ó¶˛ ܲˆÏÓ˚ ˆòÄÎ˚y
•°– ~áö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y •…yˆÏ§Ó˚ ày!î!ï˛Ü˛ §)e!ê˛Ó˚ ≤Ã!ï˛˛õyòö ܲÓ˚&ö– ~•z §)ˆÏeÓ˚ §y•yˆÏ΃ xö%Ó˚îö
ܲyˆÏ°Ó˚ ˆ§!ÓˆÏöÓ˚ Ó˚y!¢üy°y!ê˛ !öî≈Î˚ ܲÓ˚&ö–
5. Ú≤ÃyîÓhsˇÛ ܲ«˛ Ä Ú!ö‹±yîÛ Ü˛«˛ ܲyˆÏܲ ӈϰ⁄ !ö‹±yî ܲˆÏ«˛Ó˚ xö%Ó˚îö ܲy° !öî≈ˆÏÎ˚Ó˚ çöƒ í˛z˛õÎ%_´
~ܲ!ê˛ Ó˚y!¢üy°y ≤Ã!ï˛¤˛y ܲÓ˚&ö–
6. Ú¢∑ˆÏ¢y£Ïî =îyB˛Û ܲyˆÏܲ ӈϰ⁄ ~ܲ!ê˛ Ü˛ˆÏ«˛Ó˚ ï˛°=!°Ó˚ àí˛¸ ¢∑ˆÏ¢y£Ïî =îyB˛ ܲ#û˛yˆÏÓ !öî≈Î˚ ܲÓ˚y
ÎyÎ˚⁄
7. ˆ≤ë˛yà,ˆÏ•Ó˚ ¢∑ ˛õ!Ó˚ܲ“öyÎ˚ í˛z˛õÎ%_´ xö%Ó˚îö åÈyí˛¸yÄ xyÓ˚ ˆÜ˛yö‰ ˆÜ˛yö‰ !ӣψÏÎ˚ öçÓ˚ Ó˚yáy ≤ÈÏÎ˚yçö⁄
~=!°Ó˚ §¡∫ˆÏ¶˛ ˆÜ˛yö‰ ˆÜ˛yö‰ ˛õòˆÏ«˛˛õ ˆöÄÎ˚y çÓ˚&Ó˚# ӈϰ xy˛õ!ö üˆÏö ܲˆÏÓ˚ö⁄
11.10 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1
(i) (a) xö%Ó˚îö (b) §üyö (c) ܲüˆÏÓ (d) í˛zû˛Î˚•z (e) §üyö%˛õyï˛#
(ii) ˆ§!ÓˆÏöÓ˚ §)e xö%Î˚yÎ˚# TR =
0·1610·16110000300
Va
× =
= 5·4s
xö%Ó˚îö ܲy° xˆÏô≈ܲ •ˆÏï˛ •ˆÏ° ˆ¢y£Ïî !m=î •ˆÏÓ– Î!ò !öˆÏî≈Î˚ ò¢≈ˆÏܲÓ˚ §Çáƒy n •Î˚ ï˛ˆÏÓ
300 + (n × 0·4) = 300 × 2
xÌ≈yÍ n = 3000·4
= 750
305
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 2
11.19 §ü#ܲÓ˚î ˆÌˆÏܲ àí˛¸ˆÏ¢y£Ïî =îyB˛
()()
12
12
4/ VInPPa
cSTT=
−
~áy Ïö V = 1500m3, P1 = 100w, P2 = 0·5w, S = 1200m2
T1 = 3s, T2 = 2s– ôÓ˚y Îyܲ c = 340ms–1
∴
() 41500In100/0·53401200(32)
a××
=×−
= ·0147 In 200
= ·0147 ×2·303 × 2·3010 = 0·078–
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. âˆÏÓ˚Ó˚ xyÎ˚ï˛ö V = 10 × 6 × 5 = 300m3
åÈyˆÏòÓ˚ ˆ«˛eú˛° 10 × 6 = 60m2 ∴ ˆ¢y£Ïî 60 × 0·05 = 3m2
ˆüˆÏé˛Ó˚ ˆ«˛eú˛° 10 × 6 = 60m2
∴
ˆ¢y£Ïî 60 × 0·2 = 12m2
ˆòÄÎ˚yˆÏ°Ó˚ ˆ«˛eú˛° (10 + 10 + 6 + 6) × 5 = 160m2
òÓ˚çy ÈÙÈ çyöy°yÓ˚ ˆ«˛eú˛° = 160 ×
10100= 16m2
∴
ˆ¢y£Ïî = 16 × 1 = 16m2
Óy!ܲ ˆòÄÎ˚yˆÏ°Ó˚ ˆ«˛eú˛° 160 – 16 = 144m2
∴
ˆ¢y£Ïî 144 × ·02 = 2·88m2
∴
ˆüyê˛ ˆ¢y£Ïî a = 3 + 12 + 16 + 2·88m2
∴
xö%Ó˚îö ܲy° TR =
0·16130033·83
×
= 1·42s–
2. âˆÏÓ˚Ó˚ xyÎ˚ï˛ö V = 8 × 8 × 4 = 256m3
ˆòÄÎ˚y°ñ ˆüˆÏé˛Ó˚ Ä åÈyˆÏòÓ˚ ˆüyê˛ ˆ«˛eú˛° s
= 2(8 × 8 + 8 × 4 + 8 × 4) = 256m2
åÈyˆÏòÓ˚ ˆ¢y£Ïî = 8 × 8 × 0·5 = 32m2
ˆüˆÏé˛Ó˚ ˆ¢y£Ïî =
12
(8 × 8) × 0·4 = 12·8m2
306
xy§ö=!°Ó˚ ˆ¢y£Ïî = 40 × 1·25 × 0·6 = 30m2
∴
ˆüyê˛ ˆ¢y£Ïî a = 74·8m2
∴
àí˛¸ ˆ¢y£Ïî =îyB˛
74·8256
aS
α==
= 0·292
ˆ§!ÓˆÏöÓ˚ §)e xö%ÎyÎ˚# TR = 0·161 0·161 256
74·8V
a×= = 0·55s
xy•z!Ó˚ÇÈÙÈ~Ó˚ §)e xö%ÎyÎ˚# TR = ( ) ( )0·161 0·161 256
256 1 ·2925In 1V
lmα×=
− −− −= 0·47s
3. ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ 11.3 xö%ˆÏFåȈÏò xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈ–
11.3 xö%ˆÏFåȈÏòÓ˚ 11.8 §ü#ܲÓ˚î xö%ÎyÎ˚# I = Im41actVe
−⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠
Îáö I = Im/2, ï˛áö I – e–act/4v =
12
, xÌ≈yÍ
41,2
actV e
−=
t = 4Vac
In2 = 0·00815va
ˆÎáyˆÏö
c = 340ms–1 ôÓ˚y •ˆÏÎ˚ˆÏåÈ–
4. ≤ß¿!ê˛Ó˚ í˛z_Ó˚ 11.4 xö%ˆÏFåȈÏò xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5. ≤Èϟ¿Ó˚ ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ 11.2 xö%ˆÏFåȈÏò ~ÓÇ !mï˛#Î˚ xLjϢÓ˚ í˛z_Ó˚ 11.5 xö%ˆÏFåȈÏò ˛õyÄÎ˚y ÎyˆÏÓ–
6. ≤Èϟ¿Ó˚ ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ 11.2 xö%ˆÏFåȈÏò ~ÓÇ ˛õˆÏÓ˚Ó˚ xLjϢÓ˚ í˛z_Ó˚ 11.6 xö%ˆÏFåȈÏò ˛õyÄÎ˚y ÎyˆÏÓ–
7. 11.7 xö%ˆÏFåȈÏò ≤ß¿!ê˛Ó˚ í˛z_Ó˚ ˛õyÄÎy ÎyˆÏÓ– xy˛õ!ö !öçfl∫ x!û˛K˛ï˛y ˆÌˆÏÜ˛Ä !ܲå%È !°áˆÏï˛ ˛õyˆÏÓ˚ö–
307
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 12 : x!ï˛¢∑x!ï˛¢∑x!ï˛¢∑x!ï˛¢∑x!ï˛¢∑ (Ultrasonics)
àë˛ö
12.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
12.2 x!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòö
12.2.1 ˆã˛Ô¡∫ܲï˛!ï˛ !öû˛≈Ó˚ í˛zͲõyòܲ (Magnetostrictive generator)
12.2.2 ã˛y˛õÈÙÙÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲ (Piezoelectric generator)
12.3 !Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà !öî≈Î˚
12.3.1 xyˆÏ°yܲ ÓƒÓï≈˛ö (Optical diffraction) ˛õk˛!ï˛
12.3.2 ¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚ (Acoustic interference) ˛õk˛!ï˛
12.4 x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓ•yÓ˚
12.5 §yÓ˚yÇ¢
12.6 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
12.7 í˛z_Ó˚üy°y
12.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
xy˛õ!ö !öÿ˛Î˚•z ˛õˆÏí˛¸ˆÏåÈö ˆÎñ xyüÓ˚y ˆÎ ¢∑ ÷öˆÏï˛ ˛õy•z ï˛yÓ˚ ܲ¡õyˆÏB˛Ó˚ |ôù≈§#üy ≤ÃyÎ˚ 20kHz–
xˆÏöˆÏܲÓ˚ ˆ«˛ˆÏe ~•z §#üy xyÓ˚Ä !öˆÏã˛Ä •ˆÏï˛ ˛õyˆÏÓ˚– ~Ó˚ ú˛ˆÏ° xyüyˆÏòÓ˚ ˛õy!Ó˚˛õy!Ÿª≈ˆÏܲÓ˚ xˆÏöܲ ¢∑•z •Î˚ï˛
xyüyˆÏòÓ˚ ܲyˆÏåÈ x◊&ï˛ ˆÌˆÏܲ ÎyÎ˚– ˆÎ §Ó ¢ˆÏ∑Ó˚ ܲ¡õyB˛ 20kHz ~Ó˚ ˆÓ!¢ñ ˆ§=!°ˆÏܲ xyüÓ˚y x!ï˛¢∑
(Ultrasound) Ó!°– ~ çyï˛#Î˚ ï˛Ó˚ˆÏDÓ˚ ܲ¡õyB˛ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚ |ôù≈§#üy ˆÌˆÏܲ xyÓ˚Ω˛ ܲˆÏÓ˚ ܲˆÏÎ˚ܲ
ˆüyày•yÍ≈§ñ ~üö !ܲ !àày•yÍ≈§Ä (109Hz) •ˆÏï˛ ˛õyˆÏÓ˚– üyö%£Ï Óy xöƒ hflÏöƒ˛õyÎ˚# ≤Ãyî#Ó˚y x!ï˛¢∑ ÷öˆÏï˛
öy ˆ˛õˆÏ°Ä !ö¢yã˛Ó˚ Óyò%í˛¸ ܲü ˆÓ!¢ 100kHz ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ í˛zͲõߨ ܲˆÏÓ˚ ~ÓÇ ≤Ã!ï˛ôù!ö ÷ˆÏö Óyôy
~!í˛¸ˆÏÎ˚ í˛zˆÏí˛¸ ˆÓí˛¸yÎ˚ Ä !¢Ü˛yˆÏÓ˚Ó˚ §¶˛yö ܲˆÏÓ˚–
˛õÓ˚#«˛yàyˆÏÓ˚ x!ï˛¢∑ í˛zͲõyòˆÏöÓ˚ çöƒ öyöy ôÓ˚ˆÏöÓ˚ Îsf˛õy!ï˛ í˛zqy!Óï˛ •ˆÏÎ˚ˆÏåÈ– ¢∑!ÓK˛yö ã˛ã≈˛yÓ˚ ≤ÃÌü
!òˆÏܲ Îy!sfܲ ‡•z§‰ˆÏ°Ó˚ §y•yˆÏ΃ Óy ܲyÓ≈ö xyˆÏܲ≈˛ ï˛!í˛¸Í ˆüy«˛î â!ê˛ˆÏÎ˚ !öÎ˚!sfï˛ Ü˛¡õyˆÏB˛Ó˚ x!ï˛¢∑ í˛zͲõyòö
ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– !ܲv ~•z ˛õk˛!ï˛=!°Ó˚ ˙!ï˛•y!§Ü˛ =Ó˚&c ÌyܲˆÏ°Ä Óï≈˛üyˆÏö x!ï˛¢∑ í˛zͲõyòˆÏöÓ˚ çöƒ ≤Ãôyï˛ö
ˆÎ ˆû˛Ôï˛ âê˛öy=!° ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ ˆ§=!° •°ñ ˆã˛Ô¡∫ܲ ï˛!ï˛ (magnetostriction) Ä ã˛y˛õÈÙÈ!Óò%ƒÍ
(piezoelectricity)– ~áyˆÏö xy˛õ!ö x!ï˛¢∑ í˛zͲõyòˆÏöÓ˚ ~ çyï˛#Î˚ xyô%!öܲ ˛õk˛!ï˛Ó˚ §ˆÏD ˛õ!Ó˚!ã˛ï˛ •ˆÏÓö–
!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ Ä ˆÓà ˛õ!Ó˚üy˛õ ܲÓ˚yÓ˚ çöƒ ܲˆÏÎ˚ܲ!ê˛ !öû≈˛Ó˚ˆÏÎyàƒ ˛õk˛!ï˛ ã˛y°% xyˆÏåÈ–
~áyˆÏö xyüÓ˚y ~ ôÓ˚ˆÏöÓ˚ ܲˆÏÎ˚ܲ!ê˛ ˛õk˛!ï˛Ó˚ xyˆÏ°yã˛öy ܲÓ˚Ó– ܲ¡õyB˛ x!ôܲ •ÄÎ˚yÎ˚ x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ
308
fl∫yû˛y!Óܲû˛yˆÏÓ•z ܲü ~ÓÇ ~çöƒ x!ï˛¢∑ ï˛Ó˚ˆÏDÓ˚ Ó˚!Ÿ¬ !ö!ò≈‹T !òˆÏܲ ã˛y°öy ܲÓ˚y ÎyÎ˚– ~•z Ó˚!Ÿ¬Ó˚ ≤Ã!ï˛ú˛°ö
°«˛ƒ ܲˆÏÓ˚ ˆÎüö ≤Ã!ï˛ú˛°Ü˛ ÓyôyÓ˚ xÓfliyö Ä ã˛!Ó˚e !öô≈yÓ˚î ܲÓ˚y ÎyÎ˚ñ ˆï˛üö•z x!ï˛¢ˆÏ∑Ó˚ Ó˚!Ÿ¬Ó˚ üyôƒˆÏü
xˆÏöܲê˛y «˛üï˛y !ö!ò≈‹T °«˛ƒÓ› x!û˛ü%ˆÏᲠ˛õyë˛yˆÏöy ÎyÎ˚– x!ï˛¢ˆÏ∑Ó˚ ~•z ôü≈=!° ~•z ¢∑ ï˛Ó˚ˆÏDÓ˚ !Ó!û˛ß¨
ÓƒÓ•y!Ó˚ܲ ≤ÈÏÎ˚yˆÏà ܲyˆÏç °yàyˆÏöy •Î˚– ~•z ~ܲˆÏܲ xyüÓ˚y x!ï˛¢ˆÏ∑Ó˚ ܲˆÏÎ˚ܲ!ê˛ ÓƒÓ•y!Ó˚ܲ ≤ÈÏÎ˚yà !öˆÏÎ˚Ä
xyˆÏ°yã˛öy ܲÓ˚Ó–
í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠í˛zˆÏj¢ƒ ≠
~•z ~ܲܲ!ê˛ ˛õí˛¸ˆÏ° xy˛õ!öÈÙÙÙÈ
x!ï˛¢∑ ܲyˆÏܲ ӈϰ ï˛y Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
x!ï˛¢∑ ܲ#û˛yˆÏÓ í˛zͲõyòö ܲÓ˚y ÎyÎ˚ñ ï˛yÓ˚ !Ó!û˛ß¨ ˛õk˛!ï˛ Óî≈öy ܲÓ˚ˆÏï˛˛ ˛õyÓ˚ˆÏÓö–
!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ ˆÓà !öî≈Î˚ ܲ#û˛yˆÏÓ Ü˛Ó˚y •Î˚ñ ï˛y Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
x!ï˛¢∑ ܲï˛Ó˚ܲü ܲyˆÏç °yˆÏàñ !ӈϢ£Ï ܲˆÏÓ˚ xy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛ ܲ#ñ ~§Ó §yôyÓ˚î í˛zͧy•#
üyö%£ÏˆÏܲ Ó%!é˛ˆÏÎ˚ !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
12.2 x!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòöx!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòöx!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòöx!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòöx!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòö
xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ ¢∑ï˛Ó˚D xy§ˆÏ° !fli!ï˛fliy˛õܲ üyôƒˆÏü xö%˜Ïòâ≈ƒ ï˛Ó˚D– ¢∑ï˛Ó˚D ˜ï˛!Ó˚
ܲÓ˚ˆÏï˛ üyôƒˆÏüÓ˚ üˆÏôƒ ܲ¡õö¢#° ~ܲ!ê˛ í˛z͈ϧÓ˚ ≤ÈÏÎ˚yçö– §yôyÓ˚î ¢ˆÏ∑Ó˚ üˆÏï˛y x!ï˛¢∑Ä í˛z͈ϧÓ˚
ܲ¡õˆÏöÓ˚ ú˛ˆÏ° í˛zͲõߨ •Î˚– ï˛ˆÏÓ ~•z í˛z͈ϧÓ˚ ܲ¡õyB˛ ˆÎ ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ í˛zͲõyòö ܲÓ˚ˆÏï˛ •ˆÏÓñ ï˛yÓ˚
§üyö •ˆÏï˛ •ˆÏÓ– xˆÏöܲ §üÎ˚ í˛zͲõߨ x!ï˛¢∑ˆÏܲ ï˛Ó˚ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ §MÈ˛y!Ó˚ï˛ Ü˛Ó˚ˆÏï˛ •Î˚– xy˛õ!ö ~•z
˛õÎ≈yˆÏÎ˚Ó˚ x‹Tü ~ܲˆÏܲ ˛õˆÏí˛¸ˆÏåÈö ˆÎñ ï˛Ó˚ˆÏ°Ó˚ ¢∑ ˆÓ˚yô ÓyÎ˚%üyôƒˆÏüÓ˚ ï%˛ö°yÎ˚ xˆÏöܲ ˆÓ!¢– ~•z ¢∑ ˆÓ˚yˆÏôÓ˚
§ˆÏD §yüO§ƒ Ó˚«˛y ܲˆÏÓ˚ ï˛Ó˚ˆÏ° x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D ã˛y!°ï˛ ܲÓ˚ˆÏï˛ í˛zͧ!ê˛ˆÏܲ x“ §Ó˚ˆÏî ≤Ãã%˛Ó˚ Ó° ≤ÈÏÎ˚yˆÏà
§«˛ü •ˆÏï˛ •Î˚– Óï≈˛üyˆÏö x!ï˛¢∑ í˛zͲõyòˆÏöñ ˆÎ ˛õk˛!ï˛=!° §ã˛Ó˚yã˛Ó˚ ÓƒÓ•*ï˛ •Î˚ñ ~áyˆÏö xyüÓ˚y ˆ§=!°
xyˆÏ°yã˛öy ܲÓ˚Ó–
~•z ≤çˆÏD í˛zˆÏÕ‘á ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ §yôyÓ˚î ¢ˆÏ∑Ó˚ í˛zͲõyòˆÏö ÓƒÓ•*ï˛ Îsf=!°Ó˚ üï˛ x!ï˛¢ˆÏ∑Ó˚
í˛zͧ=!°Ä ï˛!í˛¸Í¢!_´ˆÏܲ•z ¢∑¢!_´ˆÏï˛ Ó˚*˛õy!hsˇ!Ó˚ï˛ Ü˛ˆÏÓ˚– ~çöƒ ~=!°ˆÏܲ ê˛Δy™!í˛í˛z§yÓ˚ (transducer) Óy
Ó˚*˛õyhsˇÓ˚ܲ Ó°y •Î˚–
12.2.1 ˆã˛Ô¡∫ܲï˛!ï˛ÈÙÈ!öû≈˛Ó˚ í˛zͲõyòܲˆã˛Ô¡∫ܲï˛!ï˛ÈÙÈ!öû≈˛Ó˚ í˛zͲõyòܲˆã˛Ô¡∫ܲï˛!ï˛ÈÙÈ!öû≈˛Ó˚ í˛zͲõyòܲˆã˛Ô¡∫ܲï˛!ï˛ÈÙÈ!öû≈˛Ó˚ í˛zͲõyòܲˆã˛Ô¡∫ܲï˛!ï˛ÈÙÈ!öû≈˛Ó˚ í˛zͲõyòܲ (Magnetostrictive generator)
ˆÜ˛yö xÎ˚ˆÏÿ˛Ô¡∫ܲ (ferromagnetic) ˛õòyÌ≈ˆÏܲ Îáö ã%˛¡∫!Ü˛ï˛ Ü˛Ó˚y •Î˚ ï˛áö ã%˛¡∫ܲˆÏöÓ˚ !òܲ ÓÓ˚yÓÓ˚ ˆ§!ê˛Ó˚
§yüyöƒ ˜òâ≈ƒ ˛õ!Ó˚Óï≈˛ö âˆÏê˛– ˛õòyÌ≈!ê˛Ó˚ üˆÏôƒ ã%˛¡∫ܲöÈÙÈï˛#Ó ï˛y Óyí˛¸yÎ˚ §ˆÏD §ˆÏD ˆ§!ê˛Ó˚ xû˛ƒhsˇÓ˚#î àë˛ˆÏö ˆÎ
˛õ!Ó˚Óï≈˛ö âˆÏê˛ ˆ§!ê˛•z ~•z ˜òâ≈ƒ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ ܲyÓ˚î– xÎ˚ˆÏÿ˛Ô¡∫ܲ ˛õòyÌ≈!ê˛Ó˚ ã%˛¡∫ܲö Îáö ã%˛¡∫ܲˆÏöÓ˚ §Ç˛õ,!_´Ó˚
309
(saturation) ˆÓ¢ !ܲå%Èê˛y !öˆÏã˛ ÌyˆÏܲñ ï˛áö ˆ§!ê˛Ó˚ ï˛!ï˛Ó˚ ˛õ!Ó˚üyî xyˆÏÓ˚y!˛õï˛ ˆã˛Ô¡∫ܲ xyˆÏӈϢÓ˚ ÓˆÏà≈Ó˚
§üyö%˛õyï˛# •Î˚– xÌ≈yÍñ ã%˛¡∫ܲˆÏöÓ˚ !òܲ ÓÓ˚yÓÓ˚ ≤ÃyÌ!üܲ ˜òâ≈ƒ
x Δ
, ˜òˆÏâ≈ƒÓ˚ ˛õ!Ó˚Óï≈˛ö
∆ξ
~ÓÇ ˆã˛Ô¡∫ܲ xyˆÏÓ¢
B •ˆÏ° ˆ°áy ÎyÎ˚ñ ˆã˛Ô¡∫ܲ ï˛!ï˛ñ
xξΔ
Δ
=KB2
...12.1
ˆÎáyˆÏö K ~ܲ!ê˛ ô &Óܲñ ÎyÓ˚ üyö ˛õ!ç!ê˛û˛ Óy ˆöˆÏà!ê˛û˛ •ˆÏï˛ ˛õyˆÏÓ˚– !öˆÏܲ° ~ÓÇ •zöû˛yÓ˚ñ ˛õyÓ˚ˆÏüöí%˛Ó˚
≤Ãû,˛!ï˛ Ü˛ˆÏÎ˚ܲ!ê˛ §ÇܲˆÏÓ˚Ó˚ (alloy) ˆ«˛ˆÏe ˆã˛Ô¡∫ܲ ï˛!ï˛Ó˚ üyö ˆÓ!¢ •ˆÏï˛ ˆòáy ÎyÎ˚– !öˆÏܲˆÏ°Ó˚ ˆ«˛ˆÏe BÈÙÈ~Ó˚
üyö 0·5Wbm–2 SÄ ÏÎÓyÓ†Óà≈!üê˛yÓV ~Ó ˆã˛ ÏΠܲü • Ï° K ô &Óܲ!ê˛Ó˚ üyö •Î˚ –1·0 × 10–4m4Wb–2–
!ÓˆÏÎ˚yà !ã˛•´!ê˛Ó˚ xÌ≈ ~•z ˆÎñ ˆã˛Ô¡∫ܲ xyˆÏӈϢÓ˚ ≤Ãû˛yˆÏÓ !öˆÏܲˆÏ°Ó˚ ˜òâ≈ƒ §ˆÏB˛yã˛ö âˆÏê˛–
xyÎ˚ˆÏÿ˛Ô¡∫ܲ Ó›ˆÏï˛ !ö!ü≈ï˛ ˆÜ˛yöÄ òˆÏ[˛Ó˚ í˛z˛õÓ˚ ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚ Ü%˛[˛°# ç!í˛¸ˆÏÎ˚ ˙ Ü%˛[˛°#ˆÏï˛ Î!ò ˛õ!Ó˚Óï˛#≈
ï˛!í˛¸Í≤ÃÓy• ˛õyë˛yˆÏöy ÎyÎ˚ñ ï˛ˆÏÓ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ≤Ã!ï˛ ˛õÎ≈yˆÏÎ˚ ˆã˛Ô¡∫ܲ ï˛!ï˛ ò%ÓyÓ˚ §ˆÏÓ≈yFã˛ üyö °yû˛ ܲÓ˚ˆÏÓ–
ˆÜ˛ööy ˆã˛Ô¡∫ܲ xyˆÏӈϢÓ˚ !ÓhflÏyÓ˚ Î!ò Bm •Î˚ ï˛ˆÏÓ B ~Ó˚ üyö +Bm Ä –Bm, í˛zû˛Î˚ ˆ«˛ˆÏe•z ˆã˛Ô¡∫ܲ ï˛!ï˛
KBm2 •ˆÏÓ– ~ˆÏï˛ ò[˛!ê˛Ó˚ Îy!sfܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ ≤ÃÓyˆÏ•Ó˚ ܲ¡õyˆÏB˛Ó˚ !m=î •ˆÏÓ– xÓ¢ƒ òˆÏ[˛Ó˚ í˛z˛õÓ˚
çí˛¸yˆÏöy !mï˛#Î˚ ~ܲ!ê˛ Ü%˛[˛°#Ó˚ üˆÏôƒ !ò‹T ï˛!í˛¸Í≤ÃÓy• ˛õy!ë˛ˆÏÎ˚ Î!ò ~ܲ!ê˛ !fliÓ˚ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ B0 ˜ï˛!Ó˚ ܲÓ˚y
ÎyÎ˚ ÎyÓ˚ üyö Bm ~Ó˚ ˆã˛ˆÏÎ˚ Óí˛¸ñ ï˛ˆÏÓ ˆüyê˛ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ ˛õ!Ó˚Óï˛#≈ ≤ÃÓyˆÏ•Ó˚ ~ܲ ˛õÎ≈yˆÏÎ˚ B0 – Bm ˆÌ Ïܲ
B0 + Bm ~Ó˚ üˆÏôƒ ~ܲÓyÓ˚ ܲ!¡õï˛ •ˆÏÓ ~ÓÇ òˆÏ[˛Ó˚ Îy!sfܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ܲ¡õyˆÏB˛Ó˚
§üyö •ˆÏÓ– !fliÓ˚ ˆã˛Ô¡∫ܲ xyˆÏÓ¢!ê˛ xÓ¢ƒ ܲˆÏÎ˚ܲ!ê˛ fliyÎ˚# ã%˛¡∫ˆÏܲÓ˚ §y•yˆÏÎƒÄ ˜ï˛!Ó˚ ܲÓ˚y ÎyÎ˚– í˛z£èï˛y Ó,!k˛Ó˚
§ˆÏD ˆã˛Ô¡∫ܲ ï˛!ï˛ Ü˛üˆÏï˛ ÌyˆÏܲ ~ÓÇ ˆ¢£Ï ˛õÎ≈hsˇ Ü%˛Ó˚# í˛z£èï˛yÎ˚ ~ˆÏ§ ˆã˛Ô¡∫ܲ ï˛!ï˛Ó˚ üyö ¢)öƒ •ˆÏÎ˚ ÎyÎ˚–
~ÓyÓ˚ ˆòáy Îyܲ ˆã˛Ô¡∫ܲ ï˛!ï˛ˆÏܲ !ܲû˛yˆÏÓ x!ï˛¢∑ í˛zͲõyòˆÏöÓ˚ ܲyˆÏç °yàyˆÏöy ÎyÎ˚– ~•z ܲyˆÏçÓ˚ í˛z˛õÎ%_´
~ܲ!ê˛ ê˛Δyö!çfiê˛yÓ˚ Óï≈˛ö# 12.1 !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
310
AB ~ܲ!ê˛ !öˆÏܲ° Ó˚í˛ñ !ë˛Ü˛ üyé˛áyˆÏö ò,ì˛¸û˛yˆÏÓ Üœ˛ƒy¡õ !òˆÏÎ˚ xyê˛ˆÏܲ Ó˚yáy •Î˚ ÎyˆÏï˛ ò%!ê˛ ≤Ãyhsˇ ü%_´ ÌyˆÏܲ–
Ó˚ˆÏí˛Ó˚ í˛z˛õÓ˚ Üœ˛ƒyˆÏ¡õÓ˚ ò%•z !òˆÏܲ ò%!ê˛ Ü%˛[˛°# L1 ~ÓÇ L2 çí˛¸yˆÏöy ÌyˆÏܲ– L1 ~ÓÇ ï˛yÓ˚ §üyhsˇÓ˚y° Ä ~ܲ!ê˛
˛õ!Ó˚Óï≈˛öˆÏÎyàƒ ôyÓ˚ܲ C, í˛zû˛ˆÏÎ˚ ~ܲ!ê˛ xö%öyò Óï≈˛ö# ˜ï˛Ó˚# ܲˆÏÓ˚ ~ܲ!ê˛ ê˛Δyö!çfiê˛yÓ˚ flõ®Ü˛ Óï≈˛ö#ˆÏï˛
(oscillator circuit) ܲyˆÏ°QˆÏÓ˚Ó˚ §ˆÏD Î%_´ ÌyˆÏܲ– L2 Ü%˛[˛°# ê˛Δyö!ê˛fiê˛yˆÏÓ˚Ó˚ ˆÓ§ ~ÓÇ ~!üê˛yˆÏÓ˚Ó˚ üˆÏôƒ Î%_´
ÌyˆÏܲ–
≤Ã̈Ïü !öˆÏܲ° Ó˚í˛!ê˛ˆÏܲ ã%˛¡∫ܲ xÌÓy !ö!ò≈‹T üyeyÓ˚ !ò‹T ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ §y•yˆÏ΃ ã%˛¡∫!Ü˛ï˛ Ü˛ˆÏÓ˚ Ó˚yáy •Î˚–
ôyÓ˚ܲ C-ˆÜ˛ ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ~üö xÓfliyÎ˚ xyöy •Î˚ñ ÎyˆÏï˛ Ó˚ˆÏí˛Ó˚ xö%˜Ïòâ≈ƒ ܲ¡õˆÏöÓ˚ fl∫yû˛y!Óܲ ܲ¡õyB˛ xyÓ˚
flõ®Ü˛ Óï≈˛ö#Ó˚ ܲ¡õyB˛ §üyö •Î˚– ï˛y •ˆÏ°•z xö%öyò •ˆÏÓ– í˛zû˛Î˚ Ü%˛[˛°#Ó˚ üˆÏôƒ Î%@¬ˆÏöÓ˚ ú˛ˆÏ° Óï≈˛ö#Ó˚ flõ®ö
ÓçyÎ˚ ÌyˆÏܲ– ܲyˆÏ°QˆÏÓ˚ ≤ÃÓyˆÏ•Ó˚ ú˛ˆÏ° ï˛yÓ˚ §ˆÏD ç!í˛¸ï˛ Ü%˛[˛°# L1 - ~Ó˚ ˆã˛Ô¡∫ܲ ˆ«˛ˆÏeÓ˚ ï˛#Ó ï˛yÓ˚ ˛õ!Ó˚Óï≈˛ö
~ÓÇ Ó˚ˆÏí˛Ó˚ ˜òâ≈ƒ ˛õ!Ó˚Óï≈˛ö •Î˚– ï˛áö x˛õÓ˚ Ü%˛[˛°# L2 ~Ó˚ úœ˛y: ˛õ!Ó˚Óï≈˛ö •Î˚ñ ÎyÓ˚ ú˛ˆÏ° ï˛yÓ˚ üˆÏôƒ ~ܲ!ê˛
xy!Ó‹T !Óû˛Ó (induced e.m.f.) §,‹T •Î˚– ˆÎˆÏ•ï%˛ ~•z Ü%˛[˛°# ê˛Δyö!çfiê˛yˆÏÓ˚Ó˚ ˆÓˆÏ§Ó˚ §ˆÏD Î%_´ñ ~•z !Óû˛Ó
xyÓyÓ˚ !ÓÓ!ô≈ï˛ •Î˚ñ ~ˆÏï˛ Ü˛yˆÏ°QÓ˚ ≤ÃÓy• ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ñ ï˛y xyÓyÓ˚ L1 ~ÓÇ L2 ~Ó˚ ˆã˛Ô¡∫ܲ
ˆ«˛eˆÏܲ ≤Ãû˛y!Óï˛ Ü˛ˆÏÓ˚– ~•zû˛yˆÏÓ flõ®ö ã˛°ˆÏï˛•z ÌyˆÏܲ– ~•z flõ®ˆÏöÓ˚ ܲ¡õyB˛ Ó˚ˆÏí˛Ó˚ ˜òˆÏâ≈ƒÓ˚ •…y§ Ó,!k˛Ó˚
ܲ¡õyˆÏB˛Ó˚ §üyö •ÄÎ˚yÎ˚ xö%öyò xÓfliyÓ˚ §,!‹T •Î˚ ~ÓÇ xö%öyò# ܲ¡õyˆÏB˛Ó˚ ¢∑ Ó˚ˆÏí˛Ó˚ ≤Ãyhsˇ ˆÌˆÏܲ í˛zͲõy!òï˛
•Î˚–
Óï≈˛ö#ˆÏï˛ ê˛Δyö!çfiê˛yÓ˚ ÓƒÓ•yˆÏÓ˚Ó˚ §%!Óôy ~•z ˆÎñ ܲü ¢!_´Ó˚ !Óû˛Ó í˛zͧ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ¢∑ í˛zͲõyòö ܲÓ˚y
ÎyÎ˚– flõ®Ü˛ Óï≈˛ö#Ó˚ ܲ¡õyB˛
f =
12LC π
...12.2
~ÓÇ xö%öyò# xÓfliyÎ˚ ~•z ܲ¡õyB˛ Ó˚ˆÏí˛Ó˚ fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyˆÏB˛Ó˚ §üyö– ˆòáyˆÏöy ã˛yÎ˚ñ ˆã˛Ô¡∫ܲ
ï˛!ï˛Ó˚ ú˛ˆÏ° Ó˚ˆÏí˛Ó˚ í˛z˛õyòyˆÏöÓ˚ •zÎ˚Ç =îyˆÏB˛Ó˚ ܲyÎ≈ܲÓ˚# üyö òyÑí˛¸yÎ˚ Y' = Y – 4 0iμ μ Y2K2B02, ˆÎáyˆÏö B0=
Ó˚ˆÏí˛Ó˚ ≤ÃyÌ!üܲ ˆã˛Ô¡∫ܲ xyˆÏÓ¢
μ
i
μ
0 = Ó˚ˆÏí˛Ó˚ ≤ÃyÌ!üܲ ˆã˛Ô¡∫ܲ xyˆÏӈϢÓ˚ xÓfliyÎ˚
0 BB
BH=
∂ ⎛⎞⎜⎟ ∂ ⎝⎠
~Ó˚ üyöñ
Y = •zÎ˚Ç =îyˆÏB˛Ó˚ fl∫yû˛y!Óܲ üyö– ü)°§%ˆÏÓ˚ ܲ!¡õï˛ •ˆÏ° Ó˚ˆÏí˛Ó˚ ܲ¡õyB˛
f = 12
Yl ρ
′...12.3
ˆÎáyˆÏö Ó˚ˆÏí˛Ó˚ ˜òâ≈ƒ l, Ó˚ˆÏí˛Ó˚ í˛z˛õyòyˆÏöÓ˚ âöc
ρ
~ÓÇ •zÎ˚Ç =öyB˛ (Young’s modulus) Y'– Ó˚ˆÏí˛Ó˚
˜òâ≈ƒ ~ÓÇ ôyÓ˚ˆÏܲÓ˚ ôyÓ˚ܲc ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ñ ≤ÃyÎ˚ 5 ˆÌˆÏܲ 60KHz xÓ!ô ¢∑ í˛zͲõyòö
ܲÓ˚y ÎyÎ˚–
311
12.2.2 ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ í˛zͲõyòܲ (Piezoelectric generator)
~•z ôÓ˚ˆÏöÓ˚ Îsf•z §ÓˆÏã˛ˆÏÎ˚ xyô%!öܲ x!ï˛¢∑ í˛zͲõyòܲ– ~Ó˚ !e´Î˚y˛õk˛!ï˛ Ó%é˛ˆÏï˛ •ˆÏ° xyˆÏà çyöˆÏï˛ •ˆÏÓ
ã˛y˛õÈÙÈ!Óò%ƒÍ !e´Î˚y (Piezoelectric effect) ܲyˆÏܲ ӈϰ–
ˆÜ˛yÎ˚yê≈˛çñ ê%˛Ó˚üƒy!°ö ≤Ãû,˛!ï˛ Ü˛ˆÏÎ˚Ü ôÓ˚ˆÏöÓ˚ ˆÜ˛°yˆÏ§Ó˚ ˜Ó!¢‹Tƒ ~•z ˆÎñ ˆÜ˛°yˆÏ§Ó˚ í˛z˛õˆÏÓ˚ !ö!ò≈‹T !òˆÏܲ˛
˛õ#í˛¸ö ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° !ö!ò≈‹T ˛õyˆÏŸª≈ xyôyö §,‹T •Î˚ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ˆÜ˛°yˆÏ§Ó˚ ò%•z !Ó˛õÓ˚#ï˛ ï˛ˆÏ°Ó˚ üˆÏôƒ !Óû˛Ó
≤ÈÏû˛ò í˛zͲõߨ •Î˚– !Óû˛Ó ≤ÈÏû˛ˆÏòÓ˚ ˛õ!Ó˚üyî ˛õ#í˛¸ˆÏöÓ˚ §üyö%˛õyï˛# •Î˚ ~ÓÇ ï˛yÓ˚ !ò¢yÄ ˛õ#í˛¸ˆÏöÓ˚ !ò¢yÓ˚ í˛z˛õˆÏÓ˚
!öû≈˛Ó˚ ܲˆÏÓ˚– ~•z !Ó˛õÓ˚#ï˛ âê˛öyÄ °«˛ƒ ܲÓ˚y ÎyÎ˚– ˆÜ˛°yˆÏ§Ó˚ í˛z˛õÓ˚ ï˛!í˛¸Í ˆ«˛e ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° !ö!ò≈‹T !òˆÏܲ
§ÇˆÏܲyã˛ö ÈÙÈ ≤çyÓ˚î •Î˚– ~•z ˛õ!Ó˚Óï≈˛ö ≤ÃÎ%_´ ï˛!í˛¸Í ˆ«˛ˆÏeÓ˚ §üyö%˛õyï˛# •Î˚– ˛õ#í˛¸ˆÏöÓ˚ ú˛ˆÏ° !Óû˛Ó ≤ÈÏû˛ò
§,!‹T ~ÓÇ !Óû˛Ó ≤ÈÏû˛ˆÏòÓ˚ ú˛ˆÏ° ˛õ#í˛¸ö í˛zͲõyòˆÏöÓ˚ ~•z âê˛öyˆÏܲ ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛˛ ≤Ãû˛yÓ ÓˆÏ°– ~áyˆÏö Ú!ö!ò≈‹T
!òÜ˛Û Ó°ˆÏï˛ Ü˛# ˆÓyé˛yˆÏöy •ˆÏÎ˚ˆÏåÈñ ï˛y çyöy òÓ˚ܲyÓ˚– ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°yˆÏ§Ó˚ ÓƒÓ•yÓ˚ §ÓˆÏã˛ˆÏÎ˚ ˆÓ!¢ •Î˚ ӈϰ
ˆÜ˛yÎ˚yê≈˛ˆÏçÓ˚ í˛zòy•Ó˚î !òˆÏÎ˚•z xyˆÏ°yã˛öy ܲÓ˚y Îyܲ–
~ܲ çyï˛#Î˚ ˆÜ˛°yˆÏ§Ó˚ üôƒ !òˆÏÎ˚ xyˆÏ°yܲ Ó˚!Ÿ¬ ˆ≤Ã!Ó˚ï˛ •ˆÏ° §yôyÓ˚îï˛ ï˛y ò%Ûû˛yˆÏà û˛yà •ˆÏÎ˚ ÎyÎ˚–
!ܲv ~ܲ Óy ~ܲy!ôܲ !ӈϢ£Ï !ò¢yÎ˚ Ó˚!Ÿ¬ ˆ≤Ã!Ó˚ï˛ •ˆÏ° ï˛y !Óû˛_´ •Î˚ öy– ˆ§•z !ò¢yˆÏܲ xyˆÏ°yܲ#Î˚ x«˛
!ã˛e 12.2 ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°yˆÏ§Ó˚ x, y ~ÓÇ z x«˛§ü)•–
(optic axis) Óy z-x«˛ Ó°y •Î˚– ˆÜ˛yÎ˚yê≈˛ˆÏçÓ˚ £Ïí˛¸û%˛çyÜ,˛!ï˛ ˆÜ˛°yˆÏ§Ó˚ ò%•z ¢#£Ï≈ §ÇˆÏÎyàܲyÓ˚# ˆÓ˚áy ÓÓ˚yÓÓ˚
z-x«˛ •Î˚ñ ˆÎüö !ã˛e 12.2(a)-~ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~•z xˆÏ«˛Ó˚ °¡∫ˆÏÓ˚áy ÓÓ˚yÓÓ˚ ˆÜ˛°y§!ê˛ˆÏܲ ܲyê˛ˆÏ°
312
≤ÃfliˆÏFåÈò!ê˛ !ã˛e 12.2(b)-~Ó˚ üˆÏï˛y £Ïí˛¸û%˛çyÜ,˛!ï˛ •ˆÏÓ– ~Ó˚ ò%!ê˛ !Ó˛õÓ˚#ï˛ ˆÜ˛Ô!îܲ !Ó®%Ó˚ §ÇˆÏÎyàܲyÓ˚#
ˆÓ˚ܲyˆÏܲ x x«˛ Ó°y •Î˚– !ã˛ˆÏe x1, x2 ~ÓÇ x3 !ï˛ö!ê˛ x«˛ ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– xyÓ˚ !Ó˛õÓ˚#ï˛ ò%!ê˛ ˛õyˆÏŸª≈Ó˚
üôƒ!Ó®%Ó˚ §ÇˆÏÎyàܲyÓ˚# ˆÓ˚áyˆÏܲ y x«˛ Ó°y •Î˚– !ã˛ˆÏe y1, y2 ~ÓÇ y3 x«˛ !ï˛ö!ê˛Ä ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
x x«˛ˆÏܲ ï˛!í˛¸Í x«˛ (electrical axis) ~ÓÇ y x«˛ˆÏܲ Îy!sfܲ x«˛ (mechanical axis) Ó°y •Î˚– ˆÜ˛yˆÏöy
x xˆÏ«˛Ó˚ °¡∫ï˛ˆÏ° ܲyê˛y ˆÜ˛°y§ˆÏܲ x ˆåÈò (x – cut) ˆÜ˛°y§ ӈϰ S!ã˛e 12.2c)–
~ܲ!ê˛ x ˆåÈò ˆÜ˛°yˆÏ§Ó˚ Îy!sfܲ x«˛
x!û˛ü%ˆÏá ˛õ#í˛¸ö ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° ï˛!í˛¸Í x«˛
x!û˛ü%ˆÏá !Óû˛Ó í˛zͲõߨ •Î˚– ~Ó˚ !Ó˛õÓ˚#ï˛!ê˛Ä
•ˆÏÎ˚ ÌyˆÏܲ– ˆÜ˛°yˆÏ§Ó˚ çƒy!ü!ï˛Ü˛ xyܲyˆÏÓ˚Ó˚
í˛z˛õÓ˚ !öû≈˛Ó˚¢#° fl∫yû˛y!Óܲ ܲ¡õB˛ xyÓ˚ ≤ÃÎ%_´
≤Ãï˛ƒyÓï˛#≈ !Óû˛ˆÏÓÓ˚ ܲ¡õyB˛ §üyö •ˆÏ°
Îy!sfܲ xö%öyò •Î˚ ~ÓÇ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚
§Ó≈y!ôܲ •Î˚– ~Ó˚ܲü ~ܲ!ê˛ ˆÜ˛°yˆÏ§Ó˚ ï˛!í˛¸Í
xˆÏ«˛Ó˚ x!û˛ü%ˆÏá ï˛!í˛¸Í ˆ«˛e ≤ÈÏÎ˚yà ܲÓ˚ˆÏ°
ˆÜ˛yÎ˚yê≈˛ˆÏç §ÇˆÏܲyã˛öÈÙÈ≤çyÓ˚ˆÏîÓ˚ ú˛ˆÏ°
ܲ¡õˆÏöÓ˚ §,!‹T •ˆÏÓ– ˆÜ˛yÎ˚yê≈˛ç â!í˛¸ˆÏï˛
Îy!sfܲ à!ï˛ í˛zͲõyòö ~•z ˛õk˛!ï˛ˆÏï˛•z
ܲÓ˚y •ˆÏÎ˚ ÌyˆÏܲ– ~áö ˆòáy Îyܲñ
x!ï˛¢∑ í˛zͲõyòˆÏöÓ˚ ܲyˆÏç ˆÜ˛yÎ˚yê≈˛ˆÏçÓ˚
~•z ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ !e´Î˚y !ܲû˛yˆÏÓ ÓƒÓ•*ï˛
•Î˚–
!ã˛e 12.3 ˆòá%ö– ~ܲ!ê˛ ˆÜ˛yÎ˚yê≈˛ç (Q) ~üöû˛yˆÏÓ Ü˛yê˛y •°ñ ÎyˆÏï˛ x x«˛ ~Ó˚ !Ó˛õÓ˚#ï˛ ˛õyŸª≈ò%!ê˛Ó˚ °¡∫
x!û˛ü%á# •Î˚– ˆÜ˛°y§!ê˛ˆÏܲ ò%!ê˛ ôyï˛Ó ˛õyï˛ A Ä B ~Ó˚ üˆÏôƒ Ó˚yáy •°– ˛õyï˛ ò%!ê˛Ó˚ §ˆÏD ~ܲ!ê˛ Ü%˛[˛°#
(L) Î%_´ xyˆÏåÈ– flõ®Ü˛ Óï≈˛ö#Ó˚ ~!üê˛yÓ˚ xLjϢ Ü%˛[˛°# L1 ~ÓÇ ˛õ!Ó˚Óï≈˛ö¢#° ôyÓ˚ܲ C !üˆÏ° ~ܲ!ê˛ xö%öyò#
Óï≈˛ö# !öü≈yî ܲˆÏÓ˚ˆÏåÈñ Îy flõ®ö ÓçyÎ˚ Ó˚yáˆÏï˛ §y•y΃ ܲÓ˚ˆÏÓ– x˛õÓ˚ Ü%˛[˛°# L2 ܲyˆÏ°QÓ˚ Óï≈˛ö#ˆÏï˛ Î%_´
xyˆÏåÈ– ôyÓ˚ܲˆÏܲ ~üö üyˆÏö !fliÓ˚ Ó˚yáy •Î˚ñ ÎyˆÏï˛ L1 Ä C myÓ˚y !öô≈y!Ó˚ï˛ Ü˛¡õyB˛
() 1 1/2LC π
ˆÜ˛°yˆÏ§Ó˚
313
fl∫yû˛y!Óܲ ܲ¡õyˆÏB˛Ó˚ §üyö •Î˚– flõ®Ü˛ Óï≈˛ö#Ó˚ ≤Ãû˛yÓ L1 Ä L2 Ü%˛[˛°#Ó˚ §ˆÏD ˛õyÓ˚flõ!Ó˚ܲ xyˆÏӈϢÓ˚ ú˛ˆÏ°
L Ü%˛[˛°#ˆÏï˛ §MÈ˛y!Ó˚ï˛ •Î˚ ~ÓÇ ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°yˆÏ§ ≤Ãï˛ƒyÓï˛#≈ ï˛!í˛¸Í ˆ«˛e xyˆÏÓ˚y!˛õï˛ •Î˚– ï˛áö ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛
!e´Î˚yÓ˚ ú˛ˆÏ° ˆÜ˛°yˆÏ§ ܲ¡õö •Î˚ ~ÓÇ ï˛y ˆÌˆÏܲ ~ܲ•z ܲ¡õyˆÏB˛Ó˚ ¢∑ !öà≈ï˛ •ˆÏï˛ ÌyˆÏܲ– ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§!ê˛
Îáö ü)°§%ˆÏÓ˚ ܲ!¡õï˛ •Î˚ ï˛áö ï˛yÓ˚ xˆÏ«˛Ó˚ §ˆÏD °¡∫ï˛° ò%!ê˛ §%flõ® ~ÓÇ üôƒï˛°!ê˛ !öflõ® ÌyˆÏܲ–
ˆÜ˛°yˆÏ§Ó˚ ˆÓÑô Î!ò l •Î˚ ~ÓÇ x x«˛ ÓÓ˚yÓÓ˚ ¢ˆÏ∑Ó˚ ˆÓà Î!ò v •Î˚ ï˛ˆÏÓ ˙ ˆÜ˛°yˆÏ§ í˛zͲõߨ fliyî%ï˛Ó˚ˆÏDÓ˚
ï˛Ó˚D˜Ïòâ≈ƒ •ˆÏÓ ï˛ 2lλ = , xÌ≈yÍñ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ •ˆÏÓ
n1 =
2vv
l λ=
ˆÜ˛°y§!ê˛ xÓ¢ƒ ï,˛ï˛#Î˚ñ ˛õMÈ˛ü ≤Ãû,˛!ï˛ xÎ%@¬ í˛z˛õ§%ˆÏÓ˚Ä Ü˛!¡õï˛ •ˆÏï˛ ˛õyÓ˚ˆÏÓ– ~•z ܲ¡õö=!°Ó˚ ˆ«˛ˆÏeÄ
ˆÜ˛°yˆÏ§Ó˚ Óy•zˆÏÓ˚Ó˚ ò%!ê˛ ï˛° §%flõ® ~ÓÇ üôƒï˛°!ê˛ !öflõ® ÌyܲˆÏÓ– ~=!°Ó˚ ܲ¡õyB˛ ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚
!ï˛ö=îñ ˛õyÑã˛=î ≤Ãû,˛!ï˛ •ˆÏÓ xÌ≈yÍñ ~=!°Ó˚ Ó˚y!¢üy°y •ˆÏÓ ÎÌye´ˆÏü n3=3n1=
32
vl
, n5 = 5n1 =
52
vl
•zï˛ƒy!ò– ï˛ˆÏÓ ˆÜ˛°y§!ê˛ !mï˛#Î˚ñ ã˛ï%˛Ì≈ ≤Ãû,˛!ï˛ Î%@¬ í˛z˛õ§%ˆÏÓ˚ ܲ!¡õï˛ •ˆÏï˛˛ ˛õyÓ˚ˆÏÓ öyñ ˆÜ˛ö öy ~=!°Ó˚ ˆ«˛ˆÏe
ˆÜ˛°yˆÏ§Ó˚ üôƒï˛ˆÏ° §%flõ® !Ó®% ÌyܲˆÏÓ ~ÓÇ ˆÜ˛°y§!ê˛Ó˚ û˛Ó˚ˆÏܲw ܲ!¡õï˛ •ˆÏï ÌyܲˆÏÓ– ˆÎˆÏ•ï%˛ ˆÜ˛°yˆÏ§Ó˚
í˛z˛õÓ˚ ˆÜ˛yöÄ °!∏˛ Ó° !e´Î˚y ܲˆÏÓ˚ öyñ xï˛~Ó ~ ôÓ˚ˆÏöÓ˚ ܲ¡õö âê˛y §Ω˛Ó öÎ˚–
ˆÜ˛°yˆÏ§Ó˚ ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ xy®yç xyüÓ˚y §•ˆÏç•z ˆ˛õˆÏï˛ ˛õy!Ó˚– x ˆåÈò ˆÜ˛yÎ˚yê≈˛ˆÏç ¢ˆÏ∑Ó˚ ˆÓà
5720ms–1– ôÓ˚&öñ ˆÜ˛°yˆÏ§Ó˚ ˆÓô 1mm– ˆ§ˆÏ«˛ˆÏe ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ n1 =
35720210− ×
Óyñ 2·86 × 106Hz–
(2n +1)-ï˛ü í˛z˛õ§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ (n = ˛õ)î≈§ÇáƒyV ~Ó˚ 2n + 1 =î •ˆÏÓ– xÌ≈yÍñ ï,˛ï˛#Î˚ í˛z˛õ§%ˆÏÓ˚Ó˚ ܲ¡õyB˛
•ˆÏÓ n3 = 3 × 2·86 × 106 Hz Óyñ 8·6 MHz, ˛õMÈ˛ü í˛z˛õ§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ •ˆÏÓ n5 = 5 × 2·86 × 106Hz
Óy 14·3MHz •zï˛ƒy!ò– ˆÜ˛°yˆÏ§Ó˚ ˆÓô xyÓ˚Ä ˆÓ!¢ •ˆÏ° fl∫“ï˛Ó˚ ܲ¡õyˆÏB˛Ó˚ ~ÓÇ ˆÓô xyÓ˚Ä Ü˛ü •ˆÏ°
!ã˛e 12.3 ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ x!ï˛¢∑ í˛zͲõyòܲ–
314
í˛zFã˛ï˛ü ܲ¡õyˆÏB˛Ó˚ ü)°§%Ó˚ ˛õyÄÎ˚y ÎyˆÏÓ– ï˛ˆÏÓ ˆÜ˛°y§!ê˛Ó˚ ˆÓô ܲ!üˆÏÎ˚ 50MHz xˆÏ˛õ«˛y x!ôܲ ܲ¡õyB˛
˛õyÄÎ˚y ÎyÎ˚ öy ˆÜ˛ööy ˆÜ˛°y§!ê˛ ï˛áö xï˛ƒhsˇ ò%Ó≈° Ä û˛D%Ó˚ •ˆÏÎ˚ ˛õˆÏí˛¸–
x!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòö ˛õk˛!ï˛ §¡∫ˆÏ¶˛ ˛õí˛¸ˆÏ°ö– ~ÓyÓ˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1 :
(a) ˆÜ˛yö ܲ¡õyˆÏB˛Ó˚ ¢∑ˆÏܲ x!ï˛¢∑ Ó°y •Î˚⁄
(b) ~ܲ!ê˛ xÎ˚ˆÏÿ˛Ô¡∫ܲ ˛õòyˆÏÌ≈ !ö!ü≈ï˛ òˆÏ[˛Ó˚ í˛z˛õÓ˚ 50KHz ܲ¡õyˆÏB˛Ó˚ Ä 0·2Wbm–2 !ÓhflÏyˆÏÓ˚Ó˚ ˛õÓ˚Óï˛#≈
ˆã˛Ô¡∫ܲ xyˆÏÓ¢ ≤ÈÏÎ˚yà ܲÓ˚y •°– ò[˛!ê˛ Î!ò ≤ÃyÌ!üܲ xÓfliyÎ˚ !Óã%˛¡∫!Ü˛ï˛ ÌyˆÏܲ ï˛ˆÏÓ ˆ§!ê˛Ó˚ ܲ¡õˆÏöÓ˚
ܲ¡õyB˛ Ü˛ï˛ •ˆÏÓ⁄ Î!ò ≤ÃyÌ!üܲ xÓfliyÎ˚ ò[˛!ê˛Ó˚ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ 0·5Wbm–2 •Î˚ ï˛áö•z Óy ܲ¡õyB˛
Ü˛ï˛ •ˆÏÓ⁄
(c) 1 MHz ܲ¡õyˆÏB˛Ó˚ ¢∑ í˛zͲõyòˆÏöÓ˚ çöƒ xy˛õ!ö ˆÜ˛yö ˆÓˆÏôÓ˚ ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏÓö⁄
12.3 !Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏà !öî≈Î˚ ≠!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏà !öî≈Î˚ ≠!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏà !öî≈Î˚ ≠!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏà !öî≈Î˚ ≠!Ó!û˛ß¨ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓˆÏà !öî≈Î˚ ≠
xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈöñ x!ï˛¢∑ Ä §yôyÓ˚î ¢ˆÏ∑Ó˚ üˆÏôƒ ˆÜ˛yöÄ ˆüÔ!°Ü˛ ˛õyÌ≈ܲƒ öy ÌyܲˆÏ°Ä
í˛zFã˛ Ü˛¡õyB˛•z x!ï˛¢ˆÏ∑Ó˚ ˜Ó!¢‹Tƒ– ˆÜ˛yöÄ üyôƒˆÏü x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà §yôyÓ˚î ¢∑ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓˆÏàÓ˚
§üyö– ܲ¡õyB˛ x!ôܲ •ÄÎ˚yÓ˚ ú˛ˆÏ° x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ xˆÏöܲ ܲü– í˛zòy•Ó˚î !•§yˆÏÓ Ó°y ÎyÎ˚ñ 1MHz
ܲ¡õyˆÏDÓ˚ x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ ÓyÎ˚%ˆÏï˛ 0·34mm, çˆÏ° ≤ÃyÎ˚ 1·5 mm(30°c) ~ÓÇ !fiê˛ˆÏ° ≤ÃyÎ˚ 6 mm–
ï˛Ó˚D˜Ïòâ≈ƒ ˆåÈyê˛ •ÄÎ˚yÓ˚ ˛ú˛ˆÏ° ~•z ¢∑ˆÏܲ xˆÏ˛õ«˛yÜ,˛ï˛ ˆåÈyê˛ ü§,î ≤Ã!ï˛ú˛°Ü˛ myÓ˚y ≤Ã!ï˛ú˛!°ï˛ ܲÓ˚y ÎyÎ˚–
xyÓyÓ˚ xyˆÏ°yܲ ï˛Ó˚ˆÏDÓ˚ üˆÏï˛y ~•z ï˛Ó˚ˆÏDÓ˚ Óƒ!ï˛ã˛yÓ˚ (interference) Ä ÓƒÓï≈˛öÄ °«˛ƒ ܲÓ˚y ÎyÎ˚– ~•z ~ܲˆÏܲ
˛õˆÏÓ˚Ó˚ xö%ˆÏFåȈÏò xy˛õ!ö x!ï˛¢ˆÏ∑Ó˚ ~üö !ܲå%È ≤ÈÏÎ˚yˆÏàÓ˚ ܲÌy ˛õí˛¸ˆÏÓö ˆÎáyˆÏö x!ï˛¢ˆÏ∑Ó˚ ˆÓà çyöy
≤ÈÏÎ˚yçö– ~áö xyüÓ˚y ~•z ˆÓà !öî≈ˆÏÎ˚Ó˚ ܲˆÏÎ˚ܲ!ê˛ ˛õk˛!ï˛ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
12.3.1 xyˆÏ°yܲ ÓƒÓï≈˛öxyˆÏ°yܲ ÓƒÓï≈˛öxyˆÏ°yܲ ÓƒÓï≈˛öxyˆÏ°yܲ ÓƒÓï≈˛öxyˆÏ°yܲ ÓƒÓï≈˛ö (Optical diffraction) ˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛
ï˛Ó˚ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ x!ï˛¢∑ ˆ≤ÃÓ˚î ܲÓ˚ˆÏ° ï˛Ó˚ˆÏ°Ó˚ ܲîy=!° ˛õÎ≈yÎ˚e´ˆÏü xyˆÏ®y!°ï˛ •Î˚– ú˛ˆÏ° ï˛Ó˚D
x!û˛ü%ˆÏá üyôƒˆÏüÓ˚ ã˛y˛õ Ä âöc fliyˆÏö fliyˆÏö˛ ˛õÎ≈yÎ˚e´ˆÏü ܲü ˆÓ!¢ •Î˚– âöˆÏcÓ˚ §ˆÏD üyôƒˆÏüÓ˚ ≤Ã!ï˛§Ó˚yB˛Ä
˛õÎ≈yÎ˚e´ˆÏü ܲˆÏü ÓyˆÏí˛¸– üyôƒˆÏüÓ˚ ~•z ܲü Ä ˆÓ!¢ ≤Ã!ï˛§Ó˚yˆÏB˛Ó˚ xÇ¢=!° !ö!ò≈‹T ò)Ó˚c ˛õˆÏÓ˚ ˛õˆÏÓ˚ ˛õÎ≈yÎ˚e´ˆÏü
•ÄÎ˚yˆÏï˛ üyôƒü!ê˛ ~ܲ!ê˛ ÓƒÓï≈˛ö ˆ@ˇÃ!ê˛Ç(grating)-~Ó˚ üˆÏï˛y ܲyç ܲˆÏÓ˚– ˆÓ!¢Ä ܲü âöˆÏcÓ˚ xÇ¢=!°
ˆ@ˇÃ!ê˛ÇˆÏÎ˚Ó˚ ÎÌye´ˆÏü xfl∫FåÈ Ä fl∫FåÈ xLjϢÓ˚ û)˛!üܲy ˆöÎ˚– ~•z xÓfliyÓ˚ üyôƒˆÏü xyˆÏ°y ˛õí˛¸ˆÏ° ï˛yÓ˚ ÓƒÓï≈˛ö
•Î˚– âê˛öy!ê˛ˆÏܲ ¢∑yˆÏ°yܲ ≤Ãû˛yÓ (Acousto—optic effect) Ó°y •Î˚– ÓƒÓï≈˛ö Óî≈y°# !ӈϟ’£Ïî ܲˆÏÓ˚
x!ï˛¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
315
xyˆÏ°yܲ ï˛Ó˚ˆÏDÓ˚ ÓƒÓï≈˛ö ≤Ãï˛ƒ«˛ ܲÓ˚yÓ˚ ÓƒÓfliy!ê˛ xy˛õ!ö 12.4 !ã˛e ˆÌˆÏܲ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~áyˆÏö ï˛Ó˚°
üyôƒˆÏü flõ®Ü˛ Óï≈˛ö#Ó˚ §ˆÏD Î%_´ ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§!ê˛ ˆí˛yÓyˆÏöy xyˆÏåÈ– ˆÜ˛°yˆÏ§Ó˚ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° üyôƒˆÏü
x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D í˛zͲõߨ •Î˚– ~•z ï˛Ó˚D
§yôyÓ˚îï˛ ï˛Ó˚ˆÏ°Ó˚ ˛õyˆÏeÓ˚ ˆòÄÎ˚yˆÏ°
≤Ã!ï˛ú˛!°ï˛ •Î˚ ~ÓÇ ï˛Ó˚ˆÏ°Ó˚ üˆÏôƒ
fliyî%ï˛Ó˚D í˛zͲõߨ ܲˆÏÓ˚– ~•z ï˛Ó˚ˆÏDÓ˚
!öflõ® !Ó®%=!°Ó˚ (N) ÓƒÓôyö •Î˚
x!ï˛¢ Ï∑Ó˚ ï˛Ó˚D ò Ïâ≈ƒÓ˚ x Ïô≈ܲñ xÌ≈yÍñ
2λ
– !öflõ® !Ó®%=!°ˆÏï˛˛ ˛õÓ˚˛õÓ˚
§ˆÏÓ≈yFã˛ âö#û˛Óö Ä §ˆÏÓ≈yFã˛ xö%û˛Óö
âˆÏê˛ xÌ≈yÍ ò%!ê˛ §ˆÏÓ≈yFã˛ âö#û˛Óö Ä
í˛zFã˛ï˛ü ≤Ã!ï˛§Ó˚yˆÏB˛Ó˚ ï˛ˆÏ°Ó˚ üˆÏôƒ
ÓƒÓôyö •Î˚
λ
– ~•z ÓƒÓôyö•z ï˛Ó˚ˆÏ°Ó˚
ܲyÎ≈ܲÓ˚# ˆ@ˇÃ!ê˛Ç ÓƒÓôyö (grating space)– xÓ¢ƒ ~áyˆÏö í˛zˆÏÕ‘á ܲÓ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ¢∑ï˛Ó˚D ˛õyˆÏeÓ˚
ˆòÄÎ˚yˆÏ° ≤Ã!ï˛ú˛!°ï˛ öy •ˆÏÎ˚ ˆ¢y!£Ïï˛ •ˆÏ° ˆÎ ç°ï˛Ó˚ˆÏDÓ˚ í˛zͲõ!_ •Î˚ñ ï˛yˆÏï˛Ä ˛õÓ˚ ˛õÓ˚ ò%!ê˛ §ˆÏÓ≈yFã˛ âö#û˛Óö
Óy §ˆÏÓ≈yFã˛ xö%û˛ÓˆÏöÓ˚ ï˛ˆÏ°Ó˚ üˆÏôƒ
λ
ÓƒÓôyö ÌyˆÏܲ ~ÓÇ ~ܲ•z Î%!_´ ≤ÈÏÎyçƒ •Î˚–
xyˆÏ°yܲ!Óòƒy §Çe´yhsˇ ˛õÎ≈yÎ˚ ˆÌˆÏܲ xy˛õ!ö çyöˆÏï˛ ˛õyÓ˚ˆÏÓö ˆÎñ
λ
i ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ~ܲÓî#≈ xyˆÏ°yܲÓ˚!Ÿ¬
Î!ò ~üö ~ܲ!ê˛ ˆ@ˇÃ!ê˛Ç ~Ó˚ í˛z˛õÓ˚ °¡∫û˛yˆÏÓ xy˛õ!ï˛ï˛ •Î˚ ÎyÓ˚ ˆ@ˇÃ!ê˛Ç ÓƒÓôyö d, ï˛ˆÏÓ ˆÎ §ü#ܲÓ˚î ˆÌˆÏܲ
˙ xyˆÏ°yܲÓ˚!Ÿ¬Ó˚ ÓƒÓï≈˛ˆÏöÓ˚ ú˛ˆÏ° í˛zͲõߨ m-e´ˆÏüÓ˚ Óî≈y!° ˆÓ˚áyÓ˚ ÓƒÓï≈˛ö ˆÜ˛yî
θ
m-˛õyÄÎ˚y ÎyˆÏÓñ ˆ§!ê˛
•°≠ dsin
θ
m= m
λ
i ...12.4
!ܲv xyüÓ˚y ˆòˆÏá!åÈñ ~ˆÏ«˛ˆÏe x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ (
λ
) ܲyÎ≈ܲÓ˚# ˆ@ˇÃ!ê˛Ç ÓƒÓôyö– §%ï˛Ó˚yÇñ xyüÓ˚y
!°áˆÏï˛ ˛õy!Ó˚
λ
sin
θ
m= m
λ
i ...12.5
~•z §)e ˆÌˆÏܲ
θ
m ˆÜ˛yî ˆüˆÏ˛õ x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ
λ
çyöy ÎyˆÏÓ– ~áö ôÓ˚&öñ x!ï˛¢ˆÏ∑Ó˚ ܲ¡õyB˛
n, ÎyÓ˚ üyö ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°yˆÏ§Ó˚ ˆÓô ˆÌˆÏܲ çyöy ÎyÎ˚– xï˛~Óñ x!ï˛¢ˆÏ∑Ó˚ ˆÓà ≠
v = n
λ
= mn
λ
i /sin
θ
m ...12.6
12.6 §)e ˆÌˆÏܲ x!ï˛¢ˆÏ∑Ó˚ ˆÓà §Ó˚y§!Ó˚ ÓyÓ˚ ܲÓ˚y ÎyÎ˚–
316
12.3.2 ¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚¢y!∑ܲ Óƒ!ï˛ã˛yÓ˚ (Acoustic interference) ˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛˛õk˛!ï˛
~•z ˛õk˛!ï˛ˆÏï˛ ò%•z ¢∑ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛ õyï˛ˆÏöÓ˚ ú˛ˆÏ° ï˛yˆÏòÓ˚ ˆÎ Óƒ!ï˛ã˛yÓ˚ âˆÏê˛ñ ˆ§!ê˛ˆÏܲ ܲyˆÏç °yàyˆÏöy
•Î˚– àƒy§ ~ÓÇ ï˛Ó˚°ñ ò%•z ôÓ˚ˆÏöÓ˚ üyôƒˆÏü•z ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà !öî≈ˆÏÎ˚ ~•z ˛õk˛!ï˛ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚–
12.5(a) !ã ˛ˆ Ïe ~ܲ!ê˛
Óƒ!ï˛ã˛yÓ˚üy˛õܲ ΈÏsfÓ˚ Ó˚*˛õˆÏÓ˚áy
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– 12.4 !ã˛ˆÏe ˆÎ
x!ï˛¢∑ í˛ zͲõyòܲ ˆòáyˆÏöy
• ÏÎ˚ ÏåÈñ § çyï˛#Î˚ ~ܲ!ê˛ Ü˛yÎ˚yê≈ ç
ˆÜ˛°y§Î%_´ flõ®Ü˛ Óï≈˛ö# ~áyˆÏö
ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ˆÜ˛°yˆÏ§Ó˚
§yüˆÏö ˆ§!ê˛Ó˚ §ˆÏD §üyhsˇÓ˚y° ~ܲ!ê˛ ü§,î ≤Ã!ï˛ú˛°Ü˛ Ó˚yáy •Î˚ñ ˆÎ!ê˛ˆÏܲ fl;$˛Ó˚ §y•yˆÏ΃ §)-û˛yˆÏÓ Î%@¬û˛yˆÏÓ
x@ˇÃ ˛õÿ˛yˆÏï˛ §Ó˚yˆÏöy ÎyÎ˚– í˛zͲõyòܲ ˆÜ˛°y§!ê˛Ó˚ ˜òâ≈ƒ Ä ≤Ãfli ¢∑ï˛Ó˚ˆÏDÓ˚ ï%˛°öyÎ˚ Óí˛¸ •ÄÎ˚yÎ˚ ˆ§!ê˛Ó˚
ܲ¡õˆÏöÓ˚ ú˛ˆÏ° §üï˛° ¢∑ï˛Ó˚D í˛zͲõy!òï˛ •Î˚– ~Ó˚ §ˆÏD ≤Ã!ï˛ú˛!°ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚ ú˛ˆÏ°
üyôƒˆÏü fliyî%ï˛Ó˚D §,!‹T •Î˚– xy˛õ!ö xyˆÏà ˆçˆÏöˆÏåÈö ˆÎñ fliyî%ï˛Ó˚ˆÏD ˛õÓ˚˛õÓ˚ ò%!ê˛ §%flõ® !Ó®% Óy !öflõ®
!Ó®%Ó˚ üˆÏôƒ
2λ
ÓƒÓôyö ÌyˆÏܲ– ˆÜ˛°y§ Ä ≤Ã!ï˛ú˛°ˆÏܲÓ˚ üˆÏôƒ ÓƒÓôyö Îáö
24nλλ +
xÌ≈yÍ 122
nλ ⎛⎞ + ⎜⎟ ⎝⎠
•Î˚ñ
ï˛áö ≤Ã!ï˛ú˛°ˆÏܲ !öflõ® !Ó®% Ä ˆÜ˛°yˆÏ§Ó˚ §yüˆÏöÓ˚ ï˛ˆÏ° §%flõ® !Ó®% ÌyˆÏܲ ~ÓÇ ~•z xÓfliyÎ˚ ˆÜ˛°yˆÏ§Ó˚
flõ®Ü˛ Óï≈˛ö#ˆÏï˛ ï˛!í˛¸Í≤ÃÓy• §ˆÏÓ≈yFã˛ üyˆÏö ˆ˛õÔÑåÈÎ˚–
(12.5a) !ã˛ˆÏe !öflõ® Ä §%flõ® !Ó®%=!°Ó˚ xÓfliyö ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~•z xÓfliyÎ˚ flõ®Ü˛ Óï≈˛ö#ˆÏï˛
ï˛!í˛¸Í≤ÃÓy• §ˆÏÓ≈yFã˛ •ˆÏÓ– ~áö ≤Ã!ï˛ú˛°Ü˛!ê˛ §Ó˚yˆÏöy •ˆÏ° ï˛!í˛¸Í≤ÃÓy• ≤Ã̈Ïü ܲüˆÏÓ !ܲv e´ü¢ Îáö
≤Ã!ï˛ú˛°Ü˛!ê˛Ó˚ §Ó˚î
2λ
•ˆÏÓ ï˛áö xyÓyÓ˚ ˆÜ˛°yˆÏ§ §%flõ® !Ó®%
˜ï˛!Ó˚ •ˆÏÓ Ä ï˛!í˛¸Í≤ÃÓy• §ˆÏÓ≈yFã˛ üyˆÏö ˆ˛õÔÑåȈÏÓ– 12.5 (b) !ã˛e
ˆÌˆÏܲ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ Äë˛yöyüyÓ˚ ôyÓ˚y!ê˛ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
~•z ˆ°á!ã˛e ˆÌˆÏܲ §•ˆÏç•z
2λ
~Ó˚ üyö !öî≈Î˚ ܲÓ˚y ÎyÎ˚ xyÓ˚ ï˛y
ˆÌˆÏܲ x!ï˛¢ˆÏ∑Ó˚ ˆÓàÄ çyöy ÎyÎ˚–
ï˛Ó˚ˆÏ°Ó˚ ˆ«˛ˆÏe Îy!sfܲ ÓƒÓfliy˛õöy ~ܲê%˛ xöƒÓ˚ܲü •Î˚– ~ܲ!ê˛
!ã˛e 12.5(a)
!ã˛e 12.5(b) ≤Ã!ï˛ú˛°ˆÏܲÓ˚ xÓfliyˆÏöÓ˚
§ˆÏD flõ®Ü˛ Óï≈˛ö#ˆÏï˛ ï˛!í˛¸ˆÏï˛Ó˚
˛õ!Ó˚Óï≈˛ö–
317
ï˛Ó˚°˛õ)î≈ ˛õyˆÏeÓ˚ !öˆÏã˛ x!ï˛¢∑ í˛zͲõyòܲ ˆÜ˛yÎ˚yê˛≈ç!ê˛ °yàyˆÏöy ÌyˆÏܲ– ˛õyˆÏeÓ˚ í˛z˛õˆÏÓ˚ ~ܲ!ê˛ ü§,î ≤Ã!ï˛ú˛°Ü˛
ï˛Ó˚ˆÏ° ˆí˛yÓyˆÏöy xyˆÏåÈ– ~•z xÓfliyö §)-û˛yˆÏÓ ˛õ!Ó˚Óï≈˛ö ܲÓ˚y ÎyÎ˚ ~ÓÇ üy•zˆÏe´y!üê˛yˆÏÓ˚Ó˚ §y•yˆÏ΃ üy˛õy
ÎyÎ˚– ˛õyˆÏeÓ˚ !öˆÏã˛ ˆÌˆÏܲ í˛zˆÏë˛ xy§y x!ï˛¢∑ ï˛Ó˚D ï˛Ó˚ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ˆ≤Ã!Ó˚ï˛ •ˆÏÎ˚ xyÓyÓ˚ ï˛Ó˚ˆÏ°Ó˚ üˆÏôƒ•z
≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ !ú˛ˆÏÓ˚ xyˆÏ§–
12.4 x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓy•Ó˚x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓy•Ó˚x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓy•Ó˚x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓy•Ó˚x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà Ä ÓƒÓy•Ó˚
xy˛õ!ö ≤Ã̈Ïü•z ˆçˆÏöˆÏåÈö ˆÎñ x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ á%Ó Ü˛ü– ˆ§ ܲyÓ˚ˆÏî ~Ó˚ §)- Ä §üï˛° Ó˚!Ÿ¬=FåÈ
˜ï˛!Ó˚ ܲÓ˚y xˆÏ˛õ«˛yÜ,˛ï˛ §•ç– §yôyÓ˚î ¢ˆÏ∑Ó˚ ï%˛°öyÎ˚ xˆÏöܲ ˆåÈyê˛ ≤Ã!ï˛ú˛°ˆÏܲ ~•z ï˛Ó˚D ≤Ã!ï˛ú˛!°ï˛ •Î˚–
üyôƒˆÏü x!ï˛¢∑ ï˛Ó˚ˆÏDÓ˚ ˆ¢y£ÏˆÏîÓ˚ ˛õ!Ó˚üy˛õ §•ˆÏç•z ܲÓ˚y ÎyÎ˚– ˆÜ˛yöÄ x§FåÈ Ó› à!ï˛˛õˆÏÌ ÌyܲˆÏ° ~•z
ï˛Ó˚D ï˛yÓ˚ åÈyÎ˚y í˛zͲõߨ ܲˆÏÓ˚– ܲ¡õyB˛ ˆÓ!¢ •ÄÎ˚yÎ˚ x!ï˛¢∑ï˛Ó˚D x!ôܲ ˛õ!Ó˚üyî ¢!_´ Ó•ö ܲˆÏÓ˚ ~ÓÇ
xï˛ƒhsˇ ˆåÈyê˛ xyܲyˆÏÓ˚Ó˚ Ó›Ó˚ IJõÓ˚ ~Ó˚ üyôƒˆÏü ¢!_´ ˆ˛õÔшÏåÈ ˆòÄÎ˚y ÎyÎ˚– ~•z çöƒ x!ï˛¢ˆÏ∑Ó˚ ≤Ã!ï˛ú˛°ö
Ä Óyôy≤Ãy!ƈÏܲ ò)Ó˚c !öî≈Î˚ñ ˆÜ˛yöÄ Ü˛!ë˛ö Ó›Ó˚ í˛z˛õ!fli!ï˛ !öî≈Î˚ •zï˛ƒy!ò öyöy ܲyˆÏç °yàyˆÏöy •Î˚– ~•z çyï˛#Î˚
ܲˆÏÎ˚ܲ!ê˛ ≤ÈÏÎ˚yà ~áyˆÏö Óî≈öy ܲÓ˚y •°–
(i) §ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y Ä çˆÏ° !öü!Iï˛ Ó›Ó˚ ò)Ó˚c !öî≈Î˚ ≠§ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y Ä çˆÏ° !öü!Iï˛ Ó›Ó˚ ò)Ó˚c !öî≈Î˚ ≠§ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y Ä çˆÏ° !öü!Iï˛ Ó›Ó˚ ò)Ó˚c !öî≈Î˚ ≠§ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y Ä çˆÏ° !öü!Iï˛ Ó›Ó˚ ò)Ó˚c !öî≈Î˚ ≠§ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y Ä çˆÏ° !öü!Iï˛ Ó›Ó˚ ò)Ó˚c !öî≈Î˚ ≠
xy˛õ!ö •Î˚ï˛ ≤Ã!ï˛ôù!öÓ˚ §y•yˆÏ΃ §ü%ˆÏoÓ˚ àû˛#Ó˚ï˛y !öî≈ˆÏÎ˚Ó˚ ˛õk˛!ï˛ §¡∫ˆÏ¶˛ ˛õˆÏí˛¸ ÌyܲˆÏÓö– ~•z ܲyˆÏçÓ˚
çöƒ x!ï˛¢∑ á%Ó•z í˛z˛õˆÏÎyà#– x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D í˛zͲõyòö ܲˆÏÓ˚ ï˛y §ü%ˆÏoÓ˚ çˆÏ° §MÈ˛y!Ó˚ï˛ Ü˛Ó˚y~ÓÇ ï˛yÓ˚
≤Ã!ï˛ôù!ö @ˇÃ•î ܲÓ˚ˆÏï˛ ˛õyÓ˚yÓ˚ üˆÏôƒ ˆÎ §üÎ˚ x!ï˛Óy!•ï˛ •Î˚ñ ï˛yÓ˚ ˛õ!Ó˚üy˛õ ˆÌˆÏܲ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ò)Ó˚c çyöy
ÎyÎ˚– xÓ¢ƒ ~çöƒ §ü%ˆÏoÓ˚ çˆÏ° !ö!ò≈‹T í˛z£èï˛yÎ˚ ¢ˆÏ∑Ó˚ à!ï˛ˆÏÓà çyöy ≤ÈÏÎ˚yçö– ¢ˆÏ∑Ó˚ ≤Ã!ï˛ôù!öÓ˚
§y•yˆÏ΃ í%˛ˆÏÓyçy•yçñ çˆÏ° û˛y§üyö öyöyôÓ˚ˆÏöÓ˚ ï˛Ó˚# Ä xöƒyöƒ Ó›Ó˚ x!hflÏc Ä ò)Ó˚c !öî≈Î˚ñ ˆ§=!°Ó˚
à!ï˛!Ó!ôÓ˚ ˛õÎ≈ˆÏÓ«˛î Ä ˆ◊î#!Óû˛yà ç°Î%ˆÏk˛Ó˚ ~ܲ!ê˛ ≤ÈÏÎ˚yçö#Î˚ xD– ~Ó˚ çöƒ ˆÎ ˛õk˛!ï˛=!° ÓƒÓ•*ï˛
•Î˚ ˆ§=!°ˆÏܲ ~ܲˆÏe ˆ§yöyÓ˚ (SONAR = Sound Navigation and Ranging) Ó°y •Î˚–
~•z ܲyˆÏçÓ˚ çöƒ ˆÎ x!ï˛¢∑ ÓƒÓ•*ï˛ •Î˚ ï˛yÓ˚ àë˛ö ~üö •ÄÎ˚y ≤ÈÏÎ˚yçö ÎyˆÏï˛ ˆ§!ê˛ çˆÏ°Ó˚ üˆÏôƒ
ΈÏÌ‹T ï˛#Ó ï˛yÓ˚ x!ï˛¢∑ í˛zͲõyòö ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚ñ ˆÜ˛ööyñ !ӈϢ£Ïï˛ ≤Ã!ï˛ú˛°Ü˛!ê˛ xˆÏöܲ ò)ˆÏÓ˚ ÌyܲˆÏ° §MÈ˛y!Ó˚ï˛
ï˛Ó˚D ¢!_´Ó˚ x!ï˛ x“ xÇ¢•z ≤Ã!ï˛ôù!öÓ˚*ˆÏ˛õ @ˇÃy•ˆÏܲ ~ˆÏ§ ˆ˛õÔÑåÈyÎ˚– ~åÈyí˛¸y ≤Ã!ï˛ôù!ö xy§yÓ˚ !òܲ!ê˛Ä §)-
û˛yˆÏÓ !öî≈Î˚ ܲÓ˚y ≤ÈÏÎ˚yçö–
318
!ã˛e 12.6
12.6 !ã˛ˆÏe çˆÏ°Ó˚ üˆÏôƒ x!ï˛¢∑ ˆ≤ÃÓ˚ˆÏîÓ˚ í˛z˛õÎ%_´ x!ôܲ «˛üï˛yÓ˚ ~ܲ!ê˛ ˆã˛Ô¡∫ܲ ï˛!ï˛ í˛zͲõyòܲ ˆòáyˆÏöy
•ˆÏÎ˚ˆÏåÈ– ~áyˆÏö ܲˆÏÎ˚ܲ ¢ï˛ !öˆÏܲˆÏ°Ó˚ Ó˚ˆÏí˛Ó˚ ~ܲ ≤Ãyhsˇ ~ܲ!ê˛ ≤âhflÏ !fiê˛ˆÏ°Ó˚ ˛õyˆÏï˛Ó˚ §ˆÏD ò,ì˛¸û˛yˆÏÓ xyÓk˛
Ä xöƒ ≤Ãyhsˇ=!° ü%_´ ÌyˆÏܲ– ≤Ã!ï˛!ê˛ Ó˚ˆÏäÓ˚ í˛z˛õÓ˚ ï˛yÓ˚ Ü%˛[˛°# çí˛¸yˆÏöy ÌyˆÏܲ ~ÓÇ ˆ§=!°Ó˚ üôƒ !òˆÏÎ˚ ~ܲ•z
ò¢yÎ˚ !öˆÏܲ° Ó˚ˆÏí˛Ó˚ xö%öyò# ܲ¡õyˆÏD ˛õÓ˚Óï˛#≈ ï˛!í˛¸Í≤ÃÓy• ˛õyë˛yˆÏöy •Î˚– !ܲå%È fliyÎ˚# ã%˛¡∫ˆÏܲÓ˚ §y•yˆÏ΃
Ó˚í˛=!°ˆÏܲ ≤ÃyÌ!üܲû˛yˆÏÓ ã%˛¡∫!Ü˛ï˛ Ü˛ˆÏÓ˚ Ó˚yáy •Î˚– §ühflÏ ÓƒÓfliy!ê˛ ç°!öˆÏÓ˚yôܲ xyôyˆÏÓ˚ Ó˚yáy ÌyˆÏܲ–
!öˆÏܲˆÏ°Ó˚ Ó˚í˛=!°Ó˚ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° !fiê˛ˆÏ°Ó˚ ˛õyï˛!ê˛ ~ܲ•z ܲ¡õyˆÏB˛ ܲ!¡õï˛ •Î˚ ~ÓÇ ˆ§!ê˛ çˆÏ°Ó˚ üˆÏôƒ
ˆí˛yÓyˆÏöy ÌyܲyÎ˚ çˆÏ°Ó˚ üˆÏôƒ x!ï˛¢∑ ï˛Ó˚D í˛zͲõߨ •Î˚– ~áyˆÏö ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ üy˛õ ¢ˆÏ∑Ó˚ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚
ï%˛°öyÎ˚ ˆÓ¢ Óí˛¸– ~çöƒ á%Ó §Çܲ#î≈ âöˆÏܲyˆÏîÓ˚ üˆÏôƒ x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D ˛õyë˛yˆÏöy §Ω˛Ó •Î˚–
≤Ã!ï˛ú˛!°ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚ @ˇÃy•Ü˛ !•§yˆÏÓÄ ˆã˛Ô¡∫ܲ ï˛!ï˛Ó˚ !Ó˛õÓ˚#ï˛ !e´Î˚yÓ˚ñ xÌ≈yÍ xy˛õ!ï˛ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚
ã˛yˆÏ˛õ !öˆÏܲ° Ó˚ˆÏí˛Ó˚ ï˛!ï˛ ~ÓÇ ï˛yÓ˚ ú˛ˆÏ° ˆã˛Ô¡∫ܲ xyˆÏӈϢÓ˚ í˛zͲõ!_Ó˚ ≤ÈÏÎ˚yà ܲÓ˚y •Î˚– ~•z çyï˛#Î˚ @ˇÃy•Ü˛
Ú•y•zˆÏí»˛yˆÏú˛yöÛ öyˆÏü ˛õ!Ó˚!ã˛ï˛–
ç°!öˆÏÓ˚yôܲ xyôyÓ˚
!öˆÏܲ° Ó˚í˛
fliyÎ˚# ã%˛¡∫ܲ
!fiê˛ˆÏ°Ó˚ ˛õyï˛
Ü% [˛°#
319
(ii) Ó›Ó˚ àë˛öàï˛ e&!ê˛ !öî≈Î˚ ≠Ó›Ó˚ àë˛öàï˛ e&!ê˛ !öî≈Î˚ ≠Ó›Ó˚ àë˛öàï˛ e&!ê˛ !öî≈Î˚ ≠Ó›Ó˚ àë˛öàï˛ e&!ê˛ !öî≈Î˚ ≠Ó›Ó˚ àë˛öàï˛ e&!ê˛ !öî≈Î˚ ≠
ì˛y°y•z ˆ°y•y Óy ôyï˛Ó ܲyë˛yˆÏüyÓ˚ üˆÏôƒ ˆÜ˛yöÄ ú˛yê˛° Óy e&!ê˛ xyˆÏåÈ !ܲöy ˆÓyé˛yÓ˚ çöƒ ï˛yÓ˚ üˆÏôƒ x!ï˛¢∑
ã˛y°öy ܲÓ˚y •Î˚ ~ÓÇ ˆ≤Ã!Ó˚ï˛ñ !Ó!«˛Æ Ä ≤Ã!ï˛ú˛!°ï˛ ï˛Ó˚DˆÏܲ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ •Ó˚y •Î˚– Ó›!ê˛Ó˚
àë˛ö Î!ò !öá%Ñï˛ •Î˚ñ ï˛ˆÏÓ ˆ§!ê˛Ó˚ §Ç°@¿ @ˇÃy•ˆÏܲ ≤Ãï˛ƒy!¢ï˛ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ ˛õyÄÎ˚y ÎyˆÏÓ– !ܲv ˆÜ˛yÌyÄ
ú˛yê˛°ñ ú˛yÑ˛õy xÇ¢ •zï˛ƒy!ò ÌyܲˆÏ°ñ ˆ§áyˆÏö x!ï˛¢∑ ï˛Ó˚D ≤Ã!ï˛ú˛!°ï˛ Óy !Ó!«˛Æ •ˆÏÓ ~ÓÇ x!öÎ˚!üï˛
§ÇˆÏÜ˛ï˛ ˛õyÄÎ˚y ÎyˆÏÓ– Ó›Ó˚ üôƒ !òˆÏÎ˚ ¢∑ï˛Ó˚D ÎyÄÎ˚yÓ˚ §üÎ˚ܲy° üy˛õˆÏï˛ ˛õyÓ˚ˆÏ° Ó›Ó˚ ˆÓô Óy ܲï˛áy!ö
àû˛#ˆÏÓ˚ àë˛öàï˛ e&!ê˛ Ó˚ˆÏÎ˚ˆÏåÈ ï˛yÄ çyöy ÎyˆÏÓ– x!ï˛§)-ñ ≤ÃyÎ˚ 10–5 cm ˆÓˆÏôÓ˚ ú˛yê˛°Ä ~û˛yˆÏÓ á%шÏç ˛õyÄÎ˚y
§Ω˛Ó– ~•z ö#!ï˛ˆÏï˛ ≤Ã›ï˛ ~ܲ!ê˛ Îsf ú˛yÎ˚yÓ˚ˆÏfiê˛yö (F.A. Firestone) xy!Ó‹,Òï˛ x!ï˛¢∑ ≤Ã!ï˛ú˛°öò¢#≈
(Ultrasonic reflectoscope)–
Ó›ˆÏܲ ˆû˛ˆÏä ö‹T öy ܲˆÏÓ˚ ï˛yÓ˚ üˆÏôƒÜ˛yÓ˚ e&!ê˛ !öô≈yÓ˚îˆÏܲ ܲy!Ó˚à!Ó˚ !ÓòƒyÓ˚ û˛y£ÏyÎ˚ Ó°y •Î˚ x!Óôùǧ#
˛õÓ˚#«˛y (non-destructive testing)– x!ï˛¢∑ ˆ§ ôÓ˚ˆÏöÓ˚ ~ܲ!ê˛ §%ˆÏÎyà ܲˆÏÓ˚ ˆòÎ˚–
≤Ã!ï˛ú˛°ö åÈyí˛¸yñ üyôƒˆÏüÓ˚ §ˆÏD !e´Î˚yÈÙÈ≤Ã!ï˛!e´Î˚yÓ˚ í˛z˛õÓ˚ !û˛!_ ܲˆÏÓ˚Ä x!ï˛¢∑ˆÏܲ ܲyˆÏç °yàyˆÏöy •Î˚–
ˆÜ˛yöÄ üyôƒˆÏüÓ˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ x!ï˛¢∑ ï˛Ó˚D üyôƒˆÏü ܲ# ˛õ!Ó˚Óï≈˛ö âê˛yˆÏÓñ ï˛y !öû≈˛Ó˚ ܲˆÏÓ˚ ï˛Ó˚ˆÏDÓ˚
¢!_´Ó˚ í˛z˛õˆÏÓ˚– ܲü ¢!_´ ï˛Ó˚D •ˆÏ° ï˛y x!ÓÜ,˛ï˛ xÓfliyÎ˚ üyôƒˆÏüÓ˚ üôƒ !òˆÏÎ˚ ã˛ˆÏ° ÎyÎ˚ñ ï˛Ó˚° Óy àƒyˆÏ§Ó˚
ˆ«˛ˆÏe xî%=!°Ó˚ üˆÏôƒ xyˆÏ°yí˛¸ö §,!‹T ܲˆÏÓ˚– xyÓ˚ ˆÓ!¢ ¢!_´§¡õߨ •ˆÏ° üyôƒˆÏüÓ˚ ÓyôyÎ˚ ï˛yÓ˚ ܲ¡õö !ÓÜ,˛ï˛
•ˆÏÎ˚ ÎyÎ˚– ï˛Ó˚ˆÏDÓ˚ ¢!_´ üyôƒˆÏüÓ˚ xî%ÈÙÈ˛õÓ˚üyî%ˆÏï˛ §MÈ˛y!Ó˚ï˛ •ˆÏÎ˚ ˆÜ˛°yˆÏ§Ó˚ í˛z£èï˛y Ó,!k˛ñ xyî!Óܲ àë˛ˆÏöÓ˚
!ÓÜ,˛!ï˛ñ Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y •zï˛ƒy!ò âê˛yÎ˚– ˆÜ˛yöÄ üyôƒˆÏüÓ˚ ˆû˛Ôï˛ Óy Ó˚y§yÎ˚!öܲ ˛õ!Ó˚Óï≈˛ö âê˛yˆÏï˛ ~•z ôÓ˚ˆÏöÓ˚
x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà •Î˚– ≤Ã̈Ïü ܲü ¢!_´Ó˚ x!ï˛¢∑ ï˛Ó˚ˆÏDÓ˚ ܲˆÏÎ˚ܲ!ê˛ ÓƒÓ•yÓ˚ í˛zˆÏÕ‘á ܲÓ˚y Îyܲ–
(iii) x!ï˛¢∑#Î˚ §ywï˛y üy˛õܲ x!ï˛¢∑#Î˚ §ywï˛y üy˛õܲ x!ï˛¢∑#Î˚ §ywï˛y üy˛õܲ x!ï˛¢∑#Î˚ §ywï˛y üy˛õܲ x!ï˛¢∑#Î˚ §ywï˛y üy˛õܲ (Ultrasonic viscometer) :
ˆÜ˛yöÄ ï˛Ó˚° ï˛yÓ˚ üôƒ !òˆÏÎ˚ §MÈ˛y!Ó˚ï˛ ¢∑ï˛Ó˚DˆÏܲ ܲï˛ê˛y ˆ¢y£Ïî ܲˆÏÓ˚ ï˛y ˙ ï˛Ó˚ˆÏ°Ó˚ §ywï˛yÓ˚ í˛z˛õÓ˚
!öû˛≈Ó˚ ܲˆÏÓ˚– ï˛Ó˚ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ d ò)Ó˚c ÎyÄÎ˚yÓ˚ ú˛ˆÏ° ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Î!ò I0 ˆÌˆÏܲ ܲˆÏü I
d •Î˚ ï˛ˆÏÓ α
=
12d
In(I0/I
d) Ó˚y!¢ˆÏܲ ¢ˆÏ∑Ó˚ ˆ¢y£ÏyîyB˛ (attenuation coefficient) Ó°y •Î˚– ~•z ˆ¢y£ÏîyB˛ ï˛Ó˚ˆÏ°Ó˚
ˆ«˛ˆÏe ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ ÓˆÏà≈Ó˚ ~ÓÇ ï˛Ó˚ˆÏDÓ˚ §ywï˛yÓ˚ §üyö%˛õyï˛# •Î˚– !ö!ò≈‹T ܲ¡õyˆÏB˛ í˛zͧ ˆÌˆÏܲ !Ó!û˛ß¨
ò)Ó˚ˆÏcÓ˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ê˛Δy™!í˛í˛z§yˆÏÓ˚Ó˚ §y•yˆÏ΃˛ ˛õ!Ó˚üy˛õ ܲˆÏÓ˚ ˆ¢y£ÏyîyˆÏB˛Ó˚ üyö !öî≈Î˚ ܲÓ˚y •Î˚– ~•zû˛yˆÏÓ
!Ó!û˛ß¨ ï˛Ó˚ˆÏ° x!ï˛¢ˆÏ∑Ó˚ ˆ¢y£ÏîyˆÏB˛Ó˚ üyö çyöy ˆàˆÏ° ˆÎ ï˛Ó˚ˆÏ°Ó˚ §ywï˛yÓ˚ üyö çyöy xyˆÏåÈñ ï˛yÓ˚ §ˆÏD
ï%˛°öy ܲˆÏÓ˚ xöƒ ï˛Ó˚ˆÏ°Ó˚ xçyöy §ywï˛yÓ˚ üyö !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
í˛zFã˛¢!_´Ó˚ x!ï˛¢ˆÏ∑Ó˚ §y•yˆÏÎƒÄ Ü˛ï˛Ü˛=!° ܲyç ܲÓ˚y •ˆÏÎ˚ ÌyˆÏܲ– ˆÎüö ≠
320
(iv) x!ï˛¢∑#Î˚ !í»˛° ≠x!ï˛¢∑#Î˚ !í»˛° ≠x!ï˛¢∑#Î˚ !í»˛° ≠x!ï˛¢∑#Î˚ !í»˛° ≠x!ï˛¢∑#Î˚ !í»˛° ≠
ôyï˛Ó Ó›ˆÏï˛ !åÈo ܲÓ˚yÓ˚ çöƒ x!ï˛¢∑ ÓƒÓ•*ï˛ •Î˚– x!ï˛¢ˆÏ∑Ó˚ ê˛Δy™!í˛í˛z§yˆÏÓ˚Ó˚ §ˆÏD !åÈo ܲÓ˚yÓ˚ !Óê˛
°yàyˆÏöy ÌyܲˆÏ° x!ï˛¢ˆÏ∑Ó˚ §üyö ܲ¡õyˆÏB˛ !Óê˛!ê˛ Ü˛yÑ˛õˆÏÓ ~ÓÇ e´ˆÏü e´ˆÏü ôyï%˛ˆÏï˛ !åÈo ܲˆÏÓ˚ ≤Ã!Ó‹T •ˆÏÓ–
§yôyÓ˚î !í»˛° ΈÏsf â)î≈ö à!ï •Î˚ñ ú˛ˆÏ° ÷ô% Ó,_yܲyÓ˚ !åÈo•z ܲÓ˚y ÎyÎ˚– !ܲv ~ ˆ«˛ˆÏe !Óê˛!ê˛ ˆâyˆÏÓ˚ öyñ
xyˆÏà !˛õˆÏåÈ Ü˛yÑ˛õˆÏï˛ Ü˛yÑ˛õˆÏï˛ ~!àˆÏÎ˚ ÎyÎ˚– ú˛ˆÏ° ˆÎ ˆÜ˛yöÄ xyÜ,˛!ï˛Ó˚ !åÈo ܲÓ˚y ÎyÎ˚– ~•z çyï˛#Î˚ ΈÏsf ܲyã˛ñ
•zflõyï˛ ~üö !ܲ •#Ó˚yÓ˚ üˆÏôƒÄ àï≈˛ ܲÓ˚y ÎyÎ˚–
(v) x!ï˛¢∑#Î˚ !Üœ˛öyÓ˚ x!ï˛¢∑#Î˚ !Üœ˛öyÓ˚ x!ï˛¢∑#Î˚ !Üœ˛öyÓ˚ x!ï˛¢∑#Î˚ !Üœ˛öyÓ˚ x!ï˛¢∑#Î˚ !Üœ˛öyÓ˚ (cleaner) :
ï˛Ó˚ˆÏ°Ó˚ üˆÏôƒ x!ï˛¢∑ í˛zͲõyòö ܲÓ˚ˆÏ° §Ó≈!ö¡¨ ã˛yˆÏ˛õÓ˚ xMÈ˛°=!°ˆÏï˛ Ó%ò‰Ó%ˆÏòÓ˚ üï˛ «%˛o «%˛o ÓyÎ˚%¢)öƒ
fliyö ˜ï˛!Ó˚ •Î˚– ~•z âê˛öyˆÏܲ à•ùÓ˚î Óy ܲƒy!û˛ˆÏê˛¢ö (cavitation) Ó°y •Î˚– ~•z ÓyÎ˚%¢)öƒï˛y ô)!°Ü˛îy ≤Ãû,˛!ï˛
«%˛o Ó›ˆÏܲ xyܲ£Ï≈î ܲˆÏÓ˚ xyˆÏö– §%ï˛Ó˚yÇñ ï˛Ó˚ˆÏ° ˆÜ˛yöÄ Ó›ˆÏܲ í%˛!ÓˆÏÎ˚ ˆ§•z ï˛Ó˚ˆÏ° x!ï˛¢∑ ï˛Ó˚D ã˛y°öy
ܲÓ˚ˆÏ° Ó›Ó˚ àyˆÏÎ˚ ˆ°ˆÏà Ìyܲy ô)ˆÏ°y üÎ˚°y í˛zˆÏë˛ ˛õ!Ó˚‹ÒyÓ˚ •ˆÏÎ˚ ÎyÎ˚– ~•zû˛yˆÏÓ §)- Ó›ñ ˆÎüö â!í˛¸Ó˚ «%˛o
ÎsfyÇ¢ Îy •yï˛ !òˆÏÎ˚ âˆÏ£Ï ˛õ!Ó˚‹ÒyÓ˚ ܲÓ˚y §Ω˛Ó öÎ˚ñ ï˛yÄ x!ï˛¢∑ ≤ÈÏÎ˚yˆÏà ˛õ!Ó˚‹ÒyÓ˚ ܲÓ˚y ÎyÎ˚–
í z_´ ˛õk˛!ï˛ Ïï˛ x!ü◊î#Î ï˛Ó° Sç° Ä ˆï˛° Óy ç° Ä ˛õyÓòV !ü!¢ ÏÎ xÓoÓ (emulsion) ˜ï˛!Ó Ü˛Óy ÎyΖ
(vi) Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y ≠Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y ≠Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y ≠Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y ≠Ó˚y§yÎ˚!öܲ !Ó!e´Î˚y ≠
Ó,•òyÜ,˛!ï˛ ˛õ!°üyˆÏÓ˚Ó˚ xyî!Óܲ ¢,C° ˆû˛ˆÏä §Ó˚° ˛õ!°üyÓ˚ ˜ï˛!Ó˚ ܲÓ˚yÓ˚ çöƒ x!ï˛¢∑ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚–
ˆÎüöñ ˛õ!Ó˚fiê˛y•z!Ó˚ö (polystyrene)-ˆÜ˛ ê˛°%•zö (toluene)-~ oÓ#û)˛ï˛ ܲˆÏÓ˚ í˛zFã˛ Ü˛¡õyB˛ (284 KHz) Ä
¢!_´Ó˚ (5W cm2) x!ï˛¢∑ ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° oÓˆÏîÓ˚ ˛õ!°üyÓ˚ xî%=!° ˆû˛ˆÏä ÎyÎ˚– ~•z ˛õk˛!ï˛ˆÏï˛ ˆŸªï˛§yˆÏÓ˚Ó˚
xî%ñ KI, H2S, H
gCl
2 ≤Ãû,˛!ï˛ xî%Ä û˛yäy §Ω˛Ó •Î˚–
(vii) !öÓ#≈çö !öÓ#≈çö !öÓ#≈çö !öÓ#≈çö !öÓ#≈çö (sterilization) ;
ˆÜ˛yöÄ üyôƒˆÏü Îáö x!ï˛¢∑ ˆ¢y!£Ïï˛ •Î˚ ï˛áö ¢∑¢!_´ ï˛y˛õ¢!_´ˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ •ˆÏÎ˚ o&ï˛ í˛z£èï˛y Ó,!k˛
ܲˆÏÓ˚– ò%ôñ ˛õyö#Î˚ ç° ≤Ãû,˛!ï˛ ï˛Ó˚ˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ¢!_´¢y°# x!ï˛¢∑ ï˛Ó˚D ˛õ!Ó˚ã˛y°öy ܲÓ˚ˆÏ° í˛zq(ï˛ í˛z£èï˛yÎ˚
öyöy ôÓ˚ˆÏîÓ˚ Ó#çyî%ñ ~üö !ܲ Óƒyäy!ã˛Ó˚ üï˛ ˆåÈyê˛ ˆåÈyê˛ ≤Ãyî# ôùǧ •Î˚–
(viii) !ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠!ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠!ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠!ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠!ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠
xyô%!öܲ !ã˛!ܲͧyÎ˚ !Ó!û˛ß¨û˛yˆÏÓ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà âˆÏê˛– !ã˛!ܲͧyÓ˚ ˆ«˛ˆÏe x!ï˛¢ˆÏ∑Ó˚ ÓƒÓ•yÓ˚ˆÏܲ
ˆüyê˛yü%!ê˛ ò%•z û˛yˆÏà û˛yà ܲÓ˚y ÎyÎ˚– ~=!° •° (a) ˆÓ˚yà !öî≈ˆÏÎ˚ Ä (b) ˆÓ˚yˆÏàÓ˚ !ã˛!ܲͧyÎ˚ x!ï˛¢ˆÏ∑Ó˚
ÓƒÓ•yÓ˚–
(a) ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠
≤Ã̈Ïü ˆÓ˚yà !öî≈ˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ÓƒÓ•yˆÏÓ˚Ó˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚y Îyܲ– xy˛õ!ö •Î˚ï˛ xy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛
öyˆÏü ˆòˆÏ•Ó˚ xû˛ƒhsˇˆÏÓ˚Ó˚ !ã˛e @ˇÃ•ˆÏîÓ˚ ˛õk˛!ï˛Ó˚ öyü ÷ˆÏöˆÏåÈö– ˆòˆÏ•Ó˚ üˆÏôƒ x!ï˛¢ˆÏ∑Ó˚ Ó˚!Ÿ¬=FåÈ ˆ≤ÃÓ˚î ~ÓÇ
≤Ã!ï˛ôù!öÓ˚ §y•yˆÏ΃ ˆòˆÏ•Ó˚ xû˛ƒhsˇˆÏÓ˚Ó˚ Îsfy!ò Ä àû≈˛fli º*ˆÏîÓ˚ !eüy!eܲ !ã˛e àë˛ö•z xy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛Ó˚
ü)° ܲyç–
321
xy˛õ!ö •z!ï˛˛õ)ˆÏÓ≈ ˆçˆÏöˆÏåÈö ˆÎñ ¢∑ï˛Ó˚D Îáö ~üö ò%•z üyôƒˆÏüÓ˚ §#üyöyÎ˚ xy˛õ!ï˛ï˛ •Î˚ñ ˆÎ=!°Ó˚
¢∑ˆÏÓ˚yô !û˛ß¨ñ ï˛áö ï˛y xyÇ!¢Ü˛û˛yˆÏÓ ≤Ã!ï˛ú˛!°ï˛ •Î˚– ¢∑ˆÏÓ˚yô üyôƒˆÏüÓ˚ âöc Ä üyôƒˆÏü ¢ˆÏ∑Ó˚ ˆÓàÈÙÙÙÈ~•z
ò%•z ~Ó˚ =îú˛°– ÓyÎ˚%üyôƒˆÏüÓ˚ ï%˛°öyÎ˚ ˆòˆÏ•Ó˚ öÓ˚ü ܲ°y=!°Ó˚ âöc xˆÏöܲ ˆÓ!¢– ~=!°Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚
ˆÓàÄ ÓyÎ˚%üyôƒˆÏüÓ˚ ï%˛ö°yÎ˚ ˆÓ!¢ñ ≤ÃyÎ˚ 1540ms–1– §%ï˛Ó˚yÇñ ÓyÎ˚% Ä öÓ˚ü ܲ°yÓ˚ ¢∑ˆÏÓ˚yô xˆÏöܲê˛y•z !û˛ß¨
~ÓÇ ÓyÎ˚% ˆÌˆÏܲ ˆòˆÏ• x!ï˛¢∑ ï˛Ó˚D xy˛õ!ï˛ï˛ •ˆÏ° ï˛yÓ˚ xyÇ!¢Ü˛ ≤Ã!ï˛ú˛°ö âˆÏê˛– xyÓyÓ˚ öÓ˚ü ܲ°y Ä
x!fliÓ˚ ¢∑ˆÏÓ˚yôÄ !û˛ß¨ •ÄÎ˚yÎ˚ x!ï˛¢∑ Îáö ˆüòÈÙÈüyǧ Ä x!fliÓ˚ §#üyöyÎ˚ xy˛õ!ï˛ï˛ •Î˚ñ ï˛áöÄ ï˛yÓ˚ !ܲå%Èê˛y
≤Ã!ï˛ú˛!°ï˛ •Î˚–
xy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛ˆÏï˛ 1 mm ˆÌˆÏܲ 3 cm ÓƒyˆÏ§Ó˚ ã˛y˛õÈÙÈ!Óò%ƒÍ ˆÜ˛°yˆÏ§Ó˚ myÓ˚y ܲˆÏÎ˚ܲ üy•zˆÏe´yˆÏ§ˆÏܲu˛
fliyÎ˚# ˆÌˆÏܲ 10 MHz ܲ¡õyˆÏB˛Ó˚ x!ï˛¢ˆÏ∑Ó˚ 鲰ܲ ˜ï˛!Ó˚ ܲˆÏÓ˚ ˆòˆÏ•Ó˚ üˆÏôƒ ˛õÓ˚˛õÓ˚ ˛õyë˛yˆÏöy •Î˚– ˙
鲰ܲ=!° ˆò•üôƒfli ˛õyܲfli°#ñ •*Í!˛õ[˛ñ x!fli ˆÌˆÏܲ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÎ˚ !ú˛ˆÏÓ˚ xyˆÏ§ ~ÓÇ í˛zͲõyòܲ ˆÜ˛°y§!ê˛•z
≤Ã!ï˛ôù!öÓ˚ @ˇÃy•Ü˛ !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚– ≤Ã!ï˛ôù!ö !ú˛ˆÏÓ˚ xy§ˆÏï˛ Ü˛ï˛ê˛y §üÎ˚ °yˆÏàñ ï˛yÓ˚ ˆÌˆÏܲ ≤Ã!ï˛ú˛°Ü˛!ê˛
ˆòˆÏ•Ó˚ üˆÏôƒ Ü˛ï˛ àû˛#Ó˚ï˛yÎ˚ xyˆÏåÈ ï˛y !öî≈Î˚ ܲÓ˚y ÎyÎ˚– ã˛y˛õÈÙÈ!Óò%ƒÍ ˆÜ˛°y§ ˆÌˆÏܲ ˛õyÄÎ˚y §ÇˆÏܲˆÏï˛Ó˚ §y•yˆÏ΃
ܲ!¡õí˛zê˛yˆÏÓ˚Ó˚ ˛õò≈yÎ˚ ˆòˆÏ•Ó˚ xû˛ƒhsˇˆÏÓ˚Ó˚ ~ܲ!ê˛ !müy!eܲ !ã˛e ú%˛!ê˛ˆÏÎ˚ ˆï˛y°y ÎyÎ˚– •zFåÈy ܲÓ˚ˆÏ° ~•z §ÇˆÏܲï˛
ܲ!¡õí˛zê˛yˆÏÓ˚Ó˚ ˆüüÓ˚#ˆÏï˛ §!MÈ˛ï˛ Ó˚yáy ÎyÎ˚ñ xyÓyÓ˚ ~Ó˚ §y•yˆÏ΃ à!ï˛¢#° ˆò•ÎˆÏsfÓ˚ SˆÎüö •*Í!˛õ[˛V ã˛°!Fã˛e
ˆòáˆÏï˛ ˛õyÄÎ˚y ÎyÎ˚–
xy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛ !ã˛!ܲͧy¢yˆÏflf ~áö ~ܲ!ê˛ Ó‡ÓƒÓ•*ï˛ !ã˛eyÎ˚ö (imaging) ˛õk˛!ï˛– ~Ó˚˛§y•yˆÏ΃
¢Ó˚#ˆÏÓ˚Ó˚ üˆÏôƒ !ê˛í˛züyˆÏÓ˚Ó˚ xÓfliyöñ •*Í!˛õˆÏ[˛Ó˚ xÓfliyñ àû≈˛fli º*ˆÏîÓ˚ xÓfliyö Ä xöƒyöƒ ï˛Ìƒ ˛õyÄÎ˚y ÎyÎ˚–
~:ÈÙȈÓ˚ !ã˛e @ˇÃ•ˆÏî !Ó!ܲÓ˚ˆÏîÓ˚ ú˛ˆÏ° üyöÓˆÏòˆÏ•Ó˚ !ӈϢ£Ïï˛ º*ˆÏîÓ˚ «˛!ï˛ •ÄÎ˚yÓ˚ xy¢B˛y ÌyˆÏܲ– !ܲv x!ï˛¢∑
~•z !òܲ !òˆÏÎ˚ §¡õ)î≈ !öÓ˚y˛õò–
(b) ˆÓ˚yà !öÓ˚yüˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öÓ˚yüˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öÓ˚yüˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öÓ˚yüˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠ˆÓ˚yà !öÓ˚yüˆÏÎ˚ x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yà ≠
ˆò•Ü˛°yÓ˚ üôƒ !òˆÏÎ˚ Îáö x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D §MÈ˛y!Ó˚ï˛ •Î˚ñ ï˛áö ˆ§!ê˛ !ܲå%Èê˛y üƒy§yˆÏçÓ˚ ܲyç ܲˆÏÓ˚– Óyï˛
Ä fl¨yÎ˚% §Çe´yhsˇ ˆÓ˚yˆÏà ˆÓòöyÓ˚ í˛z˛õ¢ˆÏü x!ï˛¢ˆÏ∑Ó˚ ï˛Ó˚D ≤ÈÏÎ˚yà ܲÓ˚y •Î˚–
üyöÓˆÏòˆÏ•Ó˚ !ܲí˛!öˆÏï˛ xˆÏöܲ §üˆÏÎ˚ ˆåÈyê˛ ˆåÈyê˛ ˛õÿÏÓ˚Ó˚ ê%˛Ü˛ˆÏÓ˚y ˜ï˛Ó˚# •Î˚– x!ï˛¢ˆÏ∑Ó˚ §y•yˆÏ΃ ~=!°
xˆÏflfy˛õã˛yÓ˚ åÈyí˛¸y•z ò)Ó˚ ܲÓ˚y ÎyÎ˚– ~•z ˛õk˛!ï˛ˆÏï˛ ˛õÿÏÓ˚Ó˚ í˛z˛õÓ˚ ~ܲy!ôܲ í˛zͧ ˆÌˆÏܲ í˛zͲõߨ x!ï˛¢ˆÏ∑Ó˚
Ó˚!Ÿ¬=FåÈ ˆÜ˛w#û)˛ï˛ ܲˆÏÓ˚ ˛õyÌÓ˚=!°ˆÏܲ ã)˛î≈ ܲÓ˚y •Î˚ ÎyˆÏï˛ x!ï˛«%˛o ܲîy=!° ˆòˆÏ•Ó˚ fl∫yû˛y!Óܲ !ö‹Òy¢ö
!e´Î˚yÓ˚ üyôƒˆÏü ÓyÓ˚ •ˆÏÎ˚ ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~•z ˛õk˛!ï˛ !°ˆÏÌy!ê˛Δ˛õ‰!§ (lithotripsy) öyˆÏü ˛õ!Ó˚!ã˛ï˛–
~ÓyÓ˚ xö%¢#°ö#Ó˚ ˛õy°y– !öˆÏã˛Ó˚ ≤ß¿=!°Ó˚ í˛z_Ó˚ xy˛õ!ö §•ˆÏç•z !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2 :
(a) ~ܲ!ê˛ 5cm ò#â≈ üyåÈ çˆÏ°Ó˚ üˆÏôƒ §yÑï˛yÓ˚ ܲyê˛ˆÏåÈ– 300Hz ܲ¡õyˆÏB˛Ó˚ ◊ÓîˆÏÎyàƒ ¢∑ üyåÈ!ê˛ ˆÌˆÏܲ
≤Ã!ï˛ú˛!°ï˛ öy •ˆÏ°Ä 300KHz ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ ˆ§!ê˛ ˆÌˆÏܲ §%flõ‹Tû˛yˆÏÓ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÓ– ~Ó˚ ܲyÓ˚î
ܲ#⁄ SçˆÏ°Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚ ˆÓà 1500ms–1)–
(b) çˆÏ°Ó˚ üˆÏôƒ 5MHz ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ ï˛Ó˚D ˛õyë˛yˆÏöy •°– ~áö ~•z çˆÏ°Ó˚ üôƒ !òˆÏÎ˚ ï˛Ó˚D
322
à!ï˛Ó˚ °¡∫ x!û˛ü%ˆÏá 600nm ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ xyˆÏ°yÓ˚ §üyhsˇÓ˚y° Ó˚!Ÿ¬=FåÈ ˛õyë˛yˆÏöy •ˆÏ° ï˛yÓ˚ ≤ÃÌü e´ˆÏüÓ˚
ÓƒÓï≈˛ö ˆÜ˛yî Ü˛ï˛ •ˆÏÓ ï˛y 12.6 §ü#ܲÓ˚î ÓƒÓ•yÓ˚ ܲˆÏÓ˚ !öî≈Î˚ ܲÓ˚&ö–
(c) §ü%ˆÏoÓ˚ çˆÏ° x!ï˛¢∑ ˆ≤ÃÓ˚ˆÏîÓ˚ çöƒ ÓƒÓ•*ï˛ ˆã˛Ô¡∫ܲ ï˛!ï˛í˛zͲõyòܲ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ~ˆÏï˛ ~ܲ!ê˛
≤âhflÏ !fiê˛ˆÏ°Ó˚ ˛õyˆÏï˛ xˆÏöܲ=!° !öˆÏܲ° ò[˛ xyÓk˛ ÌyˆÏܲ– ~•z ≤ÈÏÎ˚yçö •Î˚ ˆÜ˛ö⁄
(d) 18°C í˛z£èï˛yÎ˚ ܲƒyfiê˛Ó˚ xyˆÏÎ˚ˆÏ°Ó˚ üˆÏôƒ 1MHz ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ ï˛Ó˚D ˆ≤ÃÓ˚î ܲˆÏÓ˚ ˆòáy ˆà°
5cm ò)Ó˚ˆÏc ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y 3 =î •…y§ ˛õyÎ˚– ˙ í˛z£èï˛yÎ˚ ܲƒyfiê˛Ó˚ xˆÏ˘Î˚ˆÏ°Ó˚ §ywï˛yB˛ 1000mNsm–2– ~áö
Î!ò x!°û˛ xˆÏÎ˚ˆÏ° ~ܲ•z ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ ˆ≤ÃÓ˚î ܲˆÏÓ˚ ï˛#Ó ï˛y 50cm ò)Ó˚ˆÏc 4 =î •…y§ ˆ˛õˆÏï˛ ˆòáy
ÎyÎ˚ñ ï˛ˆÏÓ x!°û˛ xˆÏÎ˚ˆÏ°Ó˚ §ywï˛yB˛ ܲï˛⁄
12.5 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
¢ˆÏ∑Ó˚ ܲ¡õyB˛ 20Hz-~Ó˚ ܲü Óy 20000Hz-~Ó˚ ˆÓ!¢ •ˆÏï˛ ï˛y xyÓ˚ üyö%ˆÏ£ÏÓ˚ ܲyˆÏö ôÓ˚y ˛õˆÏí˛¸ öy–
ˆÜ˛yö ¢˘ˆÏ∑Ó˚ ܲ¡õyB˛ ~•z §#üyÓ˚ !öˆÏã˛ •ˆÏ° ï˛yˆÏܲ xÓ¢∑ ~ÓÇ í˛z˛õˆÏÓ˚ •ˆÏ° x!ï˛¢∑ Ó°y •Î˚– x!ï˛¢∑
öyöyû˛yˆÏÓ í˛zͲõyòö ܲÓ˚y ÎyÎ˚ ~ÓÇ ≤ÃÎ%!_´Ó˚ öyöy ˆ«˛ˆÏe x!ï˛¢∑ ÓƒÓ•*ï˛ •Î˚– xyô%!öܲܲyˆÏ° x!ï˛¢∑
í˛zͲõyòܲ ΈÏsf ˆã˛Ô¡∫ܲ ï˛!ï˛ ~ÓÇ ã˛y˛õÈÙÈ!Óò%ƒÍ !e´Î˚yˆÏܲ ܲyˆÏç °yàyˆÏöy •Î˚–
x!ï˛¢ˆÏ∑Ó˚ ≤Ã!ï˛ú˛°öˆÏܲ ܲyˆÏç °y!àˆÏÎ˚ §ü%ˆÏoÓ˚ àû˛#Ó˚ï˛yñ !öü!Iï˛ Ó›Ó˚ ò)Ó˚c Ä ôyï˛Ó Ó›Ó˚ xû˛ƒhsˇÓ˚#î
e&!ê˛ !öî≈Î˚ ܲÓ˚y •Î˚– ï˛Ó˚ˆÏ°Ó˚ §ywï˛y üy˛õyÓ˚ çöƒÄ x!ï˛¢ˆÏ∑Ó˚ ÓƒÓ•yÓ˚ •Î˚– ~ åÈyí˛¸y í˛zFã˛¢!_´Ó˚ x!ï˛¢ˆÏ∑Ó˚
≤ÈÏÎ˚yˆÏà !í»˛!°Çñ é˛y°y•z ~ÓÇ xˆÏflfy˛õã˛yÓ˚ ܲÓ˚y •Î˚– x!ï˛¢ˆÏ∑Ó˚ §ÓˆÏã˛ˆÏÎ˚ ˛õ!Ó˚!ã˛ï˛ ÓƒÓ•yÓ˚ Úxy°ê˛ΔyˆÏ§yˆÏöy@ˇÃy!ú˛Ûñ
xÌ≈yÍ x!ï˛¢ˆÏ∑Ó˚ ç#ÓˆÏò• ˆÌˆÏܲ ≤Ã!ï˛ú˛°ˆÏöÓ˚ e´üyàï˛ ˛õ!Ó˚Óï≈˛öˆÏܲ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛ˆÏÓ˚
x!§ˆÏ°y@ˇÃyˆÏú˛Ó˚ ˛õò≈yÎ˚ ¢Ó˚#ˆÏÓ˚Ó˚ xû˛ƒhsˇÓ˚û˛yˆÏàÓ˚ ò,¢ƒüyö !ã˛e ≤Ã›ï˛ Ü˛Ó˚y •Î˚– ˆÓ˚yà !öÓ˚yüˆÏÎ˚Ó˚ ˆ«˛ˆÏeÄ
xyçܲy° x!ï˛¢∑ ÓƒÓ•*ï˛ •ˆÏÎ˚ ÌyˆÏܲ–
12.6 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. x!ï˛¢∑ ܲyˆÏܲ ӈϰ⁄ çˆÏ°Ó˚ üˆÏôƒ ˆÎ ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ 5cm, ï˛yˆÏܲ !ܲ xyüÓ˚y x!ï˛¢∑ Ó°Ó⁄
2. ˆã˛Ô¡∫ܲ ï˛!ï˛ˆÏܲ x!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòˆÏö ܲ#û˛yˆÏÓ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚ ï˛yÓ˚ Óî≈öy !òö–
3. ã˛y˛õÈÙȘÓò%ƒ!ï˛Ü˛ !e´Î˚yÓ˚ §y•yˆÏ΃ x!ï˛¢ˆÏ∑Ó˚ í˛zͲõyòˆÏöÓ˚ Óî≈öy !òö– ~ܲ•z ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§ˆÏܲ ܲ#û˛yˆÏÓ
!Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ x!ï˛¢∑ í˛zͲõyòˆÏö ܲyˆÏç °yàyˆÏöy ÎyÎ˚⁄
4. x!ï˛¢∑ ï˛Ó˚ˆÏDÓ˚ à!ï˛ˆÏÓà üy˛õyÓ˚ ˆÜ˛yöÄ ~ܲ!ê˛ ˛õk˛!ï˛ Óî≈öy ܲÓ˚&ö– xy˛õöyÓ˚ Ó!î≈ï˛ ˛õk˛!ï˛!ê˛ ï˛Ó˚°
àƒy§#Î˚ÈÙÙÙÈí˛zû˛Î˚ ≤ÃܲyÓ˚ üyôƒˆÏüÓ˚ ˆ«˛ˆÏe•z ≤ÈÏÎ˚yà ܲÓ˚y ÎyÎ˚ !ܲ⁄
5. !ã˛!ܲͧy¢yˆÏflf x!ï˛¢ˆÏ∑Ó˚ ≤ÈÏÎ˚yˆÏàÓ˚ !ÓÓÓ˚î !òö–
323
6. x!ï˛¢ˆÏ∑Ó˚ ÓƒÓ•yˆÏÓ˚Ó˚ ~üö ~ܲ!ê˛ í˛zòy•Ó˚î !òö ˆÎáyˆÏö
(a) x!ï˛¢ˆÏ∑Ó˚ ≤Ã!ï˛ú˛°ö Ä
(b) x!ï˛¢ˆÏ∑Ó˚ í˛zFã˛ ï˛#Ó ï˛y ܲyˆÏç °yàyˆÏöy •Î˚–
12.7 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1
(a) ˆÎ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ ◊ÓîˆÏÎyàƒ ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ |ôù≈§#üy xÌ≈yÍ 20KHz xˆÏ˛õ«˛y ˆÓ!¢ ï˛yˆÏܲ
x!ï˛¢∑ Ó°y •Î˚–
(b) ò[˛!ê˛Ó˚ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ ≤ÃyÌ!üܲ xÓfliyÎ˚ ¢)öƒ •ˆÏ° ˆ§!ê˛Ó˚ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ •ˆÏÓ 50KHz ×
2 Óy 100KHz– ˆÜ˛ööyñ xyˆÏÓ˚y!˛õï˛ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ 0·2Wbm–2 xÌÓy –0·2Wbm–2 ÈÙÙÙÈ ˆÎ!ê˛•z ˆ•yܲ
öy ˆÜ˛ö ˆã˛Ô¡∫ܲ x!ï˛ §ˆÏÓù≈yFã˛ •ˆÏÓ–
òˆÏ[˛Ó˚ ≤ÃyÌ!üܲ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ ÈÙÈ 0·5wbm–2 •ˆÏ° ˆ§!ê˛ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ (0·5
±
0·2)Wbm–2 xÌ≈yÍñ
0·7 Ä 0·3Wbm–2 ~Ó˚ üˆÏôƒ xyˆÏ®y!°ï˛ •ˆÏÓ– ~ˆÏ«˛ˆÏe ˆã˛Ô¡∫ܲ ï˛!ï˛ ≤Ã!ï˛ ˛õÎ≈yÎ˚ܲyˆÏ° ~ܲÓyÓ˚ §ˆÏÓ≈yFã˛
~ ~ܲÓyÓ˚ §Ó≈!ö¡¨ •ˆÏÓ– §%ï˛Ó˚yÇñ òˆÏ[˛Ó˚ ܲ¡õyB˛Ä 50KHz •ˆÏÓ–
(c) ˆÎ ˆÜ˛yÎ˚yê˛≈ç ˆÜ˛°yˆÏ§Ó˚ ü%°§)ˆÏÓ˚Ó˚ ܲ¡õyB˛ 1MHz, ï˛yÓ˚ ˆÓô •ˆÏÓ ≠
l =
1
615720
2210vmsns
−
− =×
= 2·86 mm
§%ï˛Ó˚yÇñ ü)°§%Ó˚ !•§yˆÏÓ 1MHz ܲ¡õyˆÏB˛Ó˚ ¢∑ í˛zͲõyòö ܲÓ˚ˆÏï˛ 2·86mm ˆÓˆÏôÓ˚ ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§
ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– ~Ó˚ !òö=îñ ˛õyÑã˛=î •zï˛ƒy!ò ˆÓˆÏôÓ˚ ˆÜ˛°y§ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ÎÌye´ˆÏü ï,˛ï˛#Î˚ñ ˛õMÈ˛ü
•zï˛ƒy!ò §üˆÏü° !•§yˆÏÓ 1MHz ܲ¡õyB˛ í˛zͲõyòö ܲÓ˚y ÎyˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# - 2 :
(a) çˆÏ°Ó˚ üˆÏôƒ 300Hz Ä 300KHz ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ ÎÌye´ˆÏü
1500300
= 5m Ä
31500
30010 ×
= 5mm– ≤ÃÌü ˆ«˛ˆÏe ï˛Ó˚D˜Ïòâ≈ƒ üyˆÏåÈÓ˚ ˜òˆÏâ≈ƒÓ˚ ï%˛°öyÎ˚ xˆÏöܲ Óí˛¸ •ÄÎ˚yÎ˚ üyåÈ ˆÌˆÏܲ ˙ ¢∑
!ӈϢ£Ï ≤Ã!ï˛ú˛!°ï˛ •ˆÏÓ öy– !ܲv !mï˛#Î˚ ˆ«˛ˆÏe üyˆÏåÈÓ˚ xyܲyÓ˚ ï˛Ó˚D˜Ïòâ≈ƒÓ˚ ï%˛°öyÎ˚ Óí˛¸ •ÄÎ˚yÎ˚ ˙ ¢∑
§%flõ‹Tû˛yˆÏÓ ≤Ã!ï˛ú˛!°ï˛ •ˆÏÓ–
324
(b) 12.6 §ü#ܲÓ˚î xö%ÎyÎ˚# v = n
λ
= mn
λ
i / sin
θ
m
~áyˆÏö m = 1, v = n
λ
= 1500ms–1,
λ
i = 600 × 10–9, n = 5 × 106
§%ï˛Ó˚yÇñ sin
θ
=
69 510600101500
− ×××
= 0·0002 xÌ≈yÍñ θ = 6'53''
(c) xˆÏöܲ=!° !öˆÏܲ° òˆÏ[˛Ó˚ ~ܲe §¡õˆÏöÓ˚ ú˛ˆÏ° ¢!_´¢y°# ¢∑ï˛Ó˚D í˛zͲõߨ •Î˚– !fiê˛ˆÏ°Ó˚ ˛õyï˛!ê˛
≤âhflÏ •ÄÎ˚yÎ˚ ¢∑ï˛Ó˚D !∞ï˛ !òˆÏܲ §B˛#î≈ âöˆÏܲyˆÏî ˛õyë˛yˆÏöy §Ω˛Ó •Î˚–
(d) ܲƒyfiê˛Ó˚ xˆÏÎ˚ˆÏ°Ó˚ ˆ¢y£ÏîyB˛
α
c =
12·08 ×
In 3 = 11·0m–1 ~ܲ•zû˛yˆÏÓ x!°û˛ xˆÏÎ˚ˆÏ°Ó˚ ˆ¢y£ÏyîyB˛
α
0 =
12·5 ×
In4 = 1·9m–1 ~ܲ•z ܲ¡õyˆÏB˛ ˆ¢y£ÏyîyB˛ §ywï˛yÓ˚ §üyö%˛õyï˛#– §%ï˛Ó˚yÇñ x!°û˛ xˆÏÎ˚ˆÏ°Ó˚
§ywï˛yB˛ = 1000mNsm–2 ×
1·3911·0
= 126mNsm–2
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°# ≠
1. ≤ÃhflÏyÓöy ˆòá%ö– çˆÏ°Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚ ˆÓà 1500ms–1 ôˆÏÓ˚ !öˆÏ° ܲ¡õyB˛
1500·05
=30KHz– ~•z
ܲ¡õyˆÏB˛Ó˚ ¢∑ˆÏܲ x!ï˛¢∑ Ó°y •Î˚–
2. ~Ó˚ í˛z_Ó˚ 12.2.1 xö%ˆÏFåÈò ˛õyÄÎ˚y ÎyˆÏÓ–
3. 12.2.2 xö%ˆÏFåȈÏò ≤Èϟ¿Ó˚ ≤ÃÌü xLjϢÓ˚ í˛z_Ó˚ ˛õyÄÎ˚y ÎyˆÏÓ– ~ܲ!ê˛ ˆÜ˛°y§ !Ó!û˛ß¨ §üˆÏüˆÏ° ܲ!¡õï˛
•ˆÏï˛ ˛õyˆÏÓ˚– §%ï˛Ó˚yÇñ ~ܲ•z ˆÜ˛yÎ˚yê≈˛ç ˆÜ˛°y§ˆÏܲ !Ó!û˛ß¨ ܲ¡õyˆÏB˛ xö%öyò â!ê˛ˆÏÎ˚ !û˛ß¨ !û˛ß¨ ܲ¡õyˆÏB˛Ó˚
x!ï˛¢∑ í˛zͲõߨ ܲÓ˚y ÎyÎ˚–
3. 12.3 xLjϢ xy˛õ!ö ≤ß¿!ê˛Ó˚ í˛z_Ó˚ ˛õyˆÏÓö–
5. ≤ß¿!ê˛Ó˚ í˛z_Ó˚ 12.4 xLjϢ (viii) xö%ˆÏFåȈÏò ˛õyÄÎ˚y •ˆÏÎ˚ˆÏåÈ–
6. ~•z ≤ß¿!ê˛Ó˚ í˛z_Ó˚Ä 12.4 xLjϢ ˛õyÄÎ˚y ÎyˆÏÓ–
325
~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 13 ¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö (Recording and reproduc-
tion of sound)
àë˛ö
13.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
13.2 ¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§
13.3 x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛
13.3.1 Îy!sfܲ x!û˛ˆÏ°áö
13.3.2 Îy!sfܲ ˛õ%öç≈öö
13.4 ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÈÈÙÙÙÈˆê˛˛õˆÏÓ˚ܲí≈˛yÓ˚
13.4 xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛
13.5.1 ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√
13.5.2 ܲü˛õƒyQ !í˛flÒ Óy !§!í˛
13.6 xyö%£Ï!Dܲ ÎsfyÓ°#
13.6.1 üy•zˆÏe´yˆÏú˛yö
13.6.2 °yí˛zí˛!flõܲyÓ˚
13.7 §yÓ˚yÇ¢
13.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
13.9 í˛z_Ó˚üy°y
13.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
ˆÎ §Ó ≤ÃÎ%!_´ xyüÓ˚y ˜òö!®ö ç#ÓˆÏö ≤Ã!ï˛!öÎ˚ï˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ Ìy!ܲñ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚
≤ÃÎ%!_´ ˆ§=!°Ó˚ xöƒï˛ü– ~•z ≤ÃÎ%!_´Ó˚ ܲ°ƒyˆÏî xyüÓ˚y !Óàï˛ !òˆÏöÓ˚ !¢“#Ó˚ ܲt˛fl∫Ó˚ Ä §D#ï˛ Óy xyÓ,!_
÷öˆÏï˛ ˛õy•z– ˆ•ühsˇ ü%ˆÏáy˛õyôƒyˆÏÎ˚Ó˚ àyö ~üö !ܲ Ó˚Ó#wöyÌ ë˛yÜ%˛ˆÏÓ˚Ó˚ fl∫ܲˆÏt˛Ó˚ ܲ!Óï˛y xyÓ,!_ ˆÓ˚ܲˆÏí≈˛Ó˚ üyôƒˆÏü
ˆ¢yöy §Ω˛Ó– xyÓyÓ˚ ˆÎ !ÓˆÏò¢# !¢“#ˆÏܲ ã˛y«%˛£Ï ˆòáyÓ˚ ˆÜ˛yöÄ §%ˆÏÎyà ˆö•z ï˛yÑÓ˚ àyöÄ Ü˛ƒyˆÏ§ê˛ÈÙÈÓ®# •ˆÏÎ˚
xyç xyüyˆÏòÓ˚ ܲyˆÏåÈ §•ç°û˛ƒ •ˆÏÎ˚ˆÏåÈ– ˆÓ˚!í˛Ä Óy ò)Ó˚ò¢≈ˆÏö xyüÓ˚y Îy !ܲå%È ÷!ö ï˛yÓ˚ x!ôܲyÇ¢•z ˛õ)ˆÏÓ≈
ˆÓ˚ܲí≈˛ ܲÓ˚y– ã˛°!Fã˛ˆÏeÓ˚ ÎyÓï˛#Î˚ ¢∑Ä !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚•z x!û˛!°!áï˛ ÌyˆÏܲ– ~ܲ ܲÌyÎ˚ñ ¢∑ ˆÎ fliyö ܲyˆÏ°Ó˚
ÓƒÓôyö x!ï˛e´ü ܲÓ˚ˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈ ï˛y x!û˛ˆÏ°áö Ä ˛õ%öç≈öö ≤ÃÎ%!_´Ó˚ í˛zqyÓˆÏöÓ˚ ≤Ãï˛ƒ«˛ ú˛°– ~•z =Ó˚&c˛õ)î≈
≤ÃÎ%!_´Ó˚ ˛õ!Ó˚ã˛Î˚ Óƒï˛#ï˛ fl∫ö!Óòƒy §¡∫¶˛#Î˚ xyˆÏ°yã˛öy x§¡õ)î≈ ˆÌˆÏܲ ÎyÎ˚–
¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ~ܲy!ôܲ ≤ÃÎ%!_´ Óï≈˛üyˆÏö ÓƒÓ•*ï˛ •ˆÏFåÈ– ~=!°Ó˚ ü)°ö#!ï˛ ~ܲ•z–
¢∑ï˛Ó˚ˆÏD ã˛yˆÏ˛õÓ˚ ˛õ!Ó˚Óï≈˛öˆÏܲ ≤Ã̈Ïü•z üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ myÓ˚y ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛ ˛õ!Ó˚îï˛ Ü˛Ó˚y •Î˚ ~ÓÇ
326
•zˆÏ°Ü˛ê˛Δ!öܲ ˛õk˛!ï˛ˆÏï˛ Îï˛ò)Ó˚ §Ω˛Ó x!ÓÜ,˛ï˛ ˆÓ˚ˆÏá !ÓÓ!ô≈ï˛ Ü˛Ó˚y •Î˚– ~•z !ÓÓ!ô≈ï˛ §ÇˆÏÜ˛ï˛ˆÏܲ ˆÜ˛yöÄ ~ܲ
˛õk˛!ï˛ˆÏï˛ fliyÎ˚#û˛yˆÏÓ §ÇÓ˚«˛î ܲÓ˚ˆÏï˛ •Î˚ÈÙÙÙÈï˛y Îy!sfܲñ ˆã˛Ô¡∫ܲ Óy xyˆÏ°yܲ#Î˚ñ ˆÎ ˆÜ˛yöÄ í˛z˛õyˆÏÎ˚ •ˆÏï˛ ˛õyˆÏÓ˚–
§ÇÓ˚!«˛ï˛ §ÇˆÏÜ˛ï˛ˆÏܲ•z xyüÓ˚y ¢ˆÏ∑Ó˚ x!û˛!°!˛õ Óy ˆÓ˚ܲí≈˛ (record) Ó!°– ¢ˆÏ∑Ó˚ ˛õ)öç≈öˆÏöÓ˚ §üˆÏÎ˚ ~•z
x!û˛!°!˛õ!ê˛•z §ÇÓ˚«˛î õk˛!ï˛Ó˚ !Ó˛õÓ˚#ï˛ ~ܲ õk˛!ï˛Ó˚ üyôƒˆÏü õ) ÏÓ≈ §ÇÓ˚!«˛ï˛ Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛Ó˚˛˛õ%öÓ˚&ͲõyòˆÏö
ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ~•z ˛õ%öÓ˚Ͳõy!òï˛ §ÇˆÏÜ˛ï˛ˆÏܲ !ÓÓ!ô≈ï˛ Ü˛ˆÏÓ˚ ï˛yÓ˚ §y•yˆÏ΃ °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ ˛õò≈yˆÏܲ ܲ!¡õï˛
ܲÓ˚ˆÏ° x!û˛!°!áï˛ ¢∑!ê˛ˆÏܲ !ú˛ˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚–
~•z ~ܲˆÏܲ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ Ä ˛õ%öç≈öˆÏö ≤Ãã˛!°ï˛ ˛õk˛!ï˛=!° xyüyˆÏòÓ˚ ≤Ãòyö xyˆÏ°yã˛ƒ !Ó£ÏÎ˚– ~•z
≤ÃÎ%!_´Ó˚ ~ܲ!ê˛ §Ç!«˛Æ •z!ï˛•y§ ~ÓÇ ~ï˛ ÓƒÓ•*ï˛ ò%•z!ê˛ x˛õ!Ó˚•yÎ≈ ÎsfÈÙÙÙÈüy•zˆÏe´yˆÏú˛yö Ä °yí˛zí˛!flõܲyÓ˚
§¡∫ˆÏ¶˛ !ܲå%È xyˆÏ°yã˛öyÄ ~•z ~ܲˆÏܲÓ˚ xhsˇû%≈˛_´ •ˆÏÓ–
í˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒí˛zˆÏj¢ƒ
xyô%!öܲ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö ≤ÃÎ%!_´Ó˚ §ˆÏD xy˛õöyˆÏܲ ˛õ!Ó˚!ã˛ï˛ •ÄÎ˚yÓ˚ §%ˆÏÎyà ˆòÄÎ˚y•z ~•z
~ܲˆÏܲÓ˚ í˛zˆÏj¢ƒ– ~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚ˆÏ° xy˛õ!ö !ӈϢ£Ïû˛yˆÏÓ ˆÎ ܲyç=!° ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ˆ§=!° •°≠
!Óàï˛ ~ܲ ¢ï˛y∑#Ó˚ x!ôܲ §üˆÏÎ˚ ¢∑ ≤ÃÎ%!_´Ó˚ ˆÎ !Óܲy¢ âˆÏê˛ˆÏåÈ ï˛yÓ˚ !ÓÓÓ˚î !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
@ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛ ¢∑ x!û˛ˆÏ°áö Ä ï˛y ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏöÓ˚ ˛õk˛!ï˛ Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ˆê˛˛õˆÏÓ˚ܲí≈˛yÓ˚ ΈÏsfÓ˚ ܲyÎ≈ö#!ï˛ Ä àë˛ö Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏö
ˆã˛Ô¡∫ܲ ˛õk˛!ï˛Ó˚ §%!Óôy=!°Ó˚ ï˛y!°Ü˛y Ó˚ã˛öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
§Óyܲ ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ ~ÓÇ xyˆÏ°yܲ#Î˚ Óy ܲü˛õƒyQ !í˛ˆÏflÒ ¢∑ x!û˛ˆÏ°áö ˛õk˛!ï˛Ó˚ !ÓÓÓ˚î !òˆÏï˛
˛õyÓ˚ˆÏÓö ~ÓÇ ~=!° ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏöÓ˚ í˛z˛õyÎ˚ Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
¢∑ §ÇÓ˚«˛î Ä ˛õ%öç≈öˆÏöÓ˚ xyô%!öܲ !í˛!çê˛ƒy° ˛õk˛!ï˛!ê˛Ó˚ §ˆÏD ˛õ!Ó˚ã˛Î˚ °yû˛ ܲˆÏÓ˚ ~•z ˛õk˛!ï˛ Óƒyáƒy
ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
üy•zˆÏe´yˆÏú˛yö Ä °yí˛zí˛!flõܲyÓ˚ÈÙÙÙÈ~•z ò%•z §yôyÓ˚î ¢∑ΈÏsfÓ˚ àë˛ö Ä Ü˛yÎ≈˛õk˛!ï˛ Ó%!é˛ˆÏÎ˚ !òˆÏï˛
˛õyÓ˚ˆÏÓö–
ˆüyˆÏê˛Ó˚ í˛z˛õÓ˚ ~•z ~ܲܲ!ê˛ xy˛õöyˆÏܲ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%ö≈çöˆÏöÓ˚ ≤Ãã˛!°ï˛ ≤ÃÎ%!_´=!°Ó˚ §ˆÏD ˆÓ¢
!ܲå%Èê˛y ˛õ!Ó˚!ã˛ï˛ ܲÓ˚ˆÏÓñ ÎyˆÏï˛ ≤ÈÏÎ˚yçˆÏö xy˛õ!ö ~=!° §¡∫ˆÏ¶˛ •yˆÏï˛Ü˛°ˆÏü ܲyç ˆ¢áyÎ˚ x@ˇÃ§Ó˚ •ˆÏï˛ ˛õyˆÏÓ˚ö–
13.2 ¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§¢∑ ≤ÃÎ%!_´Ó˚ §Ç!«˛Æ •z!ï˛•y§
¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ˆÎ í˛zß¨ï˛ xÓfliy xyüÓ˚y Óï≈˛üyˆÏö Î%ˆÏà ˆòáˆÏï˛ ˛õy•z ï˛y Ó‡ !ÓK˛yö# Ä
≤ÃÎ%!_´!ÓˆÏòÓ˚ ò#â≈ ≤ÈÏã˛‹TyÓ˚ ú˛°– ~•z ≤ÃÎ%!_´Ó˚ •z!ï˛•y§ ≤ÃyÎ˚ ˆòí˛¸ˆÏ¢y ÓåÈÓ˚ ˛õ%Ó˚ˆÏöy– ~áyˆÏö xyüÓ˚y ¢∑ ≤ÃÎ%!_´Ó˚
í˛zˆÏÕ‘áˆÏÎyàƒ í˛zߨÎ˚ˆÏöÓ˚ ôy˛õ=!° §yÓ˚!îÓ˚ xyܲyˆÏÓ˚ Óî≈öy ܲÓ˚Ó–
327
§y° Ä àˆÏÓ£Ïܲ†í˛zqyÓܲ§y° Ä àˆÏÓ£Ïܲ†í˛zqyÓܲ§y° Ä àˆÏÓ£Ïܲ†í˛zqyÓܲ§y° Ä àˆÏÓ£Ïܲ†í˛zqyÓܲ§y° Ä àˆÏÓ£Ïܲ†í˛zqyÓܲ ≤ÃÎ%!_´Ó˚ í˛zqyÓö Ä í˛zߨÎ˚ö ≤ÃÎ%!_´Ó˚ í˛zqyÓö Ä í˛zߨÎ˚ö ≤ÃÎ%!_´Ó˚ í˛zqyÓö Ä í˛zߨÎ˚ö ≤ÃÎ%!_´Ó˚ í˛zqyÓö Ä í˛zߨÎ˚ö ≤ÃÎ%!_´Ó˚ í˛zqyÓö Ä í˛zߨÎ˚ö
1857 !°Î˚Ñ flÒê˛ Sú ˛y™V ˆú˛yöˆÏê˛y@ˇÃyú˛ ΈÏsfÓ˚ í˛zqyÓö– ~•z ΈÏsf ~ܲ!ê˛ â%Ó˚hsˆÏÓ°ˆÏöÓ˚ û)˛£Ïy
üyáyˆÏöy àyˆÏÎ˚ ¢∑ xy˛õï˛ˆÏöÓ˚ ú˛ˆÏ° ~ܲ!ê˛ ˛õò≈yÓ˚ ܲ¡õö ˆì˛í˛z
ˆá°yˆÏöy ˆÓ˚áy !•§yˆÏÓ !°!˛õÓk˛ •ï˛–
1877 ê˛üy§ xy°û˛y ~!í˛§ö ~ÑÓ˚ í˛zqy!Óï˛ ˆú˛yˆÏöy@ˇÃyú˛ (Phonograph) ΈÏsf ¢∑
SxyˆÏü!Ó˚ܲy Î%_´Ó˚y‹T…V ~ܲ!ê˛ ˛õò≈yÎ˚ xy˛õ!ï˛ï˛ •ˆÏ° ˛õò≈y!ê˛ Ü˛!¡õï˛ •ï˛ ~ÓDz ˛õò≈yÓ˚ §ˆÏD
Î%_´ ~ܲ!ê˛ §)ã˛# â%Ó˚hsˇ ˆÓ°ˆÏöÓ˚ àyˆÏÎ˚ çí˛¸yˆÏöy !ê˛ˆÏöÓ˚ ˛õyˆÏï˛Ó˚ í˛z˛õÓ˚
¢ˆÏ∑Ó˚ ã˛y˛õ xö%ÎyÎ˚# ܲüˆÏÓ!¢ àû˛#Ó˚ï˛yÓ˚ ˆ˛õÑã˛yˆÏöy ˆÓ˚áy ˆÜ˛ˆÏê˛ ã˛°ï˛–
~•z ˛õk˛!ï˛Ó˚ öyü ˆòÄÎ˚y •ˆÏÎ˚!åÈ° Ú!•°ÈÙÈˆí˛°Û (hill and dale)
x!û˛ˆÏ°áö–
1878 !ã˛ˆÏã˛fiê˛yÓ˚ ~.ˆÓ° öÓü ˆüy ÏüÓ ˆÓ° ÏöÓ í z õ ÏÓ ò,ì˛ ˛õò≈yÓ § ÏD Î%_´ ôyÓy Ï°y §)ã˛# ¢ Ï∑Ó
(xyˆÏü!Ó˚ܲy Î%_´Ó˚y‹T…V ã˛y˛õ xö%ÎyÎ˚# ˆüyü ˆÜ˛ˆÏê˛ ÓyÓ˚ ܲˆÏÓ˚ !•°ÈÙÈˆí˛° ˛õk˛!ï˛Ó˚ x!û˛ˆÏ°áö
ˆÓ˚áy ˜ï˛Ó˚# ܲÓ˚ï˛– xyˆÏ˛õ«˛yÜ,˛ï˛ ¢_´ ˆüyü !òˆÏÎ˚ ~•z ˆÓ°ö ˆÓ˚ܲˆÏí≈˛Ó˚
öܲ° ˜ï˛!Ó˚ ܲˆÏÓ˚ ˛õ%öç≈öˆÏöÓ˚ çöƒ ÓƒÓ•*ï˛ •ï˛– 1885 §yˆÏ° ~•z
˛õk˛!ï˛Ó˚ ˆ˛õˆÏê˛rê˛ ˆöö ˆÓ° Ä ˆê˛rê˛yÓ˚–
1887 ~!ü° Óy!°≈öyÓ˚ •z!ö ≤ÃÌü !í˛flÒ Óy ã˛yܲ!ï˛Ó˚ xyÜ,˛!ï˛Ó˚ ˆÓ˚ܲí≈˛ !öü≈yî
Sçyü≈y!öV ܲˆÏÓ˚ö– !çB˛ SòhflÏyV !ö!ü≈ï˛ ã˛yܲ!ï˛Ó˚ ã˛!Ó≈ çyï˛#Î˚ Ó›ˆÏï˛ xyFåÈy!òï˛
ܲˆÏÓ˚ ï˛yÓ˚ í˛z˛õÓ˚ ˛õyˆÏ¢Ó˚ !òˆÏܲ xÑyܲyÓyÑܲy ˛õуyã˛yˆÏöy ˆÓ˚áyÓ˚ xyÜ,˛!ï˛ˆÏï˛
¢∑ x!û˛!°!áï˛ •ï˛– ã˛yܲ!ï˛!ê˛Ó˚ í˛z˛õÓ˚ xƒy!§í˛ ≤ÈÏÎ˚yà ܲÓ˚ˆÏ°
ˆ§!ê˛Ó˚ xyFåÈy!òï˛ xÇ¢ x!ÓÜ,˛ï˛ ÌyÜ˛ï˛ !ܲv x!û˛ˆÏ°áö ˆÓ˚áyÓ˚ üôƒ
!òˆÏÎ˚ xƒy!§í˛ !çˆÏB˛Ó˚ §ˆÏD Ó˚y§yÎ˚!öܲ !Ó!e´Î˚yÓ˚ ú˛ˆÏ° fliyÎ˚# Ä §üyö
àû˛#Ó˚ï˛yÎ˚ ˆÓ˚áyÜ,˛!ï˛ àˆÏï≈˛Ó˚ §,!‹T ܲÓ˚ï˛– ˛õ)öç≈öˆÏöÓ˚ çöƒ òhflÏyÓ˚ ü)°
ˆÓ˚ܲí≈˛!ê˛Ó˚ ÌyˆÏüy≈≤’y!fiê˛Ü˛ !ö!ü≈ï˛ öܲ° ÓƒÓ•*ï˛ •ï˛– Óy!°≈öyÓ˚ •yˆÏï˛
ã˛y°yˆÏöy ΈÏsfÓ˚ §y•yˆÏ΃ ˆÓ˚ܲí≈˛!ê˛ ˆâyÓ˚yˆÏöyÓ˚ ÓƒÓfliy ܲˆÏÓ˚ö ~ÓÇ
§yí˛zu˛ÈÙÈÓˆÏ:Ó˚ §ˆÏD ~ܲ!ê˛ ˆåÈyê˛ ˆã˛yÇ Î%_´ ܲˆÏÓ˚ö– !ï˛!ö ~•z Îsf!ê˛
@ˇÃyüyˆÏú˛yö (Gramophone) öyˆÏü ˆ˛õˆÏê˛rê˛ Ü˛ˆÏÓ˚ö–
328
1900 Óy!°≈öyˆÏÓ˚Ó˚ í˛zqy!Óï˛ ˛õyŸª#≈Î˚ x!û˛ˆÏ°áö ˛õk˛!ï˛ xyˆÏü!Ó˚ܲy Ä
•zí˛zˆÏÓ˚yˆÏ˛õ çö!≤ÃÎ˚ •Î˚–
1907 °yí˛zhflÏ S!Ó ˆÏê˛öV ã˛°!Fã˛ˆÏeÓ˚ §ˆÏD ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ ÓƒÓfliyÓ˚ !Ó !ê˛¢ ˆ˛õˆÏê˛rê˛–
~ˆÏÏ«˛ˆÏe !ÓÓô≈ܲ ÓƒÓ•*ï˛ •Î˚!ö– ≤Ã!ï˛ !˛õë˛ 4
12
!ü!öê˛ ÓyçyˆÏöy ÎyÎ˚
~üö 78 rpm S≤Ã!ï˛ !ü!öˆÏê˛ xyÓï≈˛ö §Çáƒy 78) ˆÓ˚ܲí≈˛ ã˛y°% •Î˚–
1915 °# !í˛ ú˛ˆÏÓ˚fiê˛ ¢ˆÏ∑Ó˚ «˛üï˛y Ó,!k˛Ó˚ çöƒ ê˛ΔyˆÏÎ˚yí˛ û˛y°‰û˛ ÓƒÓ•yÓ˚
SxyˆÏü!Ó˚ܲy Î%_´Ó˚y‹T…V ܲˆÏÓ˚ x!í˛Î˚ö !ÓÓô≈ܲ !öü≈yî– ≤ÃÌü !ÓŸªÎ%ˆÏk˛Ó˚ ú˛ˆÏ° 1920 §yˆÏ°Ó˚
xyˆÏà •zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ˆÏܲÓ˚ ÓƒÓ•yÓ˚ ˛õ%ˆÏÓ˚y˛õ%!Ó˚ ã˛y°% •Î˚!ö– 1924
§yˆÏ° ˜Óò%ƒ!ï˛Ü˛ x!û˛ˆÏ°áö Ä 1925 §y° ˜Óò%ƒ!ï˛Ü˛ ˛õ%öç≈öö
≤Ãã˛!°ï˛ •Î˚–
1923 °# !í˛ ú˛ˆÏÓ˚fiê˛ !ÓÓô≈ˆÏܲÓ˚ §y•yˆÏ΃ ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ ¢∑ x!û˛ˆÏ°áö Ä ï˛yÓ˚ ˆÌˆÏܲ
¢∑ ˛õ%öç≈öˆÏöÓ˚ ˆ˛õˆÏê˛rê˛–
1927 ܲy°≈§ö Ä Ü˛yˆÏ˛õ≈rê˛yÓ˚ ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áˆÏöÓ˚ ܲyÎ≈ܲÓ˚# ˛õk˛!ï˛ S˛õ!Ó˚Óï˛#≈ ≤ÃÓy• ÓyÎ˚y§ ~Ó˚
§y•yˆÏ΃V–
1927-28 xyˆÏü!Ó˚ܲy Î%_´Ó˚y‹T… ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ ≤Ã›ï˛ ≤Ãîy°#–
Ä çyü≈y!ö
1928 •zÇ°ƒyu˛ ≤ÃÌü §Óyܲ ã˛°!Fã˛e ‘‘The Jazz Singer’’–
1936 çyü≈y!ö üƒyàˆÏöˆÏê˛yˆÏú˛yö ܲy¡õy!ö Óy!°≈ö Ó˚!í˛Ä ü°yÎ˚ ≤ÃÌü ê˛˛õˆÏÓ˚ܲí≈˛yÓ˚
≤Ãò¢≈ö ܲˆÏÓ˚–
1960 ~Ó˚ ò¢Ü˛ •°ƒyu˛ ÈÙÈ ~Ó˚ !ú˛!°˛õ‰§‰ ˆÜ˛y¡õy!ö ܲƒyˆÏ§ê˛ ˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ ã˛y°% ܲˆÏÓ˚–
≤ÃÌü !òˆÏܲ ~!ê˛ ÷ô% ܲÌy Ó°yÓ˚ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ çöƒ ÓƒÓ•*ï˛
•ï˛–
329
1970 ~Ó˚ ò¢Ü˛ ¢ˆÏ∑Ó˚ §yÇ!áƒÜ˛ Óy !í˛!çê˛ƒy° x!û˛ˆÏ°áˆÏöÓ˚ ≤Ãã˛°ö– ï˛ˆÏÓ ~ §üˆÏÎ˚
ü)° ˆê˛ˆÏ˛õ x!û˛ˆÏ°áö §yÇ!áƒÜ˛ ˛õk˛!ï˛ˆÏï˛ •ˆÏ°Ä !Óe´ˆÏÎ˚Ó˚ çöƒ
!í˛§‰Ü‰˛ Ä ˆê˛˛õ=!°ˆÏï˛ xƒyöy°à ˛õk˛!ï˛ˆÏï˛•z ¢∑ x!û˛!°!áï˛
ÌyÜ˛ï˛–
1983 çy˛õyˆÏöÓ˚ ˆ§y!ö Ä •°ƒyˆÏu˛Ó˚ !ú˛!°˛õ‰§‰ SˆÜ˛y¡õy!ö myÓ˚y ˆÎÔÌû˛yˆÏÓ
ܲü˛õƒyQ !í˛flÒ (CD Óy !§!í˛V Óy !í˛!çê˛ƒy° x!í˛Ä !í˛flÒ (DAD)-
~Ó˚ !öü≈yî Ä !Ó˛õîö–
í˛z˛õˆÏÓ˚Ó˚ §yÓ˚!îˆÏï˛ ¢∑ ≤ÃÎ%!_´Ó˚ ˆ«˛ˆÏe ˆÎ §Ó í˛zߨÎ˚ˆÏöÓ˚ ܲÌy Ó°y •ˆÏÎ˚ˆÏåÈñ ˆ§=!° åÈyí˛¸yÄ xyÓ˚Ä xˆÏöܲ
í˛zߨÎ˚îü)°Ü˛ àˆÏÓ£Ïîy Ä xy!Ó‹ÒyÓ˚ ¢∑ ≤ÃÎ%!_´ˆÏܲ Óï≈˛üyö Î%ˆÏàÓ˚ xï%˛ƒFã˛ üyˆÏö ˆ˛õÔÑåȈÏï˛ §y•y΃ ܲˆÏÓ˚ˆÏåÈ–
~=!°Ó˚ üˆÏôƒ xyˆÏåÈ üy•zˆÏe´yˆÏú˛yö Ä °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ í˛zqyÓöñ !Ó!û˛ß¨ •zˆÏ°Ü˛ê˛Δ!öܲ Óï≈˛ö#Ó˚ !öü≈yîñ ÎÌy
!ÓÓô≈ܲ Ä !ú˛°ê˛yÓ˚ Óï≈˛ö# ~ÓÇ •y•zÈÙÈú˛y•z (hifi) ≤ÃÎ%!_´Ó˚ SܲÌy!ê˛ ~ˆÏ§ˆÏåÈ high fidelity Óy í˛zFã˛ x!Óܲ°ï˛y
ˆÌˆÏܲV í˛zqÓ– ~•z ~ܲˆÏܲÓ˚ ˛õÓ˚Óï˛#≈ xÇ¢=!°ˆÏï˛ xy˛õ!ö ¢∑ ≤ÃÎ%!_´Ó˚ =Ó˚&c˛õ)î≈ í˛z˛õyòyö=!°Ó˚ ˛õ!Ó˚ã˛Î˚
˛õyˆÏÓö–
13.3 x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛
~•z ˛õk˛!ï˛ˆÏï˛ ¢∑ §ÇÓ˚«˛ˆÏîÓ˚ üyôƒˆÏü @ˇÃyüyˆÏú˛yˆÏöÓ˚ ˆÓ˚ܲí≈˛ Óy !í˛flÒ– Óï≈˛üyˆÏö ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ ܲyˆÏç
@ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛Ó˚ ÓƒÓ•yÓ˚ ≤Ãã˛!°ï˛ ˆö•z– Î!òÄ ˛õ%Ó˚ˆÏöy ˆÓ˚ܲí≈˛ ÓyçyˆÏÓ˚ °û˛ƒ ~ÓÇ Ó‡ ˆÓ˚ܲí≈˛ !Ó!û˛ß¨
≤Ã!ï˛¤˛yˆÏö §ÇÓ˚!«˛ï˛ xyˆÏåÈ– ¢∑ x!û˛ˆÏ°áö ≤ÃÎ%!_´Ó˚ •z!ï˛•yˆÏ§ ~ܲˆÏê˛ =Ó˚&c˛õ)î≈ xôƒyÎ˚ !•§yˆÏÓ xyüÓ˚y
@ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛ x!û˛ˆÏ°áö ˛õk˛!ï˛ Óî≈öy ܲÓ˚Ó–
13.3.1 Îy!sfܲ x!û˛ˆÏ°áöÎy!sfܲ x!û˛ˆÏ°áöÎy!sfܲ x!û˛ˆÏ°áöÎy!sfܲ x!û˛ˆÏ°áöÎy!sfܲ x!û˛ˆÏ°áö
¢ˆÏ∑Ó˚ Îy!sfܲ x!û˛ˆÏ°áö ˛õk˛!ï˛!ê˛ xy˛õ!ö 13.1 ÓœÜ˛ !ã˛e ˆÌˆÏܲ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– §D#ï˛ñ Ó_,´ï˛y •zï˛ƒy!ò
ˆÎ ¢∑ x!û˛!°!áï˛ •ˆÏÓ ˆ§!ê˛ ≤Ã̈Ïü üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §y•yˆÏ΃ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ä •zˆÏ°Ü˛ê˛Δ!öܲ
!ã˛e : 13.1
!ÓÓô≈ˆÏܲÓ˚ §y•yˆÏ΃ !ÓÓ!ô≈ï˛ Ü˛Ó˚y •Î˚– !ÓÓ!ô≈ï˛ §ÇˆÏÜ˛ï˛ ï˛!í˛¸Í≤ÃÓy• x!û˛ˆÏ°áö ΈÏsfÓ˚ ï˛!í˛¸Íã%˛¡∫ˆÏܲÓ˚ Ü%˛[˛°#ˆÏï˛
˛õyë˛yˆÏöy •Î˚–
330
x!û˛ˆÏ°áö Îsf!ê˛Ó˚ ~ܲ!ê˛ §Ó˚°#Ü,˛ï˛ !ã˛e S!ã˛e 13.2) ~áyˆÏö ˆòÄÎ˚y •°– ~áyˆÏö C ï˛!í˛¸Íã%˛¡∫ˆÏܲÓ˚ Ü%˛[˛°#–
Ü%˛[˛°#ˆÏï˛ ˛õ!Ó˚Óï˛#≈ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ú˛ˆÏ° ï˛!í˛¸Íã%˛¡∫ܲ!ê˛Ó˚ (M) ã%˛¡∫ܲö Ä !Óã%˛¡∫ܲö âˆÏê˛– ~Ó˚ ú˛ˆÏ° R òˆÏ[˛Ó˚
§ˆÏD §ÇÎ%_´ xyˆÏü≈ã˛yÓ˚ A-~Ó˚ í˛z˛õÓ˚ ï˛!í˛¸Íã%˛¡∫ˆÏܲÓ˚ xyܲ£Ï≈î# ê˛Ü≈˛ ܲˆÏü ÓyˆÏí˛¸– ò[˛!ê˛ !fl±Ç (s, s) !òˆÏÎ˚ xyê˛Ü˛yˆÏöy
ÌyܲyÎ˚ xyˆÏü≈ã˛yÓ˚!ê˛Ó˚ R òˆÏ[˛Ó˚ xˆÏ«˛Ó˚ í˛z˛õÓ˚ â)î≈ö ܲ¡õö âˆÏê˛– ò[˛!ê˛Ó˚ §ˆÏD
xyí˛¸yxy!í˛¸û˛yˆÏÓ °yàyˆÏöy ~ܲ!ê˛ §%ã˛#ü%á ܲ°ü P ~Ó˚ ú˛ˆÏ° ܲ!¡õï˛ •Î˚–
§%ã˛#ü%á!ê˛ˆÏï˛ ˆÜ˛yÓ˚yu˛yü Óy xöƒ ˆÜ˛yöÄ Ü˛!ë˛ö á!öˆÏçÓ˚ ¢yö ˆòÄÎ˚y ê%˛Ü˛ˆÏÓ˚y
Ó§yˆÏöy ÌyˆÏܲ ~ ˜Óò%ƒ!ï˛Ü˛ í˛z˛õyˆÏÎ˚ ˆ§!ê˛ˆÏܲ ï˛Æ Ó˚yáy •Î˚– ܲ°ü!ê˛ ˆÎ
ã˛yܲ!ï˛Ó˚ (D) í˛z˛õÓ˚ ¢∑!°!˛õ ü%!oï˛ •ˆÏÓ ï˛yÓ˚ í˛z˛õÓ˚ x“ ã˛yˆÏ˛õ Ó§yˆÏöy
ÌyˆÏܲ– ã˛yܲ!ï˛!ê˛ û˛yÓ˚# ôyï%˛Ó˚ í˛z˛õÓ˚ ˆüyˆÏüÓ˚ ≤Èϰ˛õ !òˆÏÎ˚ xÌÓy
xƒy°%!ü!öÎ˚yˆÏüÓ˚ í˛z˛õÓ˚ °y«˛yÓ˚ Óy!ö≈§ !òˆÏÎ˚ ˜ï˛!Ó˚ •Î˚– ܲ°ü!ê˛ ˆÎüö
ܲ!¡õï˛ •Î˚ñ ã˛yܲ!ï˛!ê˛Ä â%Ó˚ˆÏï˛ ÌyˆÏܲ ~ÓÇ !àÎ˚yˆÏÓ˚Ó˚ §y•yˆÏ΃ ܲ°ü!ê˛ˆÏܲ
ã˛yܲ!ï˛Ó˚ ˛õ!Ó˚!ô ˆÌˆÏܲ e´ˆÏü ˆÜ˛ˆÏwÓ˚ !òˆÏܲ !öˆÏÎ˚ ÎyÄÎ˚y •Î˚–
~Ó˚ ú˛ˆÏ° ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ ~ܲ!ê˛ ˛õƒyÑã˛yˆÏöy °yäˆÏ°Ó˚ òyˆÏàÓ˚ üï˛ ˆÓ˚áy ˜ï˛!Ó˚
•Î˚ ˆÎ!ê˛ Ü˛°ˆÏüÓ˚ §)ã˛#ü%á!ê˛Ó˚ ܲ¡õö xö%ÎyÎ˚# xÑyܲyÓyÑܲy •Î˚– ~•z ã˛yܲ!ï˛!ê˛•z
¢ˆÏ∑Ó˚ ˆÓ˚ܲí≈˛– °y«˛yÓ˚ Óy!ö≈¢ ˆòÄÎ˚y ˆÓ˚ܲí≈˛ §Ó˚y§!Ó˚ ¢ˆÏ∑Ó˚ ˛õ%öç≈öˆÏö
ÓƒÓ•*ï˛ •ï˛– xyÓ˚ ˆüyˆÏüÓ˚ ˆÓ˚ܲí≈˛ ÓƒÓ•*ï˛ •ï˛ !Ó˛õîˆÏöÓ˚ çöƒ ˆÓ˚ܲˆÏí≈˛Ó˚ ܲ!˛õ ˜ï˛!Ó˚ ܲÓ˚yÓ˚ ü)° ˆÓ˚ܲí≈˛ !•§yˆÏÓ–
xÓ¢ƒ °y«˛yÓ˚ ≤Èϰ˛õˆÏöÓ˚ üyö ~ÓÇ ˆÓ˚ܲí≈˛ ܲyê˛yÓ˚ ≤ÃÎ%!_´Ó˚ í˛zߨ!ï˛Ó˚ ú˛ˆÏ° °y«˛y ≤Èϰ!˛õï˛ ˆÓ˚ܲí≈˛•z ˛õˆÏÓ˚ ü)°
ˆÓ˚ܲí≈˛ !•§yˆÏÓ ÓƒÓ•*ï˛ •ˆÏÎ˚ˆÏåÈ–
ü)° Ó˚ܲí≈˛ ̈Ïܲ !Ó˛õîöˆÏÎyàƒ öܲ° ï˛!Ó˚ ܲÓ˚yÓ˚ çöƒ ï˛!í˛¸Í ≤Èϰ˛õö (electroplating) õk˛!ï˛ xÓ°¡∫ö
ܲÓ˚y •Î˚– ≤Ã̈Ïü ˆÓ˚ܲí≈˛!ê˛Ó˚ í˛z˛õÓ˚ ~ܲ!ê˛ ˛õyï˛°y Ó˚&ˆÏ˛õyÓ˚ xyhflÏÓ˚î ˜ï˛!Ó˚ ܲˆÏÓ˚ ï˛°!ê˛ˆÏܲ ï˛!í˛¸Í ˛õ!Ó˚Óy•# ܲÓ˚y
•Î˚– ~Ó˚ ˛õÓ˚ ü)° ˆÓ˚ܲˆÏí≈˛Ó˚ í˛z˛õÓ˚ ï˛yüy Óy !öˆÏܲˆÏ°Ó˚ ï˛!í˛¸Í ≤Èϰ˛õö ܲÓ˚y •ˆÏ° ˆÎ ôyï%˛Ó˚ ã˛yܲ!ï˛!ê˛ ˛õyÄÎ˚y
ÎyÎ˚ ˆ§!ê˛ˆÏܲ ü)° ˆÓ˚ܲˆÏí≈˛Ó˚ ˆöˆÏà!ê˛û˛ Ó°y ÎyÎ˚ xÌ≈yÍñ ü)° ˆÓ˚ܲˆÏí≈˛ ˆÎ öy°#ˆÏÓ˚áy ÌyˆÏܲñ ôyï%˛Ó˚ ã˛yܲ!ï˛ˆÏï˛
ï˛yÓ˚ x!Óܲ° ≤Ã!ï˛Ó˚*˛õ xy°ˆÏÓ˚áy ˛õyÄÎ˚y ÎyÎ˚– ~•z ã˛yܲ!ï˛!ê˛ˆÏܲ åÈyÑã˛ !•§yˆÏÓ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ xÌ≈yÍ
ÌyˆÏüy≈≤’y!fiê˛ˆÏܲÓ˚ ã˛yܲ!ï˛Ó˚ í˛z˛õÓ˚ ôyï%˛Ó˚ ã˛yܲ!ï˛!ê˛Ó˚ åÈy˛õ !òˆÏÎ˚ ÓƒÓ•yˆÏÓ˚y˛õˆÏÎyà# xˆÏöܲ=!° ˆÓ˚ܲí≈˛ ˛õyÄÎ˚y ÎyÎ˚–
Óy!î!çƒÜ˛ í˛zͲõyòˆÏöÓ˚ çöƒ ï˛!í˛¸Í ≤Èϰ˛õö ˛õk˛!ï˛ˆÏï˛ xˆÏöܲ=!° ôyï%˛Ó˚ ÚˆöˆÏà!ê˛û˛Û ˜ï˛!Ó˚ ܲÓ˚y •Î˚ñ ÎyÓ˚
≤ÈÏï˛ƒÜ˛!ê˛ ˆÌˆÏܲ xˆÏöܲ=!° Ú˛õ!ç!ê˛û˛Û ˛õyÄÎ˚y ˆÎˆÏï˛ ˛õyˆÏÓ˚–
!mï˛#Î˚ !ÓŸªÎ%ˆÏk˛Ó˚ §üÎ˚ ˛õÎ≈hsˇ !ü!öˆÏê˛ 78 ÓyÓ˚ ˆâyÓ˚yˆÏï˛ •Î˚ ~ÓÇ ~ܲ !˛õë˛ 4
12
!ü!öê˛ ÓyˆÏçñ ~Ó˚*˛õ ˆÓ˚ܲí≈˛•z
≤Ãã˛!°ï˛ !åÈ°– 1948 §yˆÏ° ≤Ã!ï˛ !˛õë˛ 30 !ü!öê˛ ÓyçyÓ˚ 33
12
rpm ˆÓ˚ܲí≈˛ ~ÓÇ ï˛yÓ˚ ˛õÓ˚ 8 !ü!öê˛ ÓyçyÓ˚
45rpm ˆÓ˚ܲˆÏí≈˛Ó˚ ≤Ãã˛°ö •Î˚– ~=!° ÎÌye´ˆÏü °Ç ˆ≤’!Î˚Ç Óy LP~ÓÇ ~:ˆÏê˛ˆÏu˛í˛ ˆ≤’ Óy EP ˆÓ˚ܲí≈˛ Ó˚*ˆÏ˛õ
˛õ!Ó˚!ã˛ï˛– ~=!°Ó˚ öy°#ˆÏÓ˚áy xï˛ƒhsˇ §)-ÈÙÙÙÈ≤ÈÏfli 56 üy•ze´ö Ä àû˛#Ó˚ï˛yÎ˚ 32 üy•ze´ö– ~çöƒ ~=!°ˆÏܲ
üy•zˆÏe´y@ˇÃ&û˛ (microgroove) ˆÓ˚ܲí≈˛Ä Ó°y •Î˚–
!ã˛e ≠ 13.2
331
13.3.2 Îy!sfܲ ˛õ%öç≈ööÎy!sfܲ ˛õ%öç≈ööÎy!sfܲ ˛õ%öç≈ööÎy!sfܲ ˛õ%öç≈ööÎy!sfܲ ˛õ%öç≈öö
¢∑ ≤ÃÎ%!_´Ó˚ ≤ÃÌü Î%ˆÏà @ˇÃyüyˆÏú˛yö ˆÓ˚ܲí≈˛ ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏö Ú§yí˛zu˛ Ó:Û S!ã˛e 13.2V öyˆÏü §¡õ)î≈
Îy!sfܲ ÓƒÓfliy @ˇÃ•î ܲÓ˚y •ï˛– ~ˆÏï˛ ¢∑!°!˛õÓ˚ öy°#ˆÏÓ˚áyÓ˚ í˛z˛õÓ˚ ~ܲ!ê˛ ôyï˛Ó §%ã˛#ü%á x“ ã˛yˆÏ˛õ Ó˚yáy •ï˛–
ˆÓ˚ܲí≈˛!ê˛ â%Ó˚ˆÏï˛ ÌyܲˆÏ° öy°#ˆÏÓ˚áyÓ˚ Óe´˛õÌ xö%ÎyÎ˚# §)ã˛#ü%á!ê˛ Ü˛!¡õï˛ •ï˛– §)ã˛#ü)á!ê˛ ~ܲ!ê˛ Ü˛#°!ܲï˛
!ã˛e ≠ 13.3
!°û˛yˆÏÓ˚Ó˚ üyôƒˆÏü §yí˛zu˛ ÓˆÏ:Ó˚ ˛õò≈yÓ˚ §ˆÏD Î%_´ ÌyܲyÎ˚ §)ã˛#ü%ˆÏáÓ˚ ܲ¡õö !ܲå%Èê˛y !ÓÓ!ô≈ï˛ •ˆÏÎ˚ ˛õò≈yÎ˚ §MÈ˛y!Ó˚ï˛
•ï˛ ~ÓÇ ˛õò≈y!ê˛ §Ó˚y§!Ó˚ x!û˛!°!áï˛ ¢ˆÏ∑Ó˚ ˛õ%öç≈öö âê˛yï˛– @ˇÃyüyˆÏú˛yö ˆÓ˚ܲí≈˛ ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏöÓ˚
˜Óò%ƒ!ï˛Ü˛ ÓƒÓfliyÓ˚ ≤Ãã˛°ˆÏöÓ˚ ˛õÓ˚ §yí˛zu˛ ÓˆÏ:Ó˚ ÓƒÓ•yÓ˚ ≤ÃyÎ˚ Ó¶˛ •ˆÏÎ˚ !àˆÏÎ˚ˆÏåÈ–
˜Óò%ƒ!ï˛Ü˛ ÓƒÓfliyÓ˚ üˆÏôƒ !ï˛ö!ê˛ ôy˛õ xyˆÏåÈ– ~=!° •° ó (1) ˜Óò%ƒ!ï˛Ü˛ !˛õܲÈÙÈxy˛õ (pick-up) ÓƒÓfliyÓ˚
§y•yˆÏ΃ ܲ¡õöˆÏܲ öy°#ˆÏÓ˚áyÓ˚ Óe´˛õÌ xö%§yÓ˚# ˜Óò%ƒ!ï˛Ü˛ !Óû˛Ó ≤ÈÏû˛ˆÏò Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚yñ (2) ˜Óò%ƒ!ï˛Ü˛
!Óû˛Ó ≤ÈÏû˛òˆÏܲ •zˆÏ°Ü˛ê˛Δ!öܲ Óï≈˛ö#Ó˚ §y•yˆÏ΃ !ÓÓ!ô≈ï˛ Ü˛Ó˚y ~ÓÇ (3) !ÓÓ!ô≈ï˛ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲï˛!ê˛
°yí˛zí˛!flõܲyˆÏÓ˚ ≤ÈÏÎ˚yà ܲˆÏÓ˚ ¢∑ í˛zͲõyòö ܲÓ˚y–
!ã˛e ≠ 13.4
˜Óò%ƒ!ï˛Ü˛ !˛õܲ xy˛õ !ê˛Ä öyöy ôÓ˚ˆÏöÓ˚ •ˆÏï˛ ˛õyˆÏÓ˚– ~=!° •°ÈÙÙÙÈ
(i) ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ~!ê˛ !fliÓ˚ Ü%˛[˛°# Óy ã˛° Ü%˛[˛°#ÈÙÈ~•z ò%•z ôÓ˚ˆÏöÓ˚ •ˆÏï˛ õyˆÏÓ˚– ï˛ˆÏ° ã˛° Ü%˛[˛°#
!˛õܲÈÙÈxyˆÏ˛õÓ˚ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ xï˛ƒhsˇ ò%Ó≈° •ÄÎ˚yÎ˚ ~!ê˛Ó˚ !ӈϢ£Ï ÓƒÓ•yÓ˚ •Î˚ öy– !fliÓ˚ Ü%˛[˛°# !˛õܲÈÙÈxyˆÏ˛õÓ˚
~ܲ!ê˛ !ã˛e ~áyˆÏö ˆòáyˆÏöy •° S!ã˛e 13.4V– ~•z !˛õܲÈÙÈxyˆÏ˛õÓ˚ §%ã˛#ü%á!ê˛ ~ܲ!ê˛ Ü˛#°!Ü˛ï˛ ˆ°y•yÓ˚
xyˆÏü≈ã˛yˆÏÓ˚Ó˚ §ˆÏD Î%_´– §)ã˛#ü%á!ê˛ Ü˛!¡õï˛ •ˆÏ° xyˆÏü≈ã˛yÓ˚!ê˛ ~ܲ!ê˛ fliyÎ˚# ã%˛¡∫ˆÏܲÓ˚ ò%•zˆÏçyí˛¸y N Ä S ˆüÓ˚&Ó˚
üôƒÓï˛#≈ ˆã˛Ô¡∫ܲˆÏ«˛ˆÏe ܲ!¡õï˛ •Î˚– ~Ó˚ ú˛ˆÏ° xyˆÏü≈ã˛yˆÏÓ˚Ó˚ üôƒ !òˆÏÎ˚ Ü%˛[˛°#Ó˚ §ˆÏD ç!í˛¸ï˛ ˆÎ Ó°ˆÏÓ˚áy=!°
ÎyÎ˚ñ ˆ§=!° !òܲ ˛õ!Ó˚Óï≈˛ö ܲˆÏÓ˚ ~ÓÇ Ü%˛[˛°#ˆÏï˛ ˜Óò%ƒ!ï˛Ü˛ xyˆÏÓ¢çyï˛ !Óû˛Ó ≤ÈÏû˛ò í˛zͲõߨ •Î˚–
332
(ii) ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ (piezo-electric) !˛õܲÈÙÈxy˛õ ≠ !˛õܲÈÙÈxy˛õ ≠ !˛õܲÈÙÈxy˛õ ≠ !˛õܲÈÙÈxy˛õ ≠ !˛õܲÈÙÈxy˛õ ≠ ~ˆÏï˛ §yôyÓ˚îï˛ ˆÓ˚yˆÏ¢° §Œê˛ (potassium
sodiumtartrate)-~Ó˚ ã˛y˛õ!Óò%ƒÍ «˛üï˛y§¡õߨ ˆÜ˛°y§ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– !˛õܲÈÙÈxyˆÏ˛õÓ˚ §)ã˛#ü%á!ê˛ ~ܲ!ê˛
ܲ#°!Ü˛ï˛ Ü˛°ˆÏüÓ˚ ü%ˆÏá °yàyˆÏöy ÌyˆÏܲ– §)ã˛#ü%ˆÏáÓ˚ ܲ¡õö âê˛ˆÏ° ܲ°ü!ê˛ ˆÜ˛°yˆÏ§Ó˚ í˛z˛õÓ˚ ˛õ#í˛¸ö §,!‹T ܲˆÏÓ˚
~ÓÇ ˆÜ˛°yˆÏ§Ó˚ ò%•z ≤ÃyˆÏhsˇÓ˚ üˆÏôƒ !Óû˛Ó ≤ÈÏû˛ˆÏòÓ˚ §,!‹T •Î˚–
(iii) ˆÓ˚yô !˛õܲÈÙÈxy˛õ ≠ ˆÓ˚yô !˛õܲÈÙÈxy˛õ ≠ ˆÓ˚yô !˛õܲÈÙÈxy˛õ ≠ ˆÓ˚yô !˛õܲÈÙÈxy˛õ ≠ ˆÓ˚yô !˛õܲÈÙÈxy˛õ ≠ ~!ê˛ˆÏï˛ Ü˛¡õö¢#° §)ã˛#ü%á!ê˛ â)î#≈Ü,˛ï˛ ܲyÓ≈ˆÏöÓ˚ í˛z˛õÓ˚ ã˛y˛õ §,!‹T ܲˆÏÓ˚ ~ÓÇ
ܲyÓ≈ˆÏöÓ˚ ˆÓ˚yô Äë˛yöyüy ܲÓ˚ˆÏï˛ ÌyˆÏܲ– ú˛ˆÏ° §)ã˛#ü%ˆÏáÓ˚ ܲ¡õˆÏöÓ˚ §ˆÏD ܲyÓ≈ˆÏöÓ˚ üôƒ !òˆÏÎ˚ ï˛!í˛¸Í≤ÃÓy•Ä ܲüˆÏï˛
Óyí˛¸ˆÏï˛ ÌyˆÏܲ–
(iv) ôyÓ˚ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ôyÓ˚ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ôyÓ˚ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ôyÓ˚ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ ôyÓ˚ܲ#Î˚ !˛õܲÈÙÈxy˛õ ≠ §%ã˛#ü%ˆÏáÓ˚ à!ï˛ ~ˆÏ«˛ˆÏe ~ܲ!ê˛ ôyÓ˚ˆÏܲÓ˚ ôyÓ˚ܲˆÏcÓ˚ ˛õ!Ó˚Óï≈˛ö §,!‹T ܲˆÏÓ˚–
í˛z˛õˆÏÓ˚ Ó!î≈ï˛ ˛õk˛!ï˛=!°Ó˚ üˆÏôƒ ≤ÃÌü ò%!ê˛•z xÌ≈yÍ ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ ~ÓÇ ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ !˛õܲÈÙÈxy˛õ•z x!ôܲ
≤Ãã˛!°ï˛–
!˛õܲÈÙÈxy˛õ myÓ˚y í˛zͲõߨ ˜Óòƒ%!ï˛Ü˛ §ÇˆÏÜ˛ï˛ •zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ˆÏܲÓ˚ §y•yˆÏ΃ !ÓÓ!ô≈ï˛ Ü˛Ó˚y •Î˚– !ÓÓ!ô≈ï˛
§ÇˆÏÜ˛ï˛ °yí˛zí˛!flõܲyˆÏÓ˚ ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° x!û˛!°!áï˛ ¢ˆÏ∑Ó˚ ˛õ%î≈çöö âˆÏê˛– ~•z ~ܲˆÏܲÓ˚ 13.6 xLjϢ xy˛õ!ö
°yí˛zí˛!flõܲyÓ˚ §¡∫ˆÏ¶˛ !ܲå%È ï˛Ìƒ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö– ˛õˆÏÓ˚Ó˚ xÇ¢!ê˛ ˛õí˛¸yÓ˚ xyˆÏà ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 1 :
(i) ˜Óò%ƒ!ï˛Ü˛ !˛õܲÈÙÈxy˛õ ˆÜ˛yö‰ ˆÜ˛yö‰ ôÓ˚ˆÏöÓ˚ •Î˚⁄
(ii) 33rpm ˆÓ˚ܲˆÏí≈˛ ˆÜ˛w ˆÌˆÏܲ 12cm ò)ˆÏÓ˚ 512Hz ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ §%Ó˚ x!û˛ˆÏ°!áï˛ •ˆÏ° ¢ˆÏ∑Ó˚
~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yÎ˚ ¢∑!°!˛õˆÏï˛ ~ܲê˛y ˜òâ≈ƒ ç%ˆÏí˛¸ ÌyܲˆÏÓ⁄
13.2 ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÙÙÙÈˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÙÙÙÈˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÙÙÙÈˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÙÙÙÈˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ÙÙÙÈˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚
@ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛Ó˚ ï%˛°öyÎ˚ ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ¢∑ §ÇÓ˚«˛î xˆÏöܲ §•ç Ä §%°û˛– ˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ ΈÏsfÓ˚
§y•yˆÏ΃ xyüÓ˚y !öˆÏçÓ˚y•z ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ¢∑ x!û˛!°!áï˛ Ü˛Ó˚ˆÏï˛ ˛õy!Ó˚ñ ~ܲ!ê˛ !ú˛ï˛yÎ˚ x!û˛!°!áï˛ ¢∑
xöƒ ~ܲ!ê˛ !ú˛ï˛yÎ˚ !°!˛õÓk˛ ܲÓ˚ˆÏï˛ ˛õy!Ó˚ ~ÓÇ •zFåÈyüï˛ ˙ ¢ˆÏ∑Ó˚ ˛õ%öç≈ööÄ âê˛yˆÏï˛ ˛õy!Ó˚– ~•z ˛õk˛!ï˛Ó˚
!ܲå%Èê˛y !Ó¢ò xyˆÏ°yã˛öy ܲÓ˚y Îyܲ–
ˆã˛Ô¡∫ܲ !ú˛ï˛y ≠ ˆã˛Ô¡∫ܲ !ú˛ï˛y ≠ ˆã˛Ô¡∫ܲ !ú˛ï˛y ≠ ˆã˛Ô¡∫ܲ !ú˛ï˛y ≠ ˆã˛Ô¡∫ܲ !ú˛ï˛y ≠ ˆê˛˛õ ˆÓ˚ܲí≈˛yˆÏÓ˚ ˆÎ ˆã˛Ô¡∫ܲ !ú˛ï˛y ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ ï˛y üy•z°yÓ˚ (Mylar) Óy xöƒ
˛õ!°ˆÏÎ˚fiê˛yÓ˚ ≤’y§!ê˛ˆÏܲÓ˚ öüö#Î˚ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ ˆã˛Ô¡∫ܲ˛˛õòyˆÏÌ≈Ó˚ ≤Èϰ˛õ °y!àˆÏÎ˚ ˜ï˛!Ó˚ ܲÓ˚y •Î˚– !ú˛ï˛yÓ˚ í˛z˛õÓ˚
~üö ~ܲ!ê˛ ˆã˛Ô¡∫ܲ ˛õòyˆÏÌ≈Ó˚ ≤Èϰ˛õ !òˆÏï˛ •Î˚ ÎyÓ˚ ˆã˛Ô¡∫ܲ ôyÓ˚î «˛üï˛y ΈÏÌ‹T ˆÓ!¢ xÌ≈yÍ ˆã˛Ô¡∫ܲ ˆ«˛e
x˛õ§,ï˛ •ˆÏ°Ä ÎyÓ˚ ˆã˛Ô¡∫ܲc xˆÏöܲê˛y•z ˆÌˆÏܲ ÎyÎ˚ xyÓyÓ˚ !ö@ˇÃy!•ï˛yÄ (coercivity) ΈÏÌ‹T ÎyˆÏï˛ !Ó˚ˆÏ°
çí˛¸yˆÏöy xÓfliyÎ˚ !ú˛ï˛yÓ˚ ~ܲ!ê˛ ˛õyˆÏܲÓ˚ ã%˛¡∫ܲˆÏöÓ˚ ≤Ãû˛yˆÏÓ xöƒ ˛õyˆÏܲÓ˚ !Óã%˛¡∫ܲö (demagnetisation) öy
âˆÏê˛– !ú˛ï˛yÓ˚ ˆÓô ܲƒyˆÏ§ˆÏê˛Ó˚ ˆ«˛ˆÏe 12 xÌÓy 8 üy•ze´ö (
mμ
)– ï˛ˆÏÓ ˆáy°y !Ó˚ˆÏ°Ó˚ !ú˛ï˛yÓ˚ ˆÓô 38 Óy
25 üy•ze´ö •ˆÏÎ˚ ÌyˆÏܲ– !ú˛ï˛yÓ˚ ≤Ãfli •Î˚ 3·78mm– ˆã˛Ô¡∫ܲ ˛õòyÌ≈ !•§yˆÏÓ §yôyÓ˚îï˛ ˆ°ÔˆÏ•Ó˚ x:y•zí˛
Fe2O
3, 0·6 ˆÌˆÏܲ 1·0 üy•ze´ö ˜òˆÏâ≈ƒÓ˚ §)ã˛#ÈÙÈxyÜ,˛!ï˛ Ü˛îyÓ˚ xyܲyˆÏÓ˚ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚– ܲƒyˆÏ§ê˛ ˆê˛˛õ
333
ˆÓ˚ܲí≈˛yˆÏÓ˚ !ú˛ï˛yÓ˚ ˆÓà ܲü •ÄÎ˚yÎ˚ í˛zFã˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ ˛õ%öç≈öˆÏö x§%!Óôy •Î˚– ~ ôÓ˚ˆÏöÓ˚ ΈÏsf ÓƒÓ•yˆÏÓ˚Ó˚
çöƒ ˆã˛Ô¡∫ܲ ˛õòyÌ≈ !•§yˆÏÓ ˆe´y!üÎ˚yü í˛y•zx:y•zí˛ (CrO2) ~ÓÇ ˆú˛!Ó˚ܲ x:y•zˆÏí˛Ó˚ ≤Èϰ˛õ !òˆÏÎ˚ !ú˛ï˛y
˜ï˛!Ó˚ ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ˆã˛Ô¡∫ܲ ˛õòyˆÏÌ≈Ó˚ ܲîy=!° !ӈϢ£Ï ôÓ˚ˆÏöÓ˚ Ó¶˛ö# xyë˛yÓ˚ §y•yˆÏ΃ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ üyáyˆÏöy
•Î˚– ~•z xyë˛y ܲîy=!°ˆÏܲ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ §üyöû˛yˆÏÓ åÈ!í˛¸ˆÏÎ˚ ˆòÎ˚ñ ÷!ܲˆÏÎ˚ ÎyÄÎ˚yÓ˚ ˛õÓ˚ !ú˛ï˛yÓ˚ öüö#Î˚ï˛y
ÓçyÎ˚ Ó˚yˆÏá ~ÓÇ !ú˛ï˛y!ê˛ ˆàyê˛yˆÏöy •ˆÏ° ˛õÓ˚Óï˛#≈ ˛õyˆÏܲÓ˚ àyˆÏÎ˚ xyê˛ˆÏܲ ÎyÎ˚ öy– ˆáy°y !Ó˚ˆÏ°Ó˚ !ú˛ï˛yÎ˚ ˆã˛Ô¡∫ܲ
≤ÈϰˆÏ˛õÓ˚ ˆÓô ≤ÃyÎ˚ 14 üy•ze´ö ~ÓÇ Ü˛ƒyˆÏ§ˆÏê˛Ó˚ !ú˛ï˛yÓ˚ ~!ê˛ ≤ÃyÎ˚ 6 üy•ze´ö •ˆÏÎ˚ ÌyˆÏܲ–
x!û˛ˆÏ°áö ≠ x!û˛ˆÏ°áö ≠ x!û˛ˆÏ°áö ≠ x!û˛ˆÏ°áö ≠ x!û˛ˆÏ°áö ≠ ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ çöƒ x˛õ!Ó˚•yÎ≈ Îsf=!° •° üy•zˆÏe´yˆÏú˛yö !ÓÓô≈ܲ Óy
xƒyü!≤’ú˛yÎ˚yÓ˚ ~ÓÇ !Ó˚ܲ!í≈˛Ç ˆ•í˛ 13.5 !ã˛ˆÏe xy˛õ!ö x!û˛ˆÏ°áˆÏöÓ˚ ˛õk˛!ï˛!ê˛ ˆòáˆÏï˛ ˛õyˆÏÓö–
ˆÓ˚ܲ!í≈˛Ç ˆ•í˛!ê˛ xy§ˆÏ° ~ܲ!ê˛ ˆ°Ô•!ö!ü≈ï˛ ˆã˛Ô¡∫ܲ ˆÜ˛yÓ˚– ~•z í˛z˛õÓ˚ ~ܲ!ê˛ ï˛yÓ˚ Ü%˛[˛°# çí˛¸yˆÏöy ÌyˆÏܲ–
ˆÎ ¢∑ x!û˛!°!áï˛ Ü˛Ó˚y •ˆÏÓñ ˆ§!ê˛ üy•ze´yˆÏú˛yˆÏöÓ˚ §y•yˆÏ΃ ˜Óò%ƒ!ï˛Ü˛ §ˆÏB˛ˆÏï˛ ˛õ!Ó˚îï˛ •Î˚ ~ÓÇ •zˆÏ°Ü˛ê˛Δ!öܲ
˛õk˛!ï˛ˆÏï˛ ˆ§!ê˛ˆÏܲ !ÓÓ!ô≈ï˛ Ü˛Ó˚y •Î˚– ~Ó˚ !ÓÓ!ô≈ï˛ §ˆÏB˛ï˛ Ü%˛[˛°#ˆÏï˛ ≤ÈÏÎ˚yà ܲÓ˚ˆÏ° ˆÜ˛yˆÏÓ˚Ó˚ ã%˛¡∫ܲö ˙ §ˆÏB˛ï˛
xö%ÎyÎ˚# ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏï˛ ÌyˆÏܲ– ˆÜ˛yˆÏÓ˚Ó˚ ~ܲfliyˆÏö ò%•z ˆüÓ˚&Ó˚ üˆÏôƒ §yüyöƒ ú˛yÑܲ ÌyˆÏܲ ~ÓÇ ˆã˛Ô¡∫ܲ !ú˛ï˛y!ê˛
˙ ú˛yÑܲ!ê˛Ó˚ !ë˛Ü˛ ˛õy¢ !òˆÏÎ˚ ã˛y!°ï˛ •Î˚– ~•z ú˛ˆÏ° ˆÜ˛yˆÏÓ˚Ó˚ ˆã˛Ô¡∫ܲ Óï≈˛ö# (magnetic circuit) !ú˛ï˛yÓ˚ ˆÎ
xÇ¢ ú˛yÑܲ!ê˛Ó˚ !ë˛Ü˛ §yüˆÏö ˛õˆÏí˛¸ñ ï˛yÓ˚ üôƒ !òˆÏÎ˚ §¡õ)î≈ •Î˚ ~ÓÇ Ü%˛[˛°#Ó˚ ï˛!í˛¸Í≤ÃÓy• xö%ÎyÎ˚# ˙ xLjϢÓ˚
ã%˛¡∫ܲö âˆÏê˛– ~•zû˛yˆÏÓ à!ï˛¢#° !ú˛ï˛yÓ˚ í˛z˛õÓ˚ ¢∑§ÇˆÏÜ˛ï˛ xö%ÎyÎ˚# ܲü ˆÓ!¢ ã%˛¡∫ܲö •ˆÏï˛ ÌyˆÏܲ ~ÓÇ ¢∑!ê˛
ã%˛¡∫ܲˆÏöÓ˚ Ó˚*ˆÏ˛õ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ x!û˛!°!áï˛ •Î˚–
!ã˛e ≠ 13.5
ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ x!û˛ˆÏ°áˆÏöÓ˚ ˆÎ ˛õk˛!ï˛!ê˛ xy˛õ!ö çyöˆÏ°öñ ˆ§!ê˛ §Ó˚° •ˆÏ°Ä ~•z ˛õk˛!ï˛ˆÏï˛ §ÇˆÏÜ˛ï˛ Ä
x˛õ¢ˆÏ∑Ó˚ xö%˛õyï˛ (signal to noise ratio) ΈÏÌ‹T x!ôܲ •Î˚ öy– ~Ó˚ ú˛ˆÏ° ˛õ%öç≈öˆÏöÓ˚ §üÎ˚ í˛zͲõߨ ¢ˆÏ∑Ó˚
üˆÏôƒ x˛õ¢∑ ˆÌˆÏܲ ÎyÎ˚– xyô%!öܲ Î%ˆÏà x!û˛ˆÏ°áˆÏöÓ˚ çöƒ ~.!§. ÓyÎ˚y§ ÓƒÓfliy ≤Ãã˛!°ï˛ •ˆÏÎ˚ˆÏåÈ– ~•z
ÓƒÓfliyÎ˚ !ú˛ï˛y!ê˛ ˆÓ˚ܲ!í≈˛Ç ˆ•í˛ÈÙÈ~ xy§yÓ˚ xyˆÏà •zˆÏÓ˚!çÇ ˆ•í˛ (erasing head)-~Ó˚ ˛õy¢ !òˆÏÎ˚ xyˆÏ§– ~!ê˛
ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚ üï˛•z ï˛ˆÏÓ xˆÏ˛õ«˛yÜ,˛ï˛ Óí˛¸ ~ܲ!ê˛ ú˛yÑܲÎ%_´ ˆã˛Ô¡∫ܲ ˆÜ˛yÓ˚– ~!ê˛Ó˚ í˛z˛õÓ˚ çí˛¸yˆÏöy Ü%˛[˛°#Ó˚
üˆÏôƒ ¢!_´¢y°# ˛õ!Ó˚Óï˛#≈ ≤ÃÓy• ˛õyë˛yˆÏöy •Î˚– !ú˛ï˛yÓ˚ ˆÎ ˆÜ˛yöÄ xÇ¢ •zˆÏÓ˚!çÇ ˆ•í˛ÈÙÈ~Ó˚ ú˛yÑܲ!ê˛Ó˚ ˛õy¢ !òˆÏÎ˚
ÎyÄÎ˚yÓ˚ §üÎ˚ o&ï˛ !òܲ ˛õ!Ó˚Óï≈˛öܲyÓ˚# ˆã˛Ô¡∫ܲˆÏ«˛ˆÏe ≤ÈÏÓ¢ ܲˆÏÓ˚ ~ÓÇ e´ü¢ ÓyÓ˚ •ˆÏÎ˚ xyˆÏ§– ~Ó˚ ú˛ˆÏ°
e´ü•…y§üyö ¢!_´Ó˚ ˆã˛Ô¡∫ܲˆÏ«˛ˆÏeÓ˚ ≤Ãû˛yˆÏÓ !ú˛ï˛yÓ˚ ˙ xLjϢÓ˚ §¡õ)î≈ !Óã%˛¡∫ܲö âˆÏê˛ xÌ≈yÍ ˆã˛Ô¡∫ܲ ܲîy=!°Ó˚
ã%˛¡∫ܲˆÏcÓ˚ !òܲ ã˛ï%˛!ò≈ˆÏܲ §üû˛yˆÏÓ Ó!rê˛ï˛ •Î˚–
!ÓÓô≈ܲ !ÓÓô≈ܲ
ˆã˛Ô¡∫ܲ ˆÜ˛yÓ˚– °yí zí˛flõ#ܲyÓ˚
ˆã˛Ô¡∫ܲ !ú˛ï˛y
üy•z Ïe´y Ïú˛yö
334
§¡õ)î≈ !Óã%˛¡∫!Ü˛ï˛ •ÄÎ˚yÓ˚ ˛õÓ˚ !ú˛ï˛y!ê˛ ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚ §B˛#î≈ ˛ú˛yшÏܲÓ˚ ˛õy¢ !òˆÏÎ˚ ÎyÎ˚– ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚
Ü%˛[˛°#ˆÏï˛ (a) üy•zˆÏe´yˆÏú˛yö Ä !ÓÓô≈ܲ ˆÌˆÏܲ xy§y §ÇˆÏÜ˛ï˛ ï˛!í˛¸Í≤ÃÓy• ~ÓÇ (b) ¢ˆÏ∑y_Ó˚ ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛
˛õ!Ó˚Óï˛#≈ ÓyÎ˚y§ ≤ÃÓy• ~ܲˆÏe ≤ÃÓy!•ï˛ •Î˚– !mï˛#Î˚ xÌ≈yÍ (a) ≤ÃÓyˆÏ•Ó˚ !ÓhflÏyÓ˚ §ÇˆÏÜ˛ï˛ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ï%˛°öyÎ˚
xˆÏöܲ ˆÓ!¢ •Î˚– ÓyÎ˚y§ ≤ÃÓyˆÏ•Ó˚ ܲ¡õyB˛Ä §ˆÏÓ≈yFã˛ ˆÎ ܲ¡õyˆÏB˛Ó˚ ¢∑ x!û˛!°!áï˛ •ˆÏÓ ï˛yÓ˚ ˆÓ¢ ܲˆÏÎ˚ܲ
=î •Î˚– ò%•z ≤ÃÓyˆÏ•Ó˚ !ü!°ï˛ ≤Ãû˛yˆÏÓ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ ˆÎ ã%˛¡∫ܲö âˆÏê˛ñ ï˛yÓ˚ üyey !ú˛ï˛yÓ˚ ˆã˛Ô¡∫ܲ ˛õòyˆÏÌ≈Ó˚
ã%˛¡∫ܲöÈÙÈï˛#Ó ï˛y (I) - ˆã˛Ô¡∫ܲÈÙÈ≤ÃÓy°ƒ (H) ˆ°ˆÏáÓ˚ ˜Ó˚!áܲ xLjϢÓ˚ üˆÏôƒ•z ÌyˆÏܲ– ~Ó˚ ú˛ˆÏ° !ú˛ï˛yÓ˚ ã%˛¡∫ܲö
ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚ Ü%˛[˛°#Ó˚ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ÎÌyÌ≈ xö%!°!˛õ •Î˚– 13.6 !ã˛e ˆÌˆÏܲ ~ !Ó£ÏÎ˚!ê˛ xy˛õöyÓ˚ ܲyˆÏåÈ
flõ‹T • ÏÓ–
!ã˛e ≠ 13.6
!ú˛ï˛yÓ˚ ã%˛¡∫ܲˆÏö ˆÎ ¢ˆÏ∑y_Ó˚ ܲ¡õyˆÏB˛Ó˚ Äë˛yöyüy ÌyˆÏܲ ï˛y fliyÎ˚# •Î˚ öy ~ÓÇ ï˛yÓ˚ xÓ!¢‹TyÇ¢ ˆÎ ¢∑
í˛zͲõߨ ܲˆÏÓ˚ ï˛y x!ï˛ í˛zFã˛ Ü˛¡õyˆÏB˛Ó˚ çöƒ ˆ¢yöy ÎyÎ˚ öy– !ܲv !ú˛ï˛yÓ˚ ˆã˛Ô¡∫ܲ ˛õòyˆÏÌ≈Ó˚ I—H ˆ°ˆÏáÓ˚
§¡õ)î≈ ˜Ó˚!áܲ xÇ¢ ÓƒÓ•yÓ˚ ܲÓ˚yÓ˚ ú˛ˆÏ° x!û˛!°!áï˛ ¢∑§ÇˆÏܲï˛!ê˛ ¢!_´¢y°# Ä xˆÏ˛õ«˛yÜ,˛ï˛ x!ÓÜ,˛ï˛ •Î˚–
x!û˛ˆÏ°áˆÏöÓ˚ í˛zÍܲ£Ï≈ Ó,!k˛ ܲÓ˚yÓ˚ çöƒ ܲˆÏÎ˚ܲ!ê˛ !ӣψÏÎ˚ §ï˛Ü≈˛ï˛y xÓ°¡∫ö ܲÓ˚ˆÏï˛ •Î˚– ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚
ú˛yÑܲ!ê˛ §¡õ)î≈ ˆ§yçy Ä §üyö •ˆÏï˛ ˛•Î˚– ˆáy°y !Ó˚ˆÏ°Ó˚ ΈÏsf ~•z ú˛yÑܲ!ê˛ ≤ÈÏfli 2.5 ˆÌ Ïܲ 13 üy•ze´ö •Î˚ñ
ï˛ˆÏÓ §yôyÓ˚î ˆê˛˛õ ˆÓ˚ܲí≈˛yˆÏÓ˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈ööÈÙÙÙÈí˛zû˛Î˚ ܲyˆÏçÓ˚ çöƒ ˆÎ ˆ•í˛ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ ï˛yÓ˚
ú˛yшÏܲÓ˚ ≤Ãfli üye 1.3 üy•ze´ö– ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ x!û˛!°!áï˛ ¢∑!°!˛õÓ˚ ≤Ãfli §yôyÓ˚î !fiê˛!Ó˚Ä Ü˛ƒyˆÏ§ˆÏê˛Ó˚
ˆ«˛ Ïe 0.53mm •Î˚ñ ï˛ˆÏÓ ˆ˛õ¢yòyÓ˚ x!û˛ˆÏ°áˆÏöÓ˚ ܲyˆÏç ~!ê˛ 6·4mm •ˆÏÎ˚ ÌyˆÏܲ– ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛Ó˚ ú˛yÑܲ!ê˛
~ÓÇ ˆ•ˆÏí˛Ó˚ ˆÎ xÇÏ¢ ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ §ÇflõˆÏ¢≈ xyˆÏ§ñˆ§!ê˛ xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛ˆÏï˛ ˛õy!°¢ ܲˆÏÓ˚ ü§,î ܲÓ˚y
•Î˚–
x!û˛!°!áï˛ ¢∑!°!˛õ ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ ≤ÈÏfliÓ˚ xˆÏô≈ܲ Óy ~ܲ ã˛ï%˛Ìy≈Ç¢ ç%ˆÏí˛¸ ÌyˆÏܲ– ~ˆÏï˛ ~ܲ•z !ú˛ï˛yÎ˚
˛õy¢y˛õy!¢ ò%•z!ê˛ Óy ã˛yÓ˚!ê˛ ¢∑!°!˛õÓ˚ §ÇÓ˚«˛î ܲÓ˚y ÎyÎ˚–
ã%˛¡
∫ܲö
ï˛#Ó
ï˛y
ˆã˛Ô¡∫ܲ ≤ÃyÓ°ƒ
x!û˛!°!áï˛
¢∑ §Ç Ïܲï˛
°∏˛ ≤ÃÓy• ¢∑ §ÇˆÏÜ˛ï˛ ≤ÃÓy• ÓyÎ˚y§ ≤ÃÓy•
335
˛õ%öç≈öö ≠ ˛õ%öç≈öö ≠ ˛õ%öç≈öö ≠ ˛õ%öç≈öö ≠ ˛õ%öç≈öö ≠ ¢ˆÏ∑Ó˚ ˛õ%öç≈öˆÏöÓ˚ çöƒ ¢∑!°!˛õÎ%_´ ˆã˛Ô¡∫ܲ !ú˛ï˛y!ê˛ x!û˛ˆÏ°áˆÏöÓ˚ çöƒ ÓƒÓ•*ï˛ ˆã˛Ô¡∫ܲ
ˆ•í˛ xÌÓy xyÓ˚Ä §B˛#î≈ ú˛yÑܲÎ%_´ ~ܲ!ê˛ !û˛ß¨ Úˆ≤’ÈÙÈÓƒyÜ˛Û ˆ•ˆÏí˛Ó˚ ˛õy¢ !òˆÏÎ˚ x!û˛ˆÏ°áˆÏöÓ˚ §üyö à!ï˛ˆÏï˛
˛õyë˛yˆÏöy •Î˚– ˆÜ˛yöÄ ˆÜ˛yöÄ x!û˛ˆÏ°áö ΈÏsf ÚˆÓ˚ܲ!í≈˛Ç ˆ•í˛Û Ä Úˆ≤’ÈÙÈÓƒyܲ ˆ•í˛Û xy°yòy •ˆÏ°Ä §ã˛Ó˚yã˛Ó˚
ÓƒÓ•*ï˛ ˆê˛˛õ ˆÓ˚ܲí≈˛yˆÏÓ˚ ÚˆÓ˚ܲ!í≈˛Ç ˆ•í˛Û!ê˛•z ˆ≤’ÈÙÈÓƒyܲ ˆ•í˛ !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚–
!ú˛ï˛yÓ˚ ã%˛¡∫ܲˆÏöÓ˚ ú˛ˆÏ° ˛õ%öç≈öˆÏö ÓƒÓ•*ï˛ ~•z ˆ•í˛!ê˛Ó˚ ˆÜ˛yˆÏÓ˚ ˆã˛Ô¡∫ܲ úœ˛yˆÏ:Ó˚ Äë˛yöyüy âˆÏê˛ ~ÓÇ ˆÜ˛yˆÏÓ˚Ó˚
í˛z˛õÓ˚ çí˛¸yˆÏöy Ü%˛[˛°#ˆÏï˛ ~ܲ!ê˛ ˛õ!Ó˚Óï˛#≈ !Óû˛Ó ˜ï˛!Ó˚ •Î˚– ~•z ˛õ!Ó˚Óï˛#≈ !Óû˛Ó úœ˛yˆÏ:Ó˚ ˛õ!Ó˚Óï≈˛ˆÏöÓ˚ •yˆÏÓ˚Ó˚
§üyö%˛õyï˛#– §%ï˛Ó˚yÇñ ~!ê˛ ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ à!ï˛ˆÏÓàñ x!û˛!°!áï˛ §ÇˆÏܲˆÏï˛Ó˚ !ÓhflÏyÓ˚ Ä Ü˛¡õyˆÏB˛Ó˚ §üyö%˛õyï˛#
•Î˚–
xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ˆ≤’ÈÙÈÓƒyܲ ˆ•ˆÏí˛ í˛zͲõߨ !Óû˛Ó x!û˛!°!áï˛ ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚
§üyö%˛õyï˛# •ÄÎ˚yÓ˚ ú˛ˆÏ° ¢ˆÏ∑Ó˚ ˛õ%öç≈öˆÏö ≤Ãã˛[˛ x§%!Óôy ˆòáy ˆòˆÏÓ– ~Ó˚ ú˛ˆÏ° 64 ˆÌ Ïܲ 128Hz ܲ¡õyˆÏB˛Ó˚
§%ˆÏÓ˚Ó˚ ï%˛°öyÎ˚ 128 ˆÌˆÏܲ 256Hz ܲ¡õyˆÏB˛Ó˚ §%ˆÏÓ˚Ó˚ ï˛#Ó ï˛y !m=î ˆ¢yöyˆÏÓñ 256 ˆÌˆÏܲ 512Hz ܲ¡õyˆÏB˛Ó˚
§%ˆÏÓ˚Ó˚ ï˛#Ó ï˛y ã˛ï%˛=≈î ˆ¢yöyˆÏÓ ~ÓÇ ~•zû˛yˆÏÓ Ü˛¡õyˆÏB˛Ó˚ §ˆÏD ï˛#Ó ï˛y Ó,!k˛Ó˚ ú˛ˆÏ° ˛õ%öç≈!öï˛ ¢∑ñ ÎyÓ˚ üˆÏôƒ
öyöy ܲ¡õyˆÏB˛Ó˚ ¢∑ !ü!◊ï˛ ÌyܲˆÏï˛ ˛õyˆÏÓ˚ñ ï˛y ˛≤Ãã˛[˛ Ó˚ܲü !ÓÜ,˛ï˛ üˆÏö •ˆÏÓ– ~•z §ü§ƒyÓ˚ §üyôyˆÏöÓ˚ çöƒ
§ühflÏ õ%öç≈öܲ ΈÏsf (Tape reproducer) xyhsˇçy≈!ï˛Ü˛ üyöܲ xö%ÎyÎ˚# §üyö#ܲÓ˚ˆÏîÓ˚ (equalisation) ÓƒÓfliy
ܲÓ˚y •Î˚– xÌ≈yÍñ !ÓÓô≈ˆÏöÓ˚ §üÎ˚ x“ ܲ¡õyˆÏB˛ ˆÓ!¢ Ä x!ôܲ ܲ¡õyˆÏB˛ ܲü !ÓÓô≈ö â!ê˛ˆÏÎ˚ ¢ˆÏ∑Ó˚ ˛õ%öç≈öö
ܲÓ˚y •Î˚–
˛õ%öç≈öˆÏö ÓƒÓ•*ï˛ ˆ≤’ÈÙÈÓƒyܲ ˆ•ˆÏí˛Ó˚ ú˛yÑܲ!ê˛ ÎˆÏÌ‹T ˆåÈyê˛ •ÄÎ˚y ≤ÈÏÎ˚yçö– ~Ó˚ ܲyÓ˚î xy˛õ!ö §•ˆÏç•z
Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– Î!ò ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ ˆÓà V ~ÓÇ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ vñ ˛õÎ≈yÎ˚ܲy° T •Î˚ñ ï˛ˆÏÓ !ú˛ï˛yÎ˚
x!û˛!°!áï˛ ã%˛¡∫ܲˆÏöÓ˚ ï˛Ó˚D˜Ïòâ≈ƒ •ˆÏÓ
λ
= VT =
Vv
~áö ˆ≤’ÈÙÈÓƒyܲ ˆ•ˆÏí˛Ó˚ ˛ú˛yÑܲ Î!ò
λ
~Ó˚ §üyö •Î˚ñ ï˛ˆÏÓ ú˛yÑܲ!ê˛Ó˚ ò%•z ≤ÃyˆÏhsˇ ã%˛¡∫ܲˆÏöÓ˚ xÓfliy xö%Ó˚*˛õ
•ˆÏÓ ~ÓÇ ˆÜ˛yˆÏÓ˚Ó˚ üˆÏôƒ ˆÜ˛yöÄ ˛õ!Ó˚Óï˛#≈ úœ˛y: öy ÎyÄÎ˚yÎ˚ Ü%˛[˛°#Ó˚ í˛zͲõߨ !Óû˛Ó ¢)öƒ •ˆÏÓ– x˛õÓ˚˛õˆÏ«˛ñ
ú˛yÑܲ!ê˛
λ
/ 2 ~Ó˚ §üyö •ˆÏ° ˛õ!Ó˚Óï˛#≈ úœ˛y: Ä í˛zͲõߨ !Óû˛Ó §Ó≈y!ôܲ •ˆÏÓ– §%ï˛Ó˚yÇñ §ˆÏÓ≈yFã˛ ˆÎ ܲ¡õyˆÏB˛Ó˚
§%Ó˚ ˛õ%öç≈öö ܲÓ˚y òÓ˚ܲyÓ˚ñ ï˛yÓ˚ ã%˛¡∫ܲöÈÙÈï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ xˆÏô≈ˆÏܲÓ˚ ˆã˛ˆÏÎ˚ ˆ≤’ÈÙÈÓƒyܲ ˆ•ˆÏí˛Ó˚ ú˛yÑܲ Óí˛¸ •ÄÎ˚y í˛z!ã˛ï˛
öÎ˚– ôÓ˚y Îyܲñ ˆã˛Ô¡∫ܲÈÙÈ!ú˛ï˛yÓ˚ ˆÓà v = ˆ§ˆÏܲˆÏu˛ 1
78
•z!MÈ˛ xÌ≈yÍ 4·8cms–1– 20000Hz ܲ¡õyˆÏB˛Ó˚ §%ˆÏÓ˚Ó˚
ˆ«˛ˆÏe
λ
=
4·820000
cm Óy 2·4 üy•ze´ö– §%ï˛Ó˚yñ 20000Hz ܲ¡õyB˛ ˛õÎ≈hsˇ §%ˆÏÓ˚Ó˚ ˛õ%öç≈öö âê˛yˆÏï˛ ˆ≤’ÈÙÈÓƒyܲ
ˆ•ˆÏí˛Ó˚ ú˛yÑܲ ~Ó˚ xˆÏô≈ܲ xÌ≈yÍ 1·2 üy•ze´ˆÏöÓ˚ ܲyåÈyܲy!åÈ •ÄÎ˚y í˛z!ã˛ï˛–
xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ !ú˛ï˛yÓ˚ ~ܲy!ôܲ ¢∑!°!˛õ ˛õy¢y˛õy!¢ ÌyˆÏܲ– !ú˛ï˛y!ê˛ ˆ§yçy !òˆÏܲ ã˛y!°ˆÏÎ˚ ~Ó˚
xˆÏô≈ܲ ˆÌˆÏܲ ~ÓÇ í˛zˆÏŒê˛y !òˆÏܲ ã˛y!°ˆÏÎ˚ Óy!ܲ xˆÏô≈ܲ ˆÌˆÏܲ ¢ˆÏ∑Ó˚ ˛õ%öç≈öö ܲÓ˚y •Î˚–
336
!fiê˛!Ó˚Ä x!û˛ˆÏ°áö ˛õk˛!ï˛x!û˛ˆÏ°áö ˛õk˛!ï˛x!û˛ˆÏ°áö ˛õk˛!ï˛x!û˛ˆÏ°áö ˛õk˛!ï˛x!û˛ˆÏ°áö ˛õk˛!ï˛ ≠ xy˛õ!ö •Î˚ï˛ çyˆÏöö ˆÎñ !fiê˛!Ó˚Ä ˛õk˛!ï˛ˆÏï˛ ~ܲ•z §ˆÏD ò%•z!ê˛ üy•zˆÏe´yˆÏú˛yö
myÓ˚y §Çà,•#ï˛ ¢∑ x!û˛!°!áï˛ •Î˚ ~ÓÇ ˛õ%öç≈öˆÏöÓ˚ §üÎ˚ ~•z ò%•z ¢∑!°!˛õ ˆÌˆÏܲ í˛zͲõߨ ¢∑ ò%•z!ê˛ ˛õ,Ìܲ
°yí˛zí˛!flõܲyˆÏÓ˚Ó˚ §y•yˆÏ΃ ôù!öï˛ •Î˚ñ ÎyˆÏï˛ ò%•z!ê˛ ¢∑ ˛õ,Ìܲ í˛zͧ ˆÌˆÏܲ xy§ˆÏåÈ ÓˆÏ° üˆÏö •Î˚– !fiê˛!Ó˚Ä
x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ çöƒ !ú˛ï˛yÎ˚ ˛õy¢y˛õy!¢ ã˛yÓ˚!ê˛ ¢∑!°!˛õ ÌyˆÏܲÈÙÙÙÈÎyÓ˚ ò%•z!ê˛ ˆ§yçyü%ˆÏá Ä xöƒ
ò%•z!ê˛ !Ó˛õÓ˚#ï˛ü%ˆÏá x!û˛!°!áï˛ •Î˚– !ã˛e 13.7 ˆÌˆÏܲ xy˛õ!ö ˆáy°y !Ó˚ˆÏ°Ó˚ ˆê˛˛õ ˆÓ˚ܲí≈˛yÓ˚ Ä Ü˛ƒyˆÏ§ê˛ ˆê˛˛õ
ˆÓ˚ܲí≈˛yˆÏÓ˚Ó˚ §yôyÓ˚î (mono) Ä !fiê˛!Ó˚Ä x!û˛ˆÏ°áö ÓƒÓfliy Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!ã˛e ≠ 13.7
ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ §%!Óôy ≠ ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ §%!Óôy ≠ ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ §%!Óôy ≠ ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ §%!Óôy ≠ ˆã˛Ô¡∫ܲ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ §%!Óôy ≠ ˛õ%Ó˚yˆÏöy !òˆÏöÓ˚ @ˇÃyüyˆÏú˛yˆÏöÓ˚ ï%˛°öyÎ˚ ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ xÌ≈yÍ ˆê˛˛õ
ˆÓ˚ܲí≈˛yˆÏÓ˚Ó˚ xˆÏöܲ=!° §%!Óôy xyˆÏåÈ– ~=!° •°≠
§•ˆÏç Ó•öˆÏÎyàƒ ÎˆÏsfÓ˚ §y•yˆÏ΃ ˆÎ ˆÜ˛í˛z ˆÎ ˆÜ˛yöÄ fliyˆÏö ¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Óy ˛õ%öç≈öö ܲÓ˚ˆÏï˛
˛õyˆÏÓ˚ö–
~ܲê˛yöy 30 Óy 45 !ü!öê˛ ÓyçyˆÏöyÓ˚ í˛z˛õˆÏÎyà# x!û˛ˆÏ°áö ܲÓ˚y §Ω˛Ó–
~ܲ•z !ú˛ï˛yÎ˚ ˛õ%Ó˚yï˛ö ¢∑!°!˛õ ü%ˆÏåÈ ˆú˛ˆÏ° öï%˛ö ܲˆÏÓ˚ x!û˛ˆÏ°áö ܲÓ˚y ÎyÎ˚– ~üö !ܲ ¢∑!°!˛õÓ˚
xÇ¢ ü%ˆÏåÈ ˆú˛ˆÏ° ˆ§!ê˛Ó˚ §¡õyòöy ܲÓ˚y ÎyÎ˚–
ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ ˆÜ˛yöÄ ≤Ãfl≥%˛ê˛ˆÏöÓ˚ (developing) ≤ÈÏÎ˚yçö •Î˚ öy– ܲyˆÏç•z x!û˛ˆÏ°áˆÏöÓ˚ !ë˛Ü˛ ˛õˆÏÓ˚•z
¢∑ ˛õ%öç≈öö ܲÓ˚y ÎyÎ˚–
ˆã˛Ô¡∫ܲ !ú˛ï˛yÓ˚ xƒyöy°à ~ÓÇ !í˛!çê˛ƒy°ÈÙÙÙÈ~•z ˛õk˛!ï˛ˆÏï˛•z ¢∑ §ÇÓ˚«˛î ܲÓ˚y ÎyÎ˚– ~•z !í˛!çê˛ƒy°
˛õk˛!ï˛ §¡∫ˆÏ¶˛ xy˛õ!ö ~Ó˚ ˛õˆÏÓ˚Ó˚ xLjϢ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
~ÓyÓ˚ xy˛õöyÓ˚ !öˆÏã˛Ó˚ xö%¢#°ö#!ê˛Ó˚ í˛z_Ó˚ ܲÓ˚ˆÏï˛ ˛õyÓ˚y í˛z!ã˛ï˛–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2 :
(i) ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ˆã˛Ô¡∫ܲ ˛õòyÌ≈ !•§yˆÏÓ ÓƒÓ•yÓ˚ •Î˚ ~üö ò%•z!ê˛ ˛õòyˆÏÌ≈Ó˚ öyü !°á%ö–
(ii) ~ !§ ÓyÎ˚y§ x!û˛ˆÏ°áö ˛õk˛!ï˛ ˛ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ܲ# §%ú˛° ˛õyÄÎ˚y ÎyÎ˚⁄
(iii) ˛õ%öç≈öܲ ΈÏsf §üyö#ܲÓ˚ˆÏîÓ˚ ≤ÈÏÎ˚yçö •Î˚ ˆÜ˛ö⁄
(iv) !fiê˛!Ó˚Ä x!û˛ˆÏ°áö ˛õk˛!ï˛ˆÏï˛ !ú˛ï˛yÓ˚ í˛z˛õÓ˚ ܲÎ˚!ê˛ ¢∑!°!˛õ ˛õy¢y˛õy!¢ ÌyˆÏܲ⁄
337
13.5 xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛
xy˛õ!ö •z!ï˛üˆÏôƒ Îy!sfܲ ˛õk˛!ï˛ˆÏï˛ xÌ≈yÍ @ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛ ~ÓÇ ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ ܲÌy˛
˛õˆÏí˛¸ˆÏåÈö– ~ ò%!ê˛ åÈyí˛¸y ï,˛ï˛#Î˚ ˆÎ ¢∑ x!û˛ˆÏ°áö ˛õk˛!ï˛Ó˚ §ˆÏD xyüÓ˚y ˛õ!Ó˚!ã˛ï˛ ˆ§!ê˛ •° xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛–
xyˆÏ°yܲÓ˚!Ÿ¬Ó˚ §y•yˆÏ΃ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ò%•z!ê˛ §¡õ)î≈ !û˛ß¨ ≤ÃÎ%!_´ xyüÓ˚y xyÎ˚_ ܲˆÏÓ˚!åÈ– ~Ó˚
üˆÏôƒ §Óyܲ ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ ¢∑ x!û˛ˆÏ°áö Ä ï˛yÓ˚ ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏöÓ˚ ≤ÃÎ%!_´ §_Ó˚ ÓåȈÏÓ˚Ó˚ ˆÓ!¢
˛õ%Ó˚ˆÏöy– §¡±!ï˛ xyˆÏ°yܲÓ˚!Ÿ¬ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ˆÎ xöƒ ≤ÃÎ%!_´ xy!Ó‹,Òï˛ •ˆÏÎ˚ˆÏåÈñ
ˆ§!ê˛ˆÏï˛ ¢∑!°!˛õ ôˆÏÓ˚ Ó˚yáyÓ˚ çöƒ xy°Ü˛#Î˚ !í˛flÒ (optical disc) Óy ܲü˛õƒyQ !í˛flÒ (compact disc) ÓƒÓ•yÓ˚
ܲÓ˚y •Î˚– ~•z xÇ¢!ê˛ˆÏï˛ xy˛õ!ö ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ~ÓÇ xyˆÏ°yܲ#Î˚ !í˛flÒÈÙÙÙÈí˛zû˛Î˚ üyôƒˆÏüÓ˚ !Ó£ÏÎ˚•z çyöˆÏï˛
˛õyÓ˚ˆÏÓö–
13.5.1 ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ≠ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ≠ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ≠ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ≠ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ ≠
!§ˆÏöüy ˆòáyÓ˚ §üÎ˚ xy˛õ!ö ˆÎ ܲÌyÓyï≈˛y Óy àyö Óyçöy ÷öˆÏï˛˛ ˛õyöñ ˆ§•z ¢∑ !§ˆÏöüyÓ˚ !ú˛ˆÏÕ√Ó˚ í˛z˛õˆÏÓ˚•z
x!û˛!°!áï˛ ÌyˆÏܲ– !ú˛ˆÏÕ√Ó˚ !ã˛ˆÏeÓ˚ x!û˛ˆÏ«˛ˆÏ˛õÓ˚ (projection) §ˆÏD §ˆÏD•z
!ú˛ˆÏÕ√Ó˚ ¢∑!°!˛õ ˆÌˆÏܲ ¢ˆÏ∑Ó˚ ˛õ%öç≈öö âê˛yˆÏöy •Î˚ñ ÎyÓ˚ ú˛ˆÏ° öy!Î˚ܲyÓ˚
ĈϤ˛Ó˚ à!ï˛ Ä ïÑ˛yÓ˚ í˛zFã˛y!Ó˚ï˛ ¢∑ñ xÌÓy ôÓ˚&öñ ôyÓüyö xˆÏŸªÓ˚ ˛õòã˛yÓ˚îy xyÓ˚
ï˛yÓ˚ á%ˆÏÓ˚Ó˚ ê˛à‰Óà‰ ôù!öÓ˚ üˆÏôƒ §üܲy°#öï˛y (synchronization) ÓçyÎ˚
ÌyˆÏܲ–
!ú˛ˆÏÕ√ ¢∑!°!˛õ ò%•z ôÓ˚ˆÏöÓ˚ •ˆÏï˛ ˛õyˆÏÓ˚ÈÙÙÙÈ˛õ!Ó˚Óï˛#≈ âöc (variable
density) Ä ˛õ!Ó˚Óï˛#≈ ≤Ãfli (variable width) – ~áyˆÏö xyüÓ˚y ~•z ò%•z
˛õk˛!ï˛ˆÏï˛ x!û˛ˆÏ°áˆÏöÓ˚ ˛õk˛!ï˛=!° §¡∫ˆÏ¶˛ §Ç!«˛Æ xyˆÏ°yã˛öy ܲÓ˚Ó–
˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ≠ 13.8(a) !ã˛e ˆÌˆÏܲ xy˛õ!ö ~•z
x!û˛ˆÏ°áˆÏöÓ˚ Ó˚*˛õ!ê˛ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~•z ˛õk˛!ï˛ˆÏï˛ !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ ¢∑!°!˛õ §Ó≈e §üyö ≤ÈÏfliÓ˚ •Î˚ !ܲv
!ú˛ˆÏÕ√ ܲy!°üyÓ˚ âöc ¢∑ï˛Ó˚ˆÏDÓ˚ §Ó˚î xö%ÎyÎ˚# ÓyˆÏí˛¸ Ä Ü˛ˆÏü– ò%•z !û˛ß¨ í˛z˛õyˆÏÎ˚ ~!ê˛ Ü˛Ó˚y ÎyÎ˚–
≤ÃÌü ˛õk˛!ï˛ˆÏï˛ üy•zˆÏe´yˆÏú˛yˆÏö í˛zͲõߨ ¢∑ §ÇˆÏܲï˛!ê˛ •zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ˆÏܲÓ˚ §y•yˆÏ΃ !ÓÓ!ô≈ï˛ Ü˛ˆÏÓ˚
í˛zͲõߨ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ myÓ˚y ~ܲ!ê˛ ˜Óò%ƒ!ï˛Ü˛ !ú˛°yˆÏürê˛ Óy ˛õyÓ˚òÈÙxyÜ≈˛ Óy!ï˛Ó˚ ÅIμ°ƒ !öÎ˚sfî ܲÓ˚y •Î˚–
Óy!ï˛!ê˛Ó˚ ÅIμ°ƒ ≤ÃyÌ!üܲ xÓfliyÎ˚ üyé˛yüy!é˛ ÌyˆÏܲ ~ÓÇ üy•zˆÏe´yˆÏú˛yˆÏö à,•#ï˛ ¢∑ï˛Ó˚D xö%ÎyÎ˚# ˆ§!ê˛ Ü˛ˆÏü
ÓyˆÏí˛¸– ~•z Óy!ï˛Ó˚ xyˆÏ°y ~ܲ!ê˛ §Ó˚& ˆÓ˚áy!åȈÏoÓ˚ (slit) üôƒ !òˆÏÎ˚ !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ ¢∑!°!˛õÓ˚ çöƒ §ÇÓ˚!«˛ï˛
fliyˆÏö ˛õˆÏí˛¸– !˘ú˛Õ√!ê˛ ˆí˛ˆÏû˛°˛õ ܲÓ˚y •ˆÏ° ˛õ!Ó˚Óï˛#≈ âöˆÏcÓ˚ ¢∑!°!˛õ!ê˛ ˛õ!Ó˚fl≥%˛ê˛ •Î˚– 13.8(b) !ã˛ˆÏe xy˛õ!ö
~•z ˛õk˛!ï˛Ó˚ ~ܲ!ê˛ Ó˚*˛õ!ã˛e ˆòáˆÏï˛ ˛õyˆÏÓö–
!ã˛e ≠ 13·8(a)
˛õ!Ó
˚Óï≈˛#
âö
c ¢
∑!°
!˛õ
338
!ã˛e ≠ 13.8(b)
!mï˛#Î˚ ˛õk˛!ï˛!ê˛ˆÏï˛ Óy!ï˛Ó˚ ÅIμ°ƒ !fliÓ˚ ÌyˆÏܲ !ܲv ˙ Óy!ï˛Ó˚ xyˆÏ°yÓ˚ §y•yˆÏ΃ ~ܲ!ê˛ xyˆÏ°yܲ myˆÏÓ˚Ó˚
(light-valve) í˛z˛õÓ˚ ~ܲ!ê˛ ˆÓ˚áy!åȈÏoÓ˚ ≤Ã!ï˛!Ó¡∫ ˆú˛°y •Î˚– xyˆÏ°yܲ û˛y°Ó‰!ê˛ xy§ˆÏ° í%˛Ó˚ƒy°%!üˆÏöÓ˚ !ú˛ï˛yÓ˚
~ܲ!ê˛ §Ó˚& Ä °¡∫y ú˛yѧ– ~•z ò%•z ˛õyˆÏê˛Ó˚ üˆÏôƒ ~ܲ!ê˛ §)- ú˛Ñyܲ Óy ˆÓ˚áy!åÈo ÌyˆÏܲ– ˛ú˛yÑܲ!ê˛ ~ܲ!ê˛ xö%û)˛!üܲ
ˆã˛Ô¡∫ܲˆÏ«˛ˆÏe í˛zÕ‘¡∫û˛yˆÏÓ ˆé˛y°yˆÏöy ÌyˆÏܲ– üy•zˆÏe´yˆÏú˛yö ˆÌˆÏܲ í˛zͲõߨ ï˛!í˛¸Í≤ÃÓy• !ú˛ï˛yÓ˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛
•ˆÏ° ˆã˛Ô¡∫ܲˆÏ«˛ˆÏe Ä ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ ˛õyÓ˚flõ!Ó˚ܲ !e´Î˚yÎ˚ í˛zͲõߨ Ó° !ú˛ï˛y!ê˛ˆÏܲ ܲ!¡õï˛ Ü˛ˆÏÓ˚ ~ÓÇ ú˛yшϧÓ˚
üˆÏôƒ ˆÓ˚áy!åÈo!ê˛Ó˚ ≤Ãfli ܲˆÏü ÓyˆÏí˛¸– ~Ó˚ ú˛ˆÏ° ˆÓ˚áy!åÈo!ê˛Ó˚ üôƒ !òˆÏÎ˚ ˆÎ xyˆÏ°y !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛
•Î˚ñ ï˛yÓ˚ ˛õ!Ó˚üyî üy•zˆÏe´yˆÏú˛yˆÏö à,•#ï˛ ¢∑ï˛Ó˚D xö%ÎyÎ˚# ܲüÈÙȈÓ!¢ •Î˚– 13·8(c) !ã˛e ˆÌˆÏܲ xy˛õ!ö ~•z
˛õk˛!ï˛Ó˚ ü)° ö#!ï˛!ê˛ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!ã˛e ≠ 13.8(c)
˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≠ ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≠ xyô%!öܲ ã˛°!Fã˛ˆÏe ˛õ)Ó≈Ó!î≈ï˛ ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áö ÓƒÓ•*ï˛ •Î˚ öy– ÓÓ˚Ç
!Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áö ≤ÃÎ%!_´•z Óï≈˛üyˆÏö ÓƒÓ•*ï˛ •ˆÏÎ˚ ÌyˆÏܲ–
˛õÓ˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áˆÏö ¢∑!°!˛õÓ˚ âöc §üyö ÌyˆÏܲ !ܲv ≤Ãfli ܲü ÓyˆÏí˛¸ [!ã˛e 13.9
(a)]– ~•z ôÓ˚ˆÏöÓ˚ x!û˛ˆÏ°áˆÏöÓ˚ ~ܲ!ê˛ §•ç ˛õk˛!ï˛ •° üy•zˆÏe´yˆÏú˛yö çyï˛ ï˛!í˛¸Í≤ÃÓy•
~ܲ!ê˛ àƒy°û˛ƒyˆÏöy!üê˛yˆÏÓ˚Ó˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ ܲÓ˚yñ ÎyˆÏï˛ àƒy°û˛ƒyˆÏöy!üê˛yˆÏÓ˚Ó˚ xyÎ˚öy
à,•#ï˛ ¢∑ï˛Ó˚D xö%ÎyÎ˚# ܲ!¡õï˛ •Î˚– xyÎ˚ï˛yܲyÓ˚ ≤ÃfliˆÏFåȈÏòÓ˚ ~ܲ!ê˛ §üyhsˇÓ˚y°
xyˆÏ°yܲÓ˚!Ÿ¬=FåÈ ˙ xyÎ˚öyÎ˚ xy˛õ!ï˛ï˛ •Î˚ ~ÓÇ ≤Ã!ï˛ú˛°ˆÏöÓ˚ ˛õÓ˚ ~ܲ!ê˛ §Ó˚&
ˆÓ˚áy!åÈoˆÏܲ xyÇ¢!ܲû˛yˆÏÓ xyˆÏ°y!Ü˛ï˛ Ü˛ˆÏÓ˚– xyÎ˚öy!ê˛ Ü˛!¡õï˛ •ˆÏ° ˆÓ˚áy!åȈÏoÓ˚
•zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ܲ
!ú˛°yˆÏürê˛ Óy!ï˛
üy•zˆÏe´yˆÏú˛yö
ˆÓ˚áy!åÈoˆ°™
ˆ°™
!ú˛°ü
¢∑!°!˛õ
ˆ°™¢∑!°!˛õ
xyˆÏ°yܲòyÓ˚
ˆÓ˚áy!åÈo
üy•zˆÏe´yˆÏú˛yö
•zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ܲ
ˆã˛Ô¡∫ܲ
ˆ°™
ˆ«˛e
Óy!ï˛
!ã˛e ≠ 13.9(a)
˛õÓ˚Óï≈˛# ≤Ãfli
¢∑!°!˛õ
339
xyˆÏ°y!Ü˛ï˛ xLjϢÓ˚ ˜òâ≈ƒ ܲüˆÏÓ!¢ •Î˚– à!ï˛¢#° !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ ˆÓ˚áy!åȈÏoÓ˚ ~ܲ!ê˛ ≤Ã!ï˛!Ó¡∫ ˛õˆÏí˛¸ ~ÓÇ à,•#ï˛
¢∑ï˛Ó˚D xö%ÎyÎ˚# ≤Ã!ï˛!ӈϡ∫Ó˚ ≤Ãfli ܲ!¡õï˛ •ÄÎ˚yÓ˚ ú˛ˆÏ° ˛õ!Ó˚Óï˛#≈ ≤ÈÏfliÓ˚ ¢∑!°!˛õ à!ë˛ï˛ •Î˚– 13.9(b) !ã˛e
ˆÌˆÏܲ xy˛õ!ö ~•z ˛õk˛!ï˛!ê˛ Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö–
!§ˆÏöüyÓ˚ !˘ú˛ˆÏÕ√Ó˚ x!û˛!°!áï˛ ¢∑!°!˛õ ˆÌˆÏܲ ü)° ¢∑!ê˛ Ü˛#û˛yˆÏÓ !ú˛ˆÏÓ˚ ˛õyÄÎ˚y ÎyÎ˚⁄ ò%•z ôÓ˚ˆÏöÓ˚
x!û˛ˆÏ°áˆÏöÓ˚ ˆ«˛ˆÏe ¢ˆÏ∑Ó˚ ˛õ%öç≈öˆÏöÓ˚ ˛õk˛!ï˛!ê˛ ~ܲ•z–
¢∑!°!˛õÓ˚ í˛z˛õÓ˚ ~ܲ!ê˛ xyˆÏ°y!Ü˛ï˛ ˆÓ˚áy!åȈÏoÓ˚ ˛≤Ã!ï˛!Ó¡∫
à!ë˛ï˛ •Î˚ ~ÓÇ !ú˛ˆÏÕ√Ó˚ üôƒ !òˆÏÎ˚ ˆ≤Ã!Ó˚ï˛ xyˆÏ°y ~ܲ!ê˛
ú˛ˆÏê˛y •zˆÏ°Ü˛!ê˛Δܲ ˆ§ˆÏ°Ó˚ í˛z˛õÓ˚ ˛õˆÏí˛¸– ~•z ú˛ˆÏê˛y •zˆÏ°Ü˛!ê˛Δܲ
ˆ§° myÓ˚y í˛zͲõߨ ï˛!í˛¸Í≤ÃÓy• ü)° x!û˛!°!áï˛ ¢ˆÏ∑Ó˚
ܲ¡õö xö%ÎyÎ˚# Äë˛yöyüy ܲˆÏÓ˚– ~•z ï˛!í˛¸Í≤ÃÓy•ˆÏܲ
!ÓÓ!ô≈ï˛ Ü˛ˆÏÓ˚ !§ˆÏöüyÓ˚ ˛õò≈yÓ˚ !˛õåȈÏö Ó˚yáy °yí˛zí˛!flõܲyˆÏÓ˚
˛õyë˛yˆÏöy •Î˚ ~ÓÇ ~•zÓ˚*˛õ ~ܲ Óy ~ܲy!ôܲ °yí˛zí˛!flõܲyÏÓ˚
˛ˆÌˆÏܲ ˛õ%öç≈!öï˛ ¢∑ í˛zͲõߨ •Î˚–
§yôyÓ˚îï˛ !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ ~ܲ!ê˛ ˛õ!Ó˚Óï˛#≈ ˛≤Ãfli ¢∑!°!˛õÓ˚
˛õ!Ó˚ÓˆÏï≈˛ ˛õy¢y˛õy!¢ ò%•z!ê˛ xö%Ó˚*˛õ ¢∑!°!˛õ x!û˛!°!áï˛
ܲÓ˚y •Î˚– ò%!ê˛Ó˚ üôƒ !òˆÏÎ˚ ˆ≤Ã!Ó˚ï˛ xyˆÏ°yܲÓ˚!Ÿ¬ ~ܲ•z ú˛ˆÏê˛y •zˆÏ°Ü˛!ê˛Δܲ ˆ§ˆÏ°Ó˚ í˛z˛õÓ˚ ˛õˆÏí˛¸– ˆòáy ˆàˆÏåÈñ
~•z ˛õk˛!ï˛ˆÏï˛ ˛õ%öç≈!öï˛ ¢ˆÏ∑Ó˚ üˆÏôƒ xÓy!N˛ï˛ x˛õ¢∑ (noise) xˆÏöܲê˛y ܲüyˆÏöy §Ω˛Ó •Î˚– ˛õˆÏÓ˚ ò%•z
¢∑!°!˛õˆÏï˛ ò%•z!ê˛ ˛õ,Ìܲ ¢∑ x!û˛!°!áï˛ Ü˛ˆÏÓ˚ ~ÓÇ ¢∑=!° ˛õ,Ìܲ ˛ú˛ˆÏê˛y •zˆÏ°Ü˛!ê˛Δܲ ˆ§°ñ !ÓÓô≈ܲ Ä
°yí˛zí˛flõ#ܲyˆÏÓ˚Ó˚ §y•yˆÏ΃ ˛õ%öç≈!öï˛ Ü˛ˆÏÓ˚ !fiê˛!Ó˚ĈÏú˛y!öܲ ¢∑ ÓƒÓfliy §,!‹T ܲÓ˚y §Ω˛Ó •ˆÏÎ˚ˆÏåÈ–
§y¡±!ï˛Ü˛Ü˛yˆÏ° !§ˆÏöüyÓ˚ !ú˛ˆÏÕ√ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ çöƒ !í˛!çê˛ƒy° ˛õk˛!ï˛ ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏFåÈ– ~•z ˛õk˛!ï˛
§¡∫ˆÏ¶˛ xy˛õ!ö ˛õˆÏÓ˚Ó˚ xLjϢ !ܲå%Èê˛y !Ó¢òû˛yˆÏÓ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
13.5.2 ܲü˛õƒyQ !í˛flÒ Óy !§!í˛Ü˛ü˛õƒyQ !í˛flÒ Óy !§!í˛Ü˛ü˛õƒyQ !í˛flÒ Óy !§!í˛Ü˛ü˛õƒyQ !í˛flÒ Óy !§!í˛Ü˛ü˛õƒyQ !í˛flÒ Óy !§!í˛ (CD)
¢∑ x!û˛ˆÏ°áˆÏöÓ˚ xöƒï˛ü xyô%!öܲ üyôƒü xyˆÏ°yܲ#Î˚ !í˛flÒ Óy ܲü˛õƒyQ !í˛flÒñ ˆÎ!ê˛ˆÏܲ xyüÓ˚y §ÇˆÏ«˛ˆÏ˛õ
!§!í˛ ÓˆÏ° Ìy!ܲ– ~•z !í˛ˆÏflÒÓ˚ í˛z˛õÓ˚ ~ܲ!ê˛ xƒy°%!ü!öÎ˚yˆÏüÓ˚ hflÏÓ˚ ÌyˆÏܲ– x!û˛!°!áï˛ ¢∑!°!˛õ xƒy°%!ü!öÎ˚yü
hflψÏÓ˚Ó˚ í˛z˛õÓ˚ !í˛ˆÏflÒÓ˚ ˛õ!Ó˚!ô ˆÌˆÏܲ ≤ÃyÎ˚ ܲˆÏwfli° ˛õÎ≈hsˇ ˛õƒyÑã˛yˆÏöy ˆÓ˚áy ÓÓ˚yÓÓ˚ «%˛o «%˛o àˆÏï≈˛Ó˚
xyܲyˆÏÓ˚ ˆáy!òï˛ ÌyˆÏܲ– xƒy°%!ü!öÎ˚yˆÏüÓ˚ hflÏÓ˚!ê˛ ÎyˆÏï˛ x«˛ï˛ ÌyˆÏܲ ˆ§çöƒ ˆ§!ê˛Ó˚ í˛z˛õˆÏÓ˚ fl∫FåÈ Ä ü§,î
≤’y!fiê˛ˆÏܲÓ˚ xyÓÓ˚î ÌyˆÏܲ– àï≈˛=!° ≤ÃyÎ˚ xyÎ˚ï˛yܲyÓ˚ÈÙÙÙÈ≤ÈÏfli 0.5 üy•ze´ö ~ ˜òâ≈ƒ 1ˆÌ Ïܲ 3 üy•ze´ö–
ˆ§=!° ˆÎ ˛õƒÑyã˛yˆÏöy ˆÓ˚áy ÓÓ˚yÓÓ˚ §yçyˆÏöy ÌyˆÏܲñ ï˛yÓ˚ ~ܲ ~ܲ ˛õƒÑyˆÏã˛Ó˚ üˆÏôƒ ú˛yÑܲ üye 1.6 üy•ze´ö–
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ~ï˛ «%˛o àï≈˛=!° ˆáy!òï˛ Ü˛Ó˚y Óy ˆ§=!°Ó˚ ˛õyˆÏë˛ymyÓ˚ ܲÓ˚yÓ˚ üï˛ §%ã˛#ü%á
xyÎ˚öy
Ü% [˛°#
ˆ°™
Óy!ï˛
xyÎ˚öy
!ã˛e : 13.9(b)
N
340
!ã˛e ≠ 13.10
˜ï˛!Ó˚ ܲÓ˚y ò%ɧyôƒ •ˆÏÓ– xy§ˆÏ° ~ ôÓ˚ˆÏöÓ˚ !í˛ˆÏflÒ ¢∑ x!û˛ˆÏ°áö xÌÓy ˛õ%öç≈öˆÏöÓ˚ çöƒ ˆ°çyÓ˚ Ó˚!Ÿ¬
ÓƒÓ•*ï˛ •Î˚–
x!û˛ˆÏ°áˆÏöÓ˚ §üÎ˚ ˆ°çyÓ˚ Ó˚!Ÿ¬!ê˛ ¢∑§ÇˆÏÜ˛ï˛ xö%ÎyÎ˚# xƒy°%!ü!öÎ˚yˆÏüÓ˚ í˛z˛õÓ˚ àï≈˛=!° ˆáy!òï˛ Ü˛ˆÏÓ˚–
˛õ%öç≈öˆÏöÓ˚ §üÎ˚ xˆÏ˛õ«˛yÜ,˛ï˛ ܲü ¢!_´Ó˚ ˆ°çyÓ˚ Ó˚!Ÿ¬ xƒy°%!ü!öÎ˚yü hflψÏÓ˚Ó˚ í˛z˛õÓ˚ ˛õˆÏí˛¸ ~ÓÇ ≤Ã!ï˛ú˛!°ï˛
Ó˚!Ÿ¬!ê˛ xyˆÏ°yܲ§ÇˆÏÓò# xô≈˛õ!Ó˚Óy•#Ó˚ §y•yˆÏ΃ à,•#ï˛ •Î˚– ≤Ã!ï˛ú˛!°ï˛ Ó˚!Ÿ¬Ó˚ ï˛#Ó ï˛yÓ˚ •…y§ Ó,!k˛ˆÏܲ
¢∑§ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y •Î˚– 13.10 !ã˛e ˆÌˆÏܲ !Ó£ÏÎ˚!ê˛ xy˛õöyÓ˚ ܲyˆÏåÈ ˛õ!Ó˚‹ÒyÓ˚ •ˆÏÓ–
@ˇÃƒyüyˆÏú˛yö ˆÓ˚ܲí≈˛ ˆÌˆÏܲ ܲü˛õƒyQ !í˛ˆÏflÒ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ ü)° ˛õyÌ≈ܲƒ •° ~•z ˆÎñ @ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛
¢∑ §ÇÓ˚!«˛ï˛ ÌyˆÏܲ xƒyöy°à ˛õk˛!ï˛ˆÏï˛ !ܲv ܲü˛õƒyQ !í˛ˆÏflÒ ¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö •Î˚ !í˛!çê˛ƒy° ˛õk˛!ï˛ˆÏï˛–
¢∑ §ÇÓ˚«˛ˆÏîÓ˚ !í˛!çê˛ƒy° ˛õk˛!ï˛ §¡∫ˆÏ¶˛ xy˛õ!ö !öÿ˛Î˚•z !ܲå%Èê˛y çyöˆÏï˛ ã˛y•zˆÏÓö–
!í˛!çê˛ƒy° §ÇÓ˚«˛î ˛õk˛!ï˛ ≠ !í˛!çê˛ƒy° §ÇÓ˚«˛î ˛õk˛!ï˛ ≠ !í˛!çê˛ƒy° §ÇÓ˚«˛î ˛õk˛!ï˛ ≠ !í˛!çê˛ƒy° §ÇÓ˚«˛î ˛õk˛!ï˛ ≠ !í˛!çê˛ƒy° §ÇÓ˚«˛î ˛õk˛!ï˛ ≠ §yôyÓ˚î @ˇÃyüyˆÏú˛yö ˆÓ˚ܲí≈˛ Óy ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ §üÎ˚ ¢∑
ï˛Ó˚ˆÏDÓ˚ ~ܲ!ê˛ ≤Ã!ï˛Ó˚*˛õ x!Ó!FåÈߨû˛yˆÏÓ !°!˛õÓk˛ •Î˚– ~•z ¢∑!°!˛õ x!û˛!°!áï˛ ü)° ¢∑ˆÏܲ Îï˛
x!Óܲ°û˛yˆÏÓ xö%§Ó˚î ܲˆÏÓ˚ñ ˛õ%öç≈!öï˛ ¢∑ ï˛ï˛ê˛y•z ü)° ¢ˆÏ∑Ó˚ xö%Ó˚*˛õ •Î˚– !í˛!çê˛ƒy° ˛õk˛!ï˛ˆÏï˛
¢∑ï˛Ó˚DˆÏܲ x!Ó!FåÈߨû˛yˆÏÓ xö%§Ó˚î ܲÓ˚y •Î˚ öy– ÓÓ˚Ç ~ܲ!ê˛ !ö!ò≈‹T §üÎ˚ xhsˇÓ˚ ¢∑ï˛Ó˚ˆÏDÓ˚ x!û˛ˆÏ°áö#Î˚
Ó˚y!¢Ó˚ öü%öy @ˇÃ•î ܲÓ˚y •Î˚ ~ÓÇ ˆ§!ê˛ˆÏܲ Óy•zöy!Ó˚ §ÇáƒyÎ˚ Ó˚*˛õyhsˇˇ!Ó˚ï˛ Ü˛Ó˚y •Î˚– xy˛õ!ö •Î˚ï˛ çyˆÏöö ˆÎñ
~ܲ!ê˛ Óy•zöy!Ó˚ xÌ≈yÍ !mÈÙÈ!û˛!_ܲ §Çáƒy ܲï˛Ü˛=!° 0 Ä 1-~Ó˚ §y•yˆÏ΃ ˆ°áy •Î˚– ¢∑ï˛Ó˚ˆÏDÓ˚ öü%öy=!°
x!û˛ˆÏ°áö üyôƒˆÏü ˛õÓ˚˛õÓ˚ Óy•zöy!Ó˚ §Çáƒy !•§yˆÏÓ !°!˛õÓk˛ •Î˚– ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ ã%˛¡∫ܲˆÏöÓ˚ ï˛yÓ˚ï˛üƒ ~ÓÇ
ܲü˛õƒyQ !í˛ˆÏflÒ xƒy°%!ü!öÎ˚yü ≤Ã!ï˛ú˛°Ü˛ Ä àï≈˛=!°Ó˚ myÓ˚y ‘0’ Ä ‘1’ §Çáƒy=!° !öˆÏò≈¢ ܲˆÏÓ˚ ˛Óy•zöy!Ó˚
§Çáƒy=!° x!û˛!°!áï˛ •Î˚– ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ fl∫FåÈ Ä xfl∫FåÈ xÇ¢=!° Óy•zöy!Ó˚ §Çáƒy !öˆÏò≈¢ ܲˆÏÓ˚– üye
ã˛yÓ˚!ê˛ !Óê˛ (bit = binary digit) myÓ˚y à!ë˛ï˛ ~ܲ!ê˛ !í˛!çê˛ƒy° ¢ˆÏ∑Ó˚ myÓ˚y ¢)öƒ (0000) ̈Ïܲ õˆÏöÓ˚ (1111)
xƒyöy°à ¢∑§ÇˆÏܲï˛
flõ#ܲyÓ˚xy Ï°yܲ§Ç ÏÓò#
xô≈˛õ!Ó˚Óy•#
!í˛!çê˛ƒy° ¢∑§ÇˆÏܲï˛
ܲü‰ õƒy܉ ê˛ !í˛§‰Ü˛
ˆ°çyÓ˚
341
˛õÎ≈hsˇ ˆüyê˛ ˆ£Ïy°!ê˛ §ÇáƒyˆÏܲ !öˆÏò≈¢ ܲÓ˚y ÎyÎ˚– ¢ˆÏ∑Ó˚ !í˛!çê˛ƒy° x!û˛ˆÏ°áˆÏö 14!ê˛ Óy 16!ê˛ !Óê˛ myÓ˚y à!ë˛ï˛
¢∑ !òˆÏÎ˚ ¢∑ï˛Ó˚ˆÏDÓ˚ ï˛yÍ«˛!îܲ üyö ˆÓyé˛yˆÏöy •Î˚– xy˛õ!ö §•ˆÏç•z !•§yÓ Ü˛ˆÏÓ˚ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö ˆÎñ 14
!ê˛ !Óê˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆüyê˛ 214 Óy 16384 !ê˛ ~ÓÇ 16 !ê˛ !Óê˛ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ˆüyê˛ 216 Óy 65536 !ê˛ §Çáƒy
ˆÓyé˛yˆÏöy ÎyÎ˚–
¢∑ x!û˛ˆÏ°áˆÏö §Ó≈!ö¡¨ Ä §ˆÏÓ≈yFã˛ ˆÎ ï˛#Ó ï˛y=!° §ÇÓ˚!«˛ï˛ •Î˚ñ ï˛yˆÏòÓ˚ ˛õyÕ‘y!ê˛ x!û˛ˆÏ°áˆÏöÓ˚ í˛zÍܲˆÏ£Ï≈Ó˚
~ܲ!ê˛ §)ã˛Ü˛– 16 !ê˛ !ÓˆÏê˛Ó˚ ¢∑ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° ò%•z ï˛#Ó ï˛yÓ˚ xö%˛õyï˛ •Î˚ 216×2 : 1– ˆÜ˛ööy ï˛#Ó ï˛y !ÓhflÏyˆÏÓ˚Ó˚
ÓˆÏà≈Ó˚ §üyö%˛õyï˛#– ˆí˛!§ˆÏӈϰÓ˚ !•§yˆÏÓ ~Ó˚ üyö 10log(232) Óyñ 10 × 32 × log2 xÌ≈yÍñ 96dB– ~•z
üyö ˆÎ ˆÜ˛yöÄ xƒyöy°à x!û˛ˆÏ°áö ˛õk˛!ï˛Ó˚ ï˛#Ó ï˛y Óƒy!ÆÓ˚ (dynamic range) ï%˛°öyÎ˚ Óí˛¸ñ ÎyÓ˚ ú˛ˆÏ°
!í˛!çê˛ƒy° ˛õk˛!ï˛ˆÏï˛ ¢ˆÏ∑Ó˚ !ÓÜ,˛!ï˛ (distortion) xƒyöy°à ˛õk˛!ï˛Ó˚ ï%˛°öyÎ˚ xˆÏöܲ ܲü •Î˚–
!ã˛e ≠ 13.11
13.11 !ã˛e ˆÌˆÏܲ xy˛õ!ö !í˛!çê˛ƒy° ˛õk˛!ï˛Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö ÓƒÓfliy!ê˛ Ó%é˛ˆÏï˛˛˛õyÓ˚ˆÏÓö–
üy•zˆÏe´yˆÏú˛yö myÓ˚y í˛zͲõߨ §ÇˆÏÜ˛ï˛ˆÏܲ !ÓÓ!ô≈ï˛ Ü˛ˆÏÓ˚ •zˆÏ°Ü˛ê˛Δ!öܲ !ú˛°ê˛yÓ˚ Óï≈˛ö#Ó˚ §y•yˆÏ΃ 20,000Hz
ܲ¡õyˆÏB˛Ó˚ í˛z˛õˆÏÓ˚Ó˚ í˛z˛õyÇ¢=!° Óyò ˆòÄÎ˚y •Î˚– §ÇˆÏܲï˛!ê˛ ~•zÓyÓ˚ !í˛!çê˛ƒy° §ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y
•Î˚– xÌ≈yÍñ ¢∑!ê˛ ~áö ‘0’ Ä ‘1’ !òˆÏÎ˚ ˜ï˛!Ó˚ Óy•zöy!Ó˚ ¢ˆÏ∑Ó˚ §ÇˆÏÜ˛ï˛ Ó˚*ˆÏ˛õ ÓyÓ˚ •Î˚– ~•z §ÇˆÏܲï˛!ê˛ˆÏܲ
≤ÈÏÎ˚yçö üï˛ §ÇˆÏ¢yôö ܲˆÏÓ˚ ˆÓ˚ܲ!í≈˛Ç ˆ•ˆÏí˛ ˛õyë˛yˆÏöy •Î˚–
¢ˆÏ∑Ó˚ ˛õ%öç≈öˆÏöÓ˚ çöƒ ˆ≤’ÈÙÈÓƒyܲ ˆ•í˛ ˆÌˆÏܲ Óy•zöy!Ó˚ §ÇˆÏܲï˛!ê˛ ≤Ã̈Ïü ~ܲ!ê˛ ˆüü!Ó˚ˆÏï˛ (memory)
çüy •Î˚ ~ÓÇ ~ܲ!ê˛ â!í˛¸Ó˚ §y•yˆÏ΃ !ö!ò≈‹T §üÎ˚ xhsˇÓ˚ û%˛° §ÇˆÏ¢yôö ≤Ã!e´Î˚yÓ˚ üyôƒˆÏü !í˛!çê˛ƒy° ˆÌˆÏܲ
xƒyöy°à §ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇÓ˚î ÓƒÓfliyÎ˚ ˛õyë˛yˆÏöy •Î˚– ~•z ÓƒÓfliy ˆÌˆÏܲ í˛zͲõߨ xƒyöy°à §ÇˆÏÜ˛ï˛ ˛õ%öÓ˚yÎ˚
20,000Hz-~Ó˚ í˛z˛õˆÏÓ˚Ó˚ ܲ¡õyB˛ Óç≈ö ܲÓ˚yÓ˚ !ú˛°ê˛yÓ˚ ~ÓÇ «˛üï˛y !ÓÓô≈ˆÏܲ (power amplifier) ÎyÎ˚ Ä
xӈϢˆÏ£Ï °yí˛zí˛!flõܲyˆÏÓ˚ ¢ˆÏ∑Ó˚ ˛õ!öç≈öö âê˛yÎ˚–
!í˛!çê˛ƒy° ˛õk˛!ï˛ˆÏï˛ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ˆÎ ÓƒÓfliyÓ˚ ܲÌy xy˛õ!ö ˛õí˛¸ˆÏ°öñ ~!ê˛ ˆã˛Ô¡∫ܲ !ú˛ï˛y
~ÓÇ Ü˛ü˛õƒyQ !í˛flÒÈÙÙÙÈí˛zû˛Î˚ üyôƒˆÏüÓ˚ ˆ«˛ˆÏe•z ≤ÈÏÎy烖 ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ ¢∑ x!û˛ˆÏ°áˆÏöÄ ~•z ˛õk˛!ï˛
¢∑ï˛Ó˚D
!ú˛°ê˛yÓ˚
Óï≈˛ö#
¢∑ï˛Ó˚ˆÏDÓ˚
öü%öy §Ç@ˇÃ• !í˛!çê˛ƒy° ¢∑!°!˛õ
Óy!•öyÓ˚# ¢ˆÏ∑ ¢∑ï˛Ó˚ˆÏDÓ˚ öü%öy
!ú˛°ê˛yÓ
Óï≈ö#
xƒyöy°à ¢∑§Ç Ïܲï˛flõ#ܲyÓ˚
xƒyöy°à ¢∑§ÇˆÏÜ˛ï˛ Ó˚*˛õyhsˇÓ˚î
342
ÓƒÓ•*ï˛ •ˆÏFåÈ ~ÓÇ ~•z ˛õyë˛ƒÓ› Ó˚ã˛öyÓ˚ §üˆÏÎ˚ ~!ê˛•z §Ó≈y!ôܲ ≤Ãã˛!°ï˛ ˛õk˛!ï˛– xyô%!öܲ 12 cm ÓƒyˆÏ§Ó˚
ܲü˛õƒyQ !í˛ˆÏflÒ 44.1 !ܲˆÏ°y•yÍ≈§‰ •yˆÏÓ˚ SxÌ≈yÍ ˆ§ˆÏܲˆÏu˛ 44100 !ê˛V à,•#ï˛ ¢ˆÏ∑Ó˚ öü%öy 16 !Óê˛ ¢ˆÏ∑Ó˚
xyܲyˆÏÓ˚ x!û˛!°!áï˛ •Î˚– ~=!°Ó˚ ï˛#Ó ï˛yÈÙÈÓƒy!Æ xhsˇï˛ 90dBñ ¢∑!ÓÜ,˛!ï˛ 0·004 ¢ï˛yLjϢÓ˚ ܲü ~ÓÇ 5Hz
ˆÌyˆÏܲ 20,000Hz ܲ¡õyˆÏB˛Ó˚ üˆÏôƒ ¢ˆÏ∑yͲõyòˆÏöÓ˚ §yí˛¸yÎ˚ (frequency response) 0·5 ˆí˛!§ˆÏӈϰÓ˚ ˆÓ!¢
ˆ•Ó˚ˆÏú˛Ó˚ •Î˚ öy– ~•z §%!Óôy=!°Ó˚ çöƒ•z ¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏö !í˛!çê˛ƒy° ≤ÃÎ%!_´ ~ܲ!ê˛ fliyÎ˚#
çyÎ˚ày ܲˆÏÓ˚ !öˆÏï˛˛ ˆ˛õˆÏÓ˚ˆÏåÈ–
ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√ Ä Ü˛ü˛õƒyQ !í˛ˆÏflÒ ¢∑ x!û˛ˆÏ°áö Ä ˆ§=!° ˆÌˆÏܲ ¢∑ ˛õ%öç≈öˆÏöÓ˚ !ӣψÏÎ˚ Îy ˛õí˛¸ˆÏ°ö
~ÓyÓ˚ ï˛yÓ˚ í˛z˛õÓ˚ ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# 3 :
(i) ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áˆÏö ÓƒÓ•*ï˛ ˆÓ˚áy!åÈo!ê˛ xï˛ƒhsˇ §%- •Î˚– ~Ó˚ ܲyÓ˚î ܲ#⁄
(ii) ˛õ!Ó˚Óï˛#≈ ≤Ãfli x!û˛ˆÏ°áˆÏö ˛õy¢y˛õy!¢ ò%•z!ê˛ ¢∑!°!˛õ x!û˛!°!áï˛ •Î˚– ~Ó˚ §%!Óôy ܲ#⁄
(iii) ¢ˆÏ∑Ó˚ !í˛!çê˛ƒy° x!û˛ˆÏ°áˆÏö 14 Óy 16 !ÓˆÏê˛Ó˚ §Çáƒy ÓƒÓ•yÓ˚ öy ܲˆÏÓ˚ xyÓ˚Ä Ü˛ü §ÇáƒÜ˛ !ÓˆÏê˛Ó˚
§Çáƒy ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° ܲ# x§%!Óôy •ï˛⁄
(iv) xyˆÏ°yܲ#Î˚ !í˛ˆÏflÒÓ˚ ˛õyˆÏë˛yk˛yˆÏÓ˚ x!ï˛ §)- ˆ°çyÓ˚ Ó˚!Ÿ¬Ó˚ ≤ÈÏÎ˚yçö •Î˚ ˆÜ˛ö⁄
13.6 xö%£Ï!Dܲ ÎsfyÓ°#xö%£Ï!Dܲ ÎsfyÓ°#xö%£Ï!Dܲ ÎsfyÓ°#xö%£Ï!Dܲ ÎsfyÓ°#xö%£Ï!Dܲ ÎsfyÓ°#
~ ˛õÎ≈hsˇ xy˛õ!ö ~•z ~ܲˆÏܲÓ˚ Îï˛ê˛y ˛õˆÏí˛¸ˆÏåÈö ï˛yˆÏï˛ ˆòˆÏáˆÏåÈö ˆÎñ ¢∑ x!û˛ˆÏ°áö Óy ˛õ%öç≈öˆÏöÓ˚ ˆÎ
ˆÜ˛yöÄ í˛zß¨ï˛ ˛õk˛!ï˛ˆÏï˛ ¢∑ˆÏܲ ≤Ã̈Ïü•z ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y •Î˚ ~ÓÇ !ÓÓô≈ö Ä xöƒyöƒ
≤Ã!e´Î˚yÓ˚ ˛õÓ˚ §ÇˆÏܲï˛!ê˛ˆÏܲ x!û˛ˆÏ°áö ÓƒÓfliyÎ˚ ˛õyë˛yˆÏöy •Î˚– ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ ï˛Ó˚ˆÏDÓ˚ üyôƒˆÏü §¡±ã˛yˆÏÓ˚Ó˚ çöƒÄ
¢∑ §ÇˆÏÜ˛ï˛ˆÏܲ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚y ≤ÈÏÎ˚yçö– ˆÎ Îsf ~•z Ó˚*˛õyhsˇÓ˚î âê˛yÎ˚ñ ˆ§!ê˛ •°
üy•zˆÏe´yˆÏú˛yö (microphone)– xyÓyÓ˚ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ˆÏܲ ˛õ%öÓ˚yÎ˚ ¢∑§ÇˆÏܲˆÏï˛ Ó˚*˛õyhsˇ!Ó˚ï˛ Ü˛Ó˚ˆÏï˛ ˆÎ
Îsf!ê˛ ÓƒÓ•*ï˛ •Î˚ñ ˆ§!ê˛ •° °yí˛zí˛!flõܲyÓ˚– üy•zˆÏe´yˆÏú˛yö Ä °yí˛zÈí˛!flõÈܲyÓ˚ÈÙÙÙÈí˛zû˛Î˚ Îsf•z ˆÜ˛yö ~ܲ üyôƒˆÏüÓ˚
~ܲ çyï˛#Î˚ ï˛Ó˚D ˆÌˆÏܲ xöƒ üyôƒˆÏü !û˛ß¨ çyï˛#Î˚ ï˛Ó˚D í˛zͲõߨ ܲˆÏÓ˚– ~ çyï˛#Î˚ ÎsfˆÏܲ ê˛Δ™!í˛í˛z§yÓ˚
(transducer) Ó°y •Î˚– ~•z xÇ¢ ˆÌˆÏܲ xy˛õ!ö ~•z ò%•z ΈÏsfÓ˚ àë˛ö Ä Ü˛yÎ≈≤Ãîy°# §¡õˆÏÜ≈˛ !ܲå%Èê˛y ôyÓ˚îy
ˆ˛õˆÏï˛ ˛õyÓ˚ˆÏÓö– ≤Ã̈Ïü xyüÓ˚y üy•zˆÏe´yˆÏú˛yö §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚–
343
13.6.1 üy•zˆÏe´yˆÏú˛yöüy•zˆÏe´yˆÏú˛yöüy•zˆÏe´yˆÏú˛yöüy•zˆÏe´yˆÏú˛yöüy•zˆÏe´yˆÏú˛yö
xy˛õ!ö xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ܲyç ¢ˆÏ∑Ó˚ ܲ¡õöˆÏܲ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛ Ó˚*˛õy!hsˇ!Ó˚ï˛
ܲÓ˚y– ~•z Ó˚*˛õyhsˇÓ˚ˆÏî öyöy ôÓ˚ˆÏöÓ˚ ˆû˛Ôï˛ âê˛öy ÓƒÓ•yÓ˚ ܲÓ˚y •ˆÏÎ˚ ÌyˆÏܲ– ˆÎüö ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ xyˆÏÓ¢ñ ôyÓ˚ˆÏܲÓ˚
ôyÓ˚ܲˆÏcÓ˚ Ä ˛õyˆÏe xyÓk˛ Ó˚yáy ܲyÓ≈ö ܲîyÓ˚ ˆÓ˚yˆÏôÓ˚ ˛õ!Ó˚Óï≈˛öñ ã˛y˛õ!Óò%ƒÍ˛ Óy !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ
(piezoelectric) x!û˛!e´Î˚y ~ÓÇ ˆã˛Ô¡∫ܲ ï˛!ï˛ Óy üƒyàˆÏöˆÏê˛y!fiê˛Δܲ¢ö (magnetostriction)– ˆÎ ˆÜ˛yöÄ
üy•zˆÏe´yˆÏú˛yö ÓƒÓ•yÓ˚ˆÏÎyàƒ •ˆÏï˛ •ˆÏ° ï˛yÓ˚ ܲˆÏÎ˚ܲ!ê˛ ÓyN˛ö#Î˚ =î Ìyܲy ≤ÈÏÎ˚yçö– xy˛õ!ö •Î˚ï˛ ~=!°
§¡∫ˆÏ¶˛ xyˆÏà•z ˆçˆÏö !öˆÏï˛ ã˛y•zˆÏÓö– ~•z =î=!° •° ≠
(a) §%ˆÏÓ!òï˛y §%ˆÏÓ!òï˛y §%ˆÏÓ!òï˛y §%ˆÏÓ!òï˛y §%ˆÏÓ!òï˛y (sensitivity) : üy•zˆÏe´yˆÏú˛yˆÏö xy˛õ!ï˛ï˛ ¢∑ã˛yˆÏ˛õÓ˚ !ÓhflÏyÓ˚ ρ0 ~ÓÇ ü%_´ Óï≈˛ö#ˆÏï˛
üy•zˆÏe´yˆÏú˛yö myÓ˚y í˛zͲõߨ !û˛ÓÓ ≤ÈÏû˛ˆÏòÓ˚ !ÓhflÏyÓ˚ E0 — ~•z ò%•zÈÙÈ~Ó˚ xö%˛õyï˛ xÌ≈yÍ M =
0
0
Eρ
Ó˚y!¢!ê˛
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §yí˛¸y (response) Óy §%ˆÏÓ!òï˛y §)!ã˛ï˛ ܲˆÏÓ˚– ~!ê˛Ó˚ ~ܲܲ ˆû˛yŒê˛†˛õƒyflÒy°
()1a Vp−
– xˆÏöܲ
§üÎ˚ 0·1 ˛õƒyflÒy° !˛õå%È ~ܲ ˆû˛yŒê˛ !Óû˛Ó ≤ÈÏû˛òˆÏܲ üyöܲ !•§yˆÏÓ ôˆÏÓ˚ ˆí˛!§ˆÏӈϰÓ˚ !•§yˆÏÓ §%ˆÏÓ!òï˛yÓ˚
üyö ˆ°áy •Î˚– ~•z !•§yˆÏÓ §%ˆÏÓ!òï˛yÓ˚ xyˆÏ˛õ!«˛Ü˛ üyö 0 0 0
0
/0·1 10
E p Ep
= – ˆÎˆÏ•ï%˛ í˛zͲõߨ «˛üï˛y !Óû˛Ó
≤à Ïû˛ ÏòÓ˚ Ó Ïà≈Ó˚ §üyö% õyï˛#ñ í˛!§ ÏÓ Ï°Ó˚ !•§y ÏÓ üy•z Ïe´y Ïú˛y ÏöÓ˚ §yí ¸y n = 10 log2
0 0
0 020log
10 10E E
p p⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
–
§yôyÓ˚îû˛yˆÏÓñ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ xöƒyöƒ =î=!° x«˛ï˛ ˆÓ˚ˆÏá §%ˆÏÓ!òï˛yÓ˚ üyö Îï˛ x!ôܲ •Î˚ ï˛ï˛•z û˛y°–
ï˛ˆÏÓ í˛zFã˛üyˆÏöÓ˚ •zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ܲ §•ç°û˛ƒ •ÄÎ˚yÎ˚ §%ˆÏÓ!òï˛yÓ˚ üyö !ܲå%Èê˛y ܲü •ˆÏ°Ä ˆ§•z âyê˛!ï˛
§•ˆÏç•z ˛õ)Ó˚î ܲÓ˚y ÎyÎ˚–
(b) ܲ¡õyB˛ÈÙȧyí˛¸yÓ˚ §%§üï˛y ܲ¡õyB˛ÈÙȧyí˛¸yÓ˚ §%§üï˛y ܲ¡õyB˛ÈÙȧyí˛¸yÓ˚ §%§üï˛y ܲ¡õyB˛ÈÙȧyí˛¸yÓ˚ §%§üï˛y ܲ¡õyB˛ÈÙȧyí˛¸yÓ˚ §%§üï˛y (uniformity of frequency-response) : üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §%ˆÏÓ!òï˛y
Óy §yí˛¸yÓ˚ üyö ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° •ˆÏ° í˛zͲõߨ !Óû˛Ó ≤ÈÏû˛ò !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ܲ¡õöˆÏܲ
ÎÌyÎÌû˛yˆÏÓ xö%§Ó˚î ܲˆÏÓ˚ öy– ú˛ˆÏ° x!û˛!°!áï˛ ¢ˆÏ∑ Óy °yí˛zí˛!flõܲyˆÏÓ˚ í˛zͲõߨ ¢ˆÏ∑ !ÓÜ,˛!ï˛ âˆÏê˛– ~•z
!ÓÜ,˛!ï˛ ~!í˛¸ˆÏÎ˚ ¢ˆÏ∑Ó˚ x!Óܲ°ï˛y (fidelity) ÓçyÎ˚ Ó˚yáyÓ˚ çöƒ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §yí˛¸yÓ˚ ÎÌy§Ω˛Ó
ܲ¡õyB˛ÈÙÈx!öû˛≈Ó˚ •ÄÎ˚y ≤ÈÏÎ˚yçö–
(c) ï˛#Ó ï˛yÈÙÈÓƒy!Æ ï˛#Ó ï˛yÈÙÈÓƒy!Æ ï˛#Ó ï˛yÈÙÈÓƒy!Æ ï˛#Ó ï˛yÈÙÈÓƒy!Æ ï˛#Ó ï˛yÈÙÈÓƒy!Æ (dynamic range) : üy•zˆÏe´yˆÏú˛yˆÏö xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ ˆÎ ˛õyÕ‘yÓ˚ üˆÏôƒ
í˛zͲõߨ !Óû˛Ó ≤ÈÏû˛ò ¢∑ã˛yˆÏ˛õÓ˚ §üyö%˛õyï˛# ÌyˆÏܲñ ˆ§!ê˛•z üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ï˛#Ó ï˛y Óƒ!Æ– x!ï˛ ò%Ó≈° ¢ˆÏ∑
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ !öçfl∫ x˛õfl∫Ó˚ ~ÓÇ x!ï˛ ï˛#Ó ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe í˛zͲõߨ !Óû˛Ó ܲ¡õˆÏöÓ˚ !ÓÜ,˛!ï˛ ï˛#Ó ï˛y Óƒy!ƈÏܲ
§#!üï˛ Ó˚yˆÏá–
344
~ÓyÓ˚ xyüÓ˚y ≤Ãã˛!°ï˛ ܲˆÏÎ˚ܲ ≤ÃܲyÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈ö#!ï˛ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó– ~=!°
•° ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yöñ ôyÓ˚ܲ Óy ܲöˆÏí˛ö§yÓ˚ üy•zˆÏe´yˆÏú˛yöñ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö Ä ã˛° Ü%˛[˛°#
üy•zˆÏe´yˆÏú˛yö–
ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≠ ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≠ ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≠ ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≠ ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≠ ~ çyï˛#Î˚ üy•zˆÏe´yˆÏú˛yˆÏö ~ܲ!ê˛ ≤ÈÏܲyˆÏ¤˛ ܲyÓ≈ˆÏöÓ˚ òyöy !ܲå%Èê˛y xy°àyû˛yˆÏÓ û˛Ó˚y
ÌyˆÏܲ (!ã˛e 13.12) – ¢∑ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ˛õò≈yÎ˚ xy˛õ!ï˛ï˛ •ˆÏ° ¢∑ã˛yˆÏ˛õÓ˚ ú˛ˆÏ° ˛õò≈y!ê˛ Ü˛!¡õï˛ •Î˚ ~ÓÇ
˛õò≈yÓ˚ §ˆÏD Î%_´ ~ܲ!ê˛ !˛õfiê˛ö ܲyÓ≈ˆÏöÓ˚ òyöy=!°Ó˚ í˛z˛õÓ˚ ã˛yˆÏ˛õÓ˚ •…y§ Ó,!k˛ âê˛yÎ˚– ~•z
ã˛yˆÏ˛õÓ˚ §ˆÏD òyöy=!°Ó˚ ˛õÓ˚flõˆÏÓ˚Ó˚ üˆÏôƒ flõ¢≈ï˛°Ä Ü˛ˆÏü ÓyˆÏí˛¸ñ ú˛ˆÏ° ܲyÓ≈ˆÏöÓ˚
òyöy=!°Ó˚ ˜Óò%ƒ!ï˛Ü˛ ˆÓ˚yôÄ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– ~•z xÓfliyÎ˚ Î!ò xy˛õ!ö Óƒyê˛y!Ó˚Ó˚
§y•yˆÏ΃ ܲyÓ≈ˆÏöÓ˚ üôƒ !òˆÏÎ˚ ˜Óò%ƒ!ï˛Ü˛ ≤ÃÓy• ˛õyë˛yˆÏï˛ ˆã˛‹Ty ܲˆÏÓ˚öñ ï˛ˆÏÓ ¢ˆÏ∑Ó˚
ܲ¡õˆÏöÓ˚ §ˆÏD ܲyÓ≈ˆÏöÓ˚ üôƒ !òˆÏÎ˚ ˜Óò%ƒ!ï˛Ü˛ ≤ÃÓy•Ä ܲ!¡õï˛ •ˆÏï˛ ÌyܲˆÏÓ–
ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛ ¢!_´¢y°#– ~Ó˚ §yí˛¸yÓ˚ üyö ÈÙÈ 45 ˆÌ Ïܲ
- 40 ˆí˛!§ˆÏӈϰÓ˚ üï˛– ~!ê˛ ˆê˛Ü˛§•z xÌã˛ fl∫“ü)°ƒ ó ï˛ˆÏÓ ~!ê˛ˆÏï˛ Ü˛¡õyB˛ §yí˛¸y §%§ü
öy •ÄÎ˚yÎ˚ ¢ˆÏ∑Ó˚ x!Óܲ°ï˛y ÓçyÎ˚ ÌyˆÏܲ öy– ˆê˛!°ˆÏú˛yö Ä ˆÓï˛yÓ˚ ˆÎyàyˆÏÎyˆÏà
ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yö ≤ÃyÎ˚•z ÓƒÓ•*ï˛ •Î˚–
ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yö ≠ ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yö ≠ ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yö ≠ ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yö ≠ ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yö ≠ ~•z üy•zˆÏe´yˆÏú˛yˆÏö ~ܲ!ê˛ ê˛yö ˆòÄÎ˚y ôyï%˛Ó˚ ˛õò≈y Ä ~ܲ!ê˛ ò,ì˛¸û˛yˆÏÓ §ÇÓk˛ ˛õyï˛
!ü!°ï˛û˛yˆÏÓ ~ܲ!ê˛ §üyhsˇÓ˚y° ˛õyï˛ ôyÓ˚ˆÏܲÓ˚ üï˛ Ü˛yç ܲˆÏÓ˚– xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚ ܲ¡õˆÏöÓ˚ §ˆÏD ôyï%˛Ó˚ ˛õò≈y!ê˛
ܲ!¡õï˛ •Î˚ñ ôyÓ˚ˆÏܲÓ˚ ôyÓ˚ܲˆÏcÓ˚Ä •…y§ Ó,!k˛ •Î˚– ~•z ôyÓ˚ˆÏܲÓ˚ §ˆÏD ~ܲ!ê˛ Óƒyê˛y!Ó˚ Ä
û˛yÓ˚ ˆÓ˚yô ˆ◊î# §üÓyˆÏÎ˚ Î%_´ ܲÓ˚ˆÏ° ôyÓ˚ܲˆÏcÓ˚ Äë˛yöyüyÓ˚ ú˛ˆÏ° û˛yÓ˚ˆÏÓ˚yˆÏôÓ˚ üôƒ !òˆÏÎ˚
˛õ!Ó˚Óï˛#≈ ï˛!í˛¸Í≤ÃÓyˆÏ•Ó˚ §,!‹T •Î˚– 13.13 !ã˛ˆÏe xy˛õ!ö ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ~ܲ!ê˛
àë˛ö!ã˛e ˆòáˆÏï˛ ˛õyˆÏÓö–
ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ~ܲ!ê˛ Óí˛¸ §%!Óôy ~•z ˆÎñ ~Ó˚ §%ˆÏÓ!òï˛y ≤ÃyÎ˚ 6-7 KHz
˛õ¡õyB˛ ˛õÎ≈hsˇ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ– xÓ¢ƒ ~Ó˚ ˆÓ!¢ ܲ¡õyˆÏB˛ ê˛yö ˆòÄÎ˚y ôyï%˛Ó˚ ˛õò≈yÓ˚
!öçfl∫ ܲ¡õyˆÏB˛ xö%öyˆÏòÓ˚ ˛ú˛ˆÏ° ܲ¡õyB˛ÈÙȧyí˛¸y Ó,!k˛ ˛õyÎ˚– ~Ó˚Ä x!ôܲ ܲ¡õyˆÏB˛ §yí˛¸yÓ˚
üyö ˛õò≈yÓ˚ û˛ˆÏÓ˚Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° •Î˚ ~ÓÇ e´ü¢ ܲˆÏü ÎyÎ˚– ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §%ˆÏÓ!òï˛yÓ˚ üyö xï˛ƒhsˇ
x“– ˆí˛!§ˆÏӈϰÓ˚ !•§yˆÏÓ ~!ê˛ ÈÙÈ 50db ~Ó˚ ܲyåÈyܲy!åÈ– ï˛ˆÏÓ •zˆÏ°Ü˛ê˛Δ!öܲ !ÓÓô≈ܲ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ~•z x§%!Óôy
ܲy!ê˛ˆÏÎ˚ Äë˛y ÎyÎ˚– ôyÓ˚ˆÏܲÓ˚ ôyÓ˚ܲc §yôyÓ˚îï˛ xï˛ƒhsˇ x“ •Î˚ñ ≤ÃyÎ˚ 100 pF ~Ó˚ üˆÏï˛y– ~çöƒ ò#â≈ ˆÎyçܲ
ï˛yÓ˚ ÓƒÓ•yÓ˚ öy ܲˆÏÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §ˆÏD•z ≤Ãy܉˛ÈÙÈ!ÓÓô≈ܲ (pre-amplifier) ˆÓ˚ˆÏá í˛zͲõߨ !Óû˛Ó ≤ÈÏû˛òˆÏܲ
!ÓÓ!ô≈ï˛ Ü˛Ó˚ˆÏï˛ •Î˚– ÓƒÓ•yˆÏÓ˚Ó˚ §üÎ˚ ~•z üy•zˆÏe´yˆÏú˛yˆÏö 200 ˆÌˆÏܲ 400 ˆû˛yˆÏŒê˛Ó˚ Óƒyê˛y!Ó˚ xÌÓy §¡õ)î≈
!fliÓ˚ !Óû˛ˆÏÓÓ˚ Óƒyê˛y!Ó˚ ~!°!üˆÏöê˛Ó˚ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏï˛ •Î˚–
!˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö ≠ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö ≠ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö ≠ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö ≠ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yö ≠ ~ ôÓ˚ ÏöÓ˚ üy•z Ïe´y Ïú˛y ÏöÓ˚ !e´Î˚y Ó%é˛ Ïï˛ • Ï° ≤ÃÌ Ïü•z !˛õ Ïçy•z Ï°Ü˛!ê Δܲ
Óy ã˛y˛õ !Óòƒ%Í ôˆÏü≈Ó˚ !Ó£ÏÎ˚!ê˛ xyˆÏ°yã˛öy ܲˆÏÓ˚ !öˆÏï˛ •ˆÏÓ– myò¢ ~ܲˆÏܲ xy˛õ!ö ~ §¡∫ˆÏ¶˛ !ܲå%Èê˛y ˆçˆÏöˆÏåÈö–
ܲyÓ≈ ÏöÓ˚ òyöy
!ã˛e ≠ 13.12
˛õò≈y
§ÇÓk˛˛ ˛õyï˛
!ã˛e ≠ 13.13
˛õò≈y
345
ˆÜ˛yÎ˚yê≈˛çñ ˆÓ˚yˆÏ¢Ó˚ §Œê˛ (Rochelle Salt), ~ !í˛ !˛õ (ammonium dihydrogen phosphate) ≤Ãû, !ï˛
ˆÜ˛°yˆÏ§ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ ôü≈ ˆòáy ÎyÎ˚– ~•z ˆÜ˛°y§=!° ˆÌˆÏܲ !ӈϢ£Ïû˛yˆÏÓ Ü˛yê˛y xLjϢ Îy!sfܲ ˛õ#í˛¸ö
≤ÈÏÎ˚yà ܲÓ˚ˆÏ° ˆ§!ê˛Ó˚ ò%•z ï˛ˆÏ°Ó˚ üˆÏôƒ !Óû˛Ó ≤ÈÏû˛ò ˜ï˛!Ó˚ •Î˚ ~ÓÇ ò%•z !Ó˛õÓ˚#ï˛ ï˛ˆÏ° ˛õÓ˚flõÓ˚ !Ó˛õÓ˚#ï˛
˜Óò%ƒ!ï˛Ü˛ xyôyö ˆòáy ˆòÎ˚– !Óû˛Ó ≤ÈÏû˛ˆÏòÓ˚ öyü xÓ¢ƒ ˆÜ˛°y§!ê˛ Ü˛#û˛yˆÏÓ Ü˛yê˛y •ˆÏÎ˚ˆÏåÈ ~ÓÇ ˆÜ˛yö ôÓ˚ˆÏöÓ˚
˛õ#í˛¸ö ≤ÈÏÎ˚yà ܲÓ˚y •ˆÏÎ˚ˆÏåÈ xÌ≈yÍñ ï˛y §Çöüöñ ≤çyÓ˚î Óy ÓÇܲöÈÙÙÙÈ~Ó˚ ˆÜ˛yö çyï˛#Î˚ñ ï˛yÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚–
¢∑ï˛Ó˚D !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ ˆÜ˛°yˆÏ§Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •ˆÏ° ¢∑ã˛yˆÏ˛õÓ˚ Äë˛yöyüyÓ˚ §ˆÏD ˆÜ˛°yˆÏ§Ó˚ ò%•z !Ó˛õÓ˚#ï˛
ï˛ˆÏ°Ó˚ üˆÏôƒ ˆÎ !Óû˛Ó ≤ÈÏû˛ò §,‹T •Î˚ñ ˆ§!ê˛ !ÓÓô≈ö ܲˆÏÓ˚ °yí˛z!flõܲyˆÏÓ˚ xÌÓy ¢∑ x!û˛ˆÏ°áö ÓƒÓfliyÎ˚
˛õyë˛yˆÏöy •Î˚– ~•z çyï˛#Î˚ üy•zˆÏe´yˆÏú˛yöˆÏܲ ˆÜ˛°y§ üy•zˆÏe´yˆÏú˛yö Ó°y •Î˚– ˆÓ!Ó˚Î˚yü ê˛y•zê˛ƒyˆÏöê˛ çyï˛#Î˚
ˆ§Ó˚y!üܲ ˛õòyˆÏÌ≈ !ӈϢ£Ï ≤Ã!e´Î˚yÎ˚ fliyÎ˚# !Óò%ƒÍÈÙȈüÓ˚&û˛Óö (electric polarization) âê˛yˆÏ° ~!ê˛
!˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ ôü≈ °yû˛ ܲˆÏÓ˚– ~•z çyï˛#Î˚ ˆ§Ó˚y!üܲ ˛õòyÌ≈ ˆÎ üy•zˆÏe´yˆÏú˛yˆÏö ÓƒÓ•*ï˛ •Î˚ñ ï˛yˆÏܲ ˆ§Ó˚y!üܲ
üy•zˆÏe´yˆÏú˛yö Ó°y •Î˚–
§yôyÓ˚î ¢∑ã˛yˆÏ˛õ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ üy•zˆÏe´yˆÏú˛yˆÏö í˛zͲõߨ !Óû˛Ó ≤ÈÏû˛ò x!ï˛ x“ üyˆÏöÓ˚ •Î˚– ~•z !Óû˛Ó
≤ÈÏû˛ò Óyí˛¸yˆÏöyÓ˚ çöƒ ˆÎ !ӈϢ£Ï àë˛ö ˛õk˛!ï˛ xÓ°¡∫ö ܲÓ˚y •Î˚ ˆ§!ê˛ xy˛õ!ö 13.14 !ã˛ˆÏe ˆòáˆÏï˛ ˛õyˆÏÓö–
¢∑ï˛Ó˚D ~ܲ!ê˛ •y°Ü˛y ˛õò≈yÓ˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ •ˆÏÎ˚ ˆ§!ê˛ˆÏܲ ܲ!¡õï˛ Ü˛ˆÏÓ˚– ˛õò≈yÓ˚ ˆÜ˛ˆÏwÓ˚ §Ç°@¿ ~ܲ!ê˛
ܲyÑê˛y ˙ ܲ¡õöˆÏܲ ò%•z!ê˛ !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ ˆÜ˛°yˆÏ§Ó˚ ~ܲ ≤ÃyˆÏhsˇ §MÈ˛y!°ï˛ ܲˆÏÓ˚–
ˆÜ˛°y§ ò%•z!ê˛ ˛õy¢y˛õy!¢ ˆçyí˛¸y °yàyˆÏöy ÌyˆÏܲ ~ÓÇ ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÎyˆÏàÓ˚ §%!ÓôyÓ˚
çöƒ ˆ§=!°Ó˚ í˛zû˛Î˚!òˆÏܲ ôyï%˛Ó˚ ≤Èϰ˛õ °yàyˆÏöy ÌyˆÏܲ– ˆÜ˛°y§Î%ˆÏ@¬Ó˚ (bimorph)
í˛z˛õÓ˚ ˛õ#í˛¸ö ≤ÃÎ%_´ •ˆÏ° ˆ§!ê˛ ï˛yÓ˚ !ÓÜ,˛!ï˛Ó˚ §üyö%˛õyï˛# !Óû˛Ó ≤ÈÏû˛ò í˛zͲõߨ ܲˆÏÓ˚–
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ˛õò≈yñ ܲyÑê˛y Ä ˆÜ˛°y§Î%@¬ ~üöû˛yˆÏÓ ˜ï˛!Ó˚ •Î˚ ˆÎñ ˆ§!ê˛Ó˚
ܲ¡õö¢#° xLjϢÓ˚ xö%öyò ܲ¡õyB˛ §ã˛Ó˚yã˛Ó˚ ˆÎ ܲ¡õyB˛ ˛õyÕ‘yÓ˚ üˆÏôƒ
üy•zˆÏe´yˆÏú˛yö!ê˛ ÓƒÓ•*ï˛ •Î˚ñ ï˛yÓ˚ í˛z˛õˆÏÓ˚ ÌyˆÏܲ– ú˛ˆÏ° ~!ê˛Ó˚ ܲ¡õyB˛ §yí˛¸y ≤ÃyÎ˚
§%§ü ÌyˆÏܲ– ~•z ˛õk˛!ï˛ˆÏï˛ !ö!ü≈ï˛ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §yí˛¸y §yôyÓ˚îï˛ ◊yÓƒ ܲ¡õyˆÏB˛
–50
±
5 ˆí˛!§ˆÏӈϰÓ˚ üˆÏôƒ ÌyˆÏܲ– ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ï%˛°öyÎ˚ ˆÜ˛°y§
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ~ܲ!ê˛ Óyí˛¸!ï˛ §%!Óôy ~•z ˆÎñ ~!ê˛ˆÏï˛ ˆÜ˛yöÄ §Ç°@¿ ≤Ãy܉˛ÈÙÈ!ÓÓô≈ܲ ÓƒÓ•yˆÏÓ˚Ó˚ ≤ÈÏÎ˚yçö •Î˚
öy– x!û˛û˛y£Ïî ÓƒÓfliyÎ˚ñ ◊Óî §•yÎ˚ܲ ΈÏsf Ä ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y üy˛õܲ ΈÏsf ≤ÃyÎ˚•z !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ
üy•zˆÏe´yˆÏú˛yö ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚–
ã˛°Ü%˛[˛°# ã˛°Ü%˛[˛°# ã˛°Ü%˛[˛°# ã˛°Ü%˛[˛°# ã˛°Ü%˛[˛°# (moving coil) üy•zˆÏe´yˆÏú˛yö ≠ üy•zˆÏe´yˆÏú˛yö ≠ üy•zˆÏe´yˆÏú˛yö ≠ üy•zˆÏe´yˆÏú˛yö ≠ üy•zˆÏe´yˆÏú˛yö ≠ 13.3.2 xLjϢ ˆÎ ã˛°Ü%˛[˛°# !˛õܲÈÙÈxyˆÏ˛õÓ˚ í˛zˆÏÕ‘á ܲÓ˚y
•ˆÏÎ˚ˆÏåÈñ ~•z üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ܲyÎ≈ö#!ï˛ ï˛yÓ˚•z xö%Ó˚*˛õ– 13.15 !ã˛e ˆÌˆÏܲ xy˛õ!ö ~•z üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛ö!ê˛
Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~áyˆÏö M ~ܲ!ê˛ Óy!ê˛Ó˚ xyܲyˆÏÓ˚Ó˚ fliyÎ˚# ã%˛¡∫ܲ ÎyÓ˚ ~ܲ!ê˛ ˆüÓ˚& (N) Ó°yÎ˚Ü,˛!ï˛ ~ÓÇ xöƒ
ˆüÓ˚&!ê˛ Óy!ê˛Ó˚ ˆÜ˛ˆÏw hflψÏΩ˛Ó˚ xyÜ,˛!ï˛!Ó!¢‹T (S) D ˛õò≈y!ê˛ ò,ì˛¸ Ä xô≈ˆÏày°Ü˛yÜ,˛!ï˛ñ ~Ó˚ Óy•zˆÏÓ˚Ó˚ xÇ¢ ˆì˛í˛z
ˆá°yˆÏöy ˛õyˆÏï˛Ó˚ §y•yˆÏ΃ !öÓk˛ñ ÎyˆÏï˛ ˛õò≈y!ê˛ §•ˆÏç Äë˛yöyüy ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– ˛˛õò≈yÓ˚ §ˆÏD ~ܲ!ê˛ ï˛yˆÏÓ˚Ó˚
Ü%˛[˛°# xyê˛Ü˛yˆÏöy ÌyˆÏܲñ ˆÎ!ê˛ ˛õò≈yÓ˚ ܲ¡õˆÏöÓ˚ §ˆÏD fliyÎ˚# ã%˛¡∫ܲˆÏܲÓ˚ ˆã˛Ô¡∫ˆÏ«˛ˆÏe ã˛°yˆÏú˛Ó˚y ܲˆÏÓ˚– ~Ó˚ ú˛ˆÏ°
!ã˛e ≠ 13.14
ˆÜ˛°y§Î%@¬
˛õò≈y
346
~•z Ü%˛[˛°#ˆÏï˛ ˆã˛Ô¡∫ܲ xyˆÏÓ¢ç!öï˛ !Óû˛Ó ≤ÈÏû˛ˆÏòÓ˚ §,!‹T •Î˚– ã%˛¡∫ˆÏܲÓ˚ ˆüÓ˚& ò%•z!ê˛Ó˚ üˆÏôƒ ˆÓ˚¢ˆÏüÓ˚ ܲy˛õˆÏí˛¸Ó˚
~ܲ!ê˛ ¢B%˛ xyê˛Ü˛yˆÏöy ÌyˆÏܲ– ˛õò≈y Ä ˆÓ˚¢ü# ¢B%˛Ó˚ üˆÏôƒ xyÓk˛ ÓyÎ˚% §Çöüö Ä
ï˛ö%û˛ÓˆÏöÓ˚ ú˛ˆÏ° ˛õò≈yÓ˚ í˛z˛õÓ˚ ≤Ãï˛ƒyöÎ˚ܲ Ó° ˜ï˛!Ó˚ ܲˆÏÓ˚– ˆì˛í˛z ˆá°yˆÏöy ˛õyï˛!ê˛Ä
˛õò≈yˆÏܲ §yüƒyÓfliyÎ˚ ˆú˛Ó˚yˆÏï˛ §y•y΃ ܲˆÏÓ˚– ~åÈyí˛¸y ˆÓ˚¢ü# ¢B%˛!ê˛ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚
üˆÏôƒ ˛õò≈yÓ˚ xö%öyò# ܲ¡õö ˆÓ˚yô ܲˆÏÓ˚ ܲ¡õyB˛ÈÙÙÙȧyí˛¸yˆÏܲ xˆÏöܲê˛y §%£Ïü ܲˆÏÓ˚–
~•z üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ §yí˛¸yÓ˚ üyö xï˛ƒhsˇ ܲüñ ≤ÃyÎ˚È-90 ˆÌˆÏܲ ÈÙÈ95 ˆÈí˛!§ˆÏӈϰÓ˚
üˆÏôƒñ ï˛ˆÏÓ ~Ó˚ xû˛ƒhsˇÓ˚#î ≤Ã!ï˛ˆÏÓ˚yô (internal impedance) á%Ó Ü˛üñ ≤ÃyÎ˚
10
Ω
~Ó˚ üˆÏï˛y– ú˛ˆÏ° í˛zͲõߨ ˛õ!Ó˚Óï˛#≈ !Óû˛Ó ≤ÈÏû˛òˆÏܲ §•ˆÏç•z ê˛Δy™ú˛Ó˚üyÓ˚ myÓ˚y
Ó!ô≈ï˛ Ü˛ˆÏÓ˚ !ÓÓô≈ˆÏܲ ˛õyë˛yˆÏöy ÎyÎ˚–
!ò‹Tï˛y !ò‹Tï˛y !ò‹Tï˛y !ò‹Tï˛y !ò‹Tï˛y (directionality) : ~áyˆÏö xy˛õ!ö ã˛yÓ˚ ôÓ˚ˆÏöÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛öñ ܲyÎ≈˛õk˛!ï˛ Ä Ü˛ˆÏÎ˚ܲ!ê˛
ôü≈ §¡∫ˆÏ¶˛ çyöˆÏï˛ ˛õyÓ˚ˆÏ°ö– !ܲv üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ xyÓ˚Ä ~ܲ!ê˛ =Ó˚&c˛õ)î≈ ôü≈ §¡∫ˆÏ¶˛ xy˛õöyÓ˚ üˆÏö !çK˛y§y
ÌyܲˆÏï˛ ˛õyˆÏÓ˚– xyüÓ˚y öyê˛ˆÏܲÓ˚ üˆÏMÈ˛ ˆÎ üy•zˆÏe´yˆÏú˛yö ÓƒÓ•yÓ˚ ܲ!Ó˚ ï˛y üˆÏMÈ˛Ó˚ !Ó!û˛ß¨ xÇ¢ ˆÌˆÏܲ xy§y
¢∑ ôÓ˚ˆÏï˛ §«˛ü •ÄÎ˚y í˛z!ã˛ï˛– !ܲv ~ܲçö §yÇÓy!òܲ !û˛ˆÏí˛¸Ó˚ üˆÏôƒ ˆÜ˛yöÄ ~ܲ !ö!ò≈‹T Óƒ!_´Ó˚ ܲt˛fl∫Ó˚
ˆÓ˚ܲí≈˛ ܲÓ˚ˆÏï˛ ã˛y•zˆÏ° ï˛yÑÓ˚ üy•zˆÏe´yˆÏú˛yö Îï˛ ~ܲü%á# (unidirectional) •Î˚ ï˛ï˛•z û˛y°– ~•z ò%•z ˆ«˛ˆÏe
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ˆÎ ôü≈!ê˛ xyüÓ˚y !ÓˆÏÓã˛öy ܲÓ˚!åÈ ˆ§!ê˛ •° ï˛yÓ˚ !ò‹Tï˛y (directionality)– ¢∑
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ IJõÓ˚ §Ó˚y§!Ó˚ §yüˆÏö ˆÌˆÏܲ xy˛õ!ï˛ï˛ •ˆÏ° ˆÎ §yí˛¸y ˛õyÄÎ˚y ÎyÎ˚ñ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ xˆÏ«˛Ó˚
§ˆÏD ¢∑ xy˛õï˛ˆÏöÓ˚ !òˆÏá xyöï˛ ÌyܲˆÏ° §yí˛¸y ï˛yÓ˚ ˆã˛ˆÏÎ˚ ܲü •Î˚– ~!ê˛Ó˚ ≤ÃÌü ܲyÓ˚î üy•zˆÏe´yˆÏú˛yˆÏöÓ˚
myÓ˚y ¢∑ï˛Ó˚ˆÏDÓ˚ ÓƒÓï≈˛ö (diffraction), ÎyÓ˚ ú˛ˆÏ° ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ xyܲyˆÏÓ˚Ó˚ §ˆÏD ï%˛°ö#Î˚
•ˆÏ° !ï˛Î≈ܲû˛yˆÏÓ xy˛õ!ï˛ï˛ ¢∑ üy•zˆÏe´yˆÏú˛yˆÏö xˆÏ˛õ«˛yÜ,˛ï˛ x“ ¢∑ã˛y˛õ í˛zͲõߨ ܲˆÏÓ˚– !mï˛#Î˚ ܲyÓ˚î!ê˛ ~•z
ˆÎñ !ï˛Î≈ܲû˛yˆÏÓ xy˛õ!ï˛ï˛ ¢∑ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ˛õò≈yÓ˚ !Ó!û˛ß¨ xLjϢ ˆÎ ˛õ!Ó˚Óï˛#≈ ¢∑ã˛y˛õ ≤ÈÏÎ˚yà ܲˆÏÓ˚
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ xyܲyÓ˚ ï˛Ó˚D ˜òˆÏâ≈ƒÓ˚ ï%˛°öyÎ˚ ˆö•yï˛ ˆåÈyê˛ öy •ˆÏ° ï˛yÓ˚ ò¢yˆÏܲyî ˛õò≈yÓ˚ !Ó!û˛ß¨ xLjϢ
!Ó!û˛ß¨ •Î˚– ú˛ˆÏ° ˆ§=!°Ó˚ ≤Ãû˛yÓ Óƒ!ï˛ã˛yˆÏÓ˚Ó˚ ú˛ˆÏ° «˛Î˚≤ÃyÆ •Î˚– ï˛ˆÏÓ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ !ò‹Tï˛y ï˛yÓ˚ àë˛öñ
xyܲyÓ˚ñ xyÜ,˛!ï˛ Ä ¢ˆÏ∑Ó˚ ï˛Ó˚D˜ÏòˆÏâ≈ƒÓ˚ IJõÓ˚ !öû≈˛Ó˚¢#° ~ÓÇ !Ó!û˛ß¨ ≤ÈÏÎ˚yˆÏàÓ˚ ˆ«˛ˆÏe !Ó!û˛ß¨ ≤ÃܲyÓ˚ !ò‹Tï˛y
ÓyN˛ö#Î˚ ӈϰ üˆÏö ܲÓ˚y •Î˚– ~ÓyÓ˚ xyüÓ˚y üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ !Ó˛õÓ˚#ï˛ôü#≈ ~ܲ!ê˛ ÎˆÏsfÓ˚ ܲÌy xyˆÏ°yã˛öy ܲÓ˚Ó–
13.6.2 °yí˛zí˛!flõܲyÓ˚°yí˛zí˛!flõܲyÓ˚°yí˛zí˛!flõܲyÓ˚°yí˛zí˛!flõܲyÓ˚°yí˛zí˛!flõܲyÓ˚ (Loudspeaker)
üy•zˆÏe´yˆÏú˛yö ˆÎüö ¢∑ï˛Ó˚ˆÏDÓ˚ ܲ¡õöˆÏܲ ˜Óò%ƒ!ï˛Ü˛ !Óû˛Ó Óy ≤ÃÓyˆÏ•Ó˚ ܲ¡õˆÏö ˛õ!Ó˚îï˛ Ü˛ˆÏÓ˚ñ ˆï˛üö•z
°yí˛zí˛!flõܲyÓ˚ ˜Óò%ƒ!ï˛Ü˛ !Óû˛Ó Óy ≤ÃÓyˆÏ•Ó˚ ܲ¡õö ˆÌˆÏܲ ¢∑ í˛zͲõߨ ܲˆÏÓ˚– xÌ≈yÍñ ~!ê˛Ó˚ ܲyÎ≈
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ܲyˆÏÎ≈Ó˚ !ë˛Ü˛ !Ó˛õÓ˚#ï˛– xyò¢≈ °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ ˆÎ =î=!° Ìyܲy í˛z!ã˛ï˛ ˆ§=!° xyˆÏà ˆçˆÏö
ˆöÄÎ˚y Îyܲ– ~=!° •° ≠
(a) ˜Óò%ƒ!ï˛Ü˛ ¢!_´ˆÏܲ ¢∑¢!_´ˆÏï˛ Ó˚*˛õyhsˇÓ˚ˆÏîÓ˚ ò«˛ï˛y ÎÌy§Ω˛Ó x!ôܲ •ÄÎ˚y í˛z!ã˛ï˛–
(b) ¢∑ í˛zͲõyòö §yí˛¸y (acoustic output response) §ü@ˇÃ ◊yÓƒ ܲ¡õyˆÏB˛ ܲ¡õyB˛ÈÙÈx!öû≈˛Ó˚ •ÄÎ˚y
í z!ã˛ï˛–
fliyÎ˚# ã%˛¡∫ܲ
!ã˛e ≠ 13.15
347
(c) í˛zͲõߨ ¢ˆÏ∑ Îö ܲyöÄÓ˚*˛õ !ÓÜ,˛!ï˛ öy ÌyˆÏܲ ~ÓÇ fliyÎ˚# Ä «˛îfliyÎ˚#ñ í˛zû˛Î˚ ≤ÃܲyÓ˚ Óò%ƒ!ï˛Ü˛ §ÇˆÏܲˆÏï˛Ó˚
çöƒ•z ˆÎö í˛zͲõߨ ¢ˆÏ∑Ó˚ x!Óܲ°ï˛y ÓçyÎ˚ ÌyˆÏܲ–
(d) í˛zͲõߨ ¢ˆÏ∑Ó˚ !Ó!ܲÓ˚î x!ò‹Tñ xÌ≈yÍ !Ó!û˛ß¨ !òˆÏÜ §üû˛yˆÏÓ Ó!rê˛ï˛ •ÄÎ˚y ã˛y•z–
(e) !ö!ò≈‹T ¢∑ í˛zͲõyòö «˛üï˛yÓ˚ çöƒ °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ xyܲyÓ˚ ˆÎö ÎÌy§Ω˛Ó ˆåÈyê˛ •Î˚–
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ üˆÏôy °yí˛zí˛!flõܲyÓ˚ñ ÎyˆÏܲ xˆÏöܲ §üÎ˚ Ú!flõܲyÓ˚Û Ó°y •Î˚ñ !Ó!û˛ß¨ ˆû˛Ôï˛ âê˛öyˆÏܲ !û˛!_
ܲˆÏÓ˚ !ö!ü≈ï˛ •ˆÏï˛ ˛õyˆÏÓ˚– ~áyˆÏö xyüÓ˚y §Ó≈y!ôܲ˛ ≤Ãã˛!°ï˛ ã˛°Ü%˛[˛°# °yí˛zí˛ !flõܲyˆÏÓ˚Ó˚ àë˛ö Ä Ü˛yÎ≈ö#!ï˛Ó˚
!ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
ã˛°Ü%˛[˛°# °yí˛zí˛!flõܲyÓ˚ ã˛°Ü%˛[˛°# °yí˛zí˛!flõܲyÓ˚ ã˛°Ü%˛[˛°# °yí˛zí˛!flõܲyÓ˚ ã˛°Ü%˛[˛°# °yí˛zí˛!flõܲyÓ˚ ã˛°Ü%˛[˛°# °yí˛zí˛!flõܲyÓ˚ (Moving coil loudspeaker) : ~!ê˛Ó˚ àë˛ö xˆÏöܲê˛y ã˛°Ü%˛[˛°#
üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ üˆÏï˛y– !ã˛e 13.16 °«˛ƒ ܲÓ˚ˆÏ° xy˛õ!ö ˆòáˆÏï˛ ˛õeyˆÏÓö ˆÎñ ~Ó˚ ã%˛¡∫ܲ!ê˛ Óy!ê˛Ó˚ xyÜ,˛!ï˛ñ
~ܲ!ê˛ ˆüÓ˚& Ó°Î˚yÜ,˛!ï˛ ~ÓÇ Óy!ê˛Ó˚ x«˛ ÓÓ˚yÓÓ˚ ~ܲ!ê˛ ò[˛•z xöƒ ˆüÓ˚&– ¢_´
ܲyàˆÏçÓ˚ ˜ï˛!Ó˚ ~ܲ!ê˛ ¢B%˛ xyÜ,˛!ï˛Ó˚ ˛õò≈yÓ˚ §ˆÏD xyê˛Ü˛yˆÏöy fl∫Ó˚Ü%˛[˛°# ò%!ê˛ ˆã˛Ô¡∫ܲ
ˆüÓ˚&Ó˚ üˆÏôƒ §üy«˛û˛yˆÏÓ ÌyˆÏܲ– ¢B%˛Ó˚ Óy•zˆÏÓ˚Ó˚ ≤Ãyhsˇ ì˛í˛z á°yˆÏöy (corrugated)
~ÓÇ ï˛yÓ˚ ˆÓí˛¸!ê˛ xyÓk˛ ÌyˆÏܲ– ˛õò≈yÓ˚ ˆÜ˛w#Î˚ xMÈ˛°!ê˛Ä ˆì˛í˛z ˆá°yˆÏöy ܲyàç
Óy xöƒ ˆÜ˛yöÄ öÓ˚ü ˛õò≈yÓ˚ §y•yˆÏ΃ !flõܲyˆÏÓ˚Ó˚ ≤ÈÏܲyˆÏ¤˛Ó˚ §ˆÏD xyê˛ˆÏܲ Ó˚yáy
•Î˚–
fl∫Ó˚Ü%˛[˛°#ˆÏï˛ ¢∑§ÇˆÏܲˆÏï˛Ó˚ !Óò%ƒÍ≤ÃÓy• ˛õyë˛yˆÏöy •ˆÏ° ˆã˛Ô¡∫ܲˆÏ«˛e ≤ÃÓyˆÏ•Ó˚
!òܲ Ä üyö xö%ÎyÎ˚# Ü%˛[˛°#Ó˚ í˛z˛õÓ˚ Ó° ≤ÈÏÎ˚yà ܲˆÏÓ˚– ~Ó˚ ú˛ˆÏ° fl∫Ó˚Ü%˛[˛°#!ê˛ ~ÓÇ
ï˛yÓ˚ §ˆÏD §ü@ˇÃ ˛õò≈y!ê˛ §üy«˛û˛yˆÏÓ §yüˆÏö !˛õåȈÏö ܲ!¡õï˛ •Î˚– ~•z ܲ¡õö•z
!flõܲyÓ˚ ˆÌˆÏܲ ¢∑ í˛zͲõߨ ܲˆÏÓ˚– ¢B%˛ ˛õò≈y!ê˛Ó˚ Óƒy§ ˆÓ¢ Óí˛¸ñ ≤ÃyÎ˚ 20-25 cm
˛õÎ≈hsˇ •ÄÎ˚yÎ˚ ˛õò≈yÓ˚ ܲ¡õö xˆÏöܲê˛y ÓyÎ˚%ˆÏï˛ §MÈ˛y!°ï˛ •Î˚ñ ÎyÓ˚ ú˛ˆÏ° í˛zͲõߨ ¢∑!ê˛ xyüÓ˚y ˆÓ¢ ˆçyˆÏÓ˚
÷öˆÏï˛ ˛õy•z–
!ö¡¨ ܲ¡õyˆÏB˛ ¢B%˛ ˛õò≈y!ê˛Ó˚ §Óê˛y•z ~ܲˆÏe ܲ!¡õï˛ •Î˚– !ܲv í˛zFã˛ Ü˛¡õyˆÏB˛ ˛õò≈y!ê˛ ~ܲ Óy ~ܲy!ôܲ !öflõ®
Ó,_ myÓ˚y !Óû˛_´ •Î˚– Óy•zˆÏÓ˚Ó˚ xMÈ˛°=!°ˆÏï˛ Ü˛¡õˆÏöÓ˚ !ÓhflÏyÓ˚ ܲü ÌyˆÏܲ– ¢∑ ≤Ãôyöï˛ ˆÜ˛w#Î˚ xMÈ˛° ˆÌˆÏܲ•z
!Óܲ#î≈ •Î˚– §yôyÓ˚îï˛ ~ܲ!ê˛ °yí˛zí˛!flõܲyÓ˚ í˛zFã˛ Ä !ö¡¨ í˛zû˛Î˚ ܲ¡õyˆÏB˛ ΈÏÌ‹T ¢∑«˛üï˛y !Óܲ#î≈ ܲÓ˚ˆÏï˛
˛õyˆÏÓ˚ öy– ~çöƒ ~ܲ•z §ˆÏD xhsˇï˛ ò%•z!ê˛ !flõܲyÓ˚ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ ÎyÓ˚ üˆÏôƒ !ö¡¨ ܲ¡õyˆÏB˛Ó˚ !flõܲyÓ˚!ê˛Ó˚
¢B%˛ ˛õò≈y Óí˛¸ ~ÓÇ í˛zFã˛ Ü˛¡õyˆÏB˛Ó˚ !flõܲyÓ˚!ê˛Ó˚ ¢B%˛˛õò≈y xˆÏ˛õ«˛yÜ,˛ï˛ ˆåÈyê˛ •Î˚– ~ܲ!ê˛ •zˆÏ°Ü˛!ê˛Δ!öܲ !öÓ≈yã˛Ü˛
Óï≈˛ö# !ÓÓô≈ܲ ˆÌˆÏܲ xy§y ¢∑ §ÇˆÏܲˆÏï˛Ó˚ !ö¡¨ Ä í˛zFã˛ Ü˛¡õyB˛ í˛z˛õyÇ¢=!°ˆÏܲ ˛õ,Ìܲ ܲˆÏÓ˚ ˆ§=!°ˆÏܲ
ÎÌyÎÌû˛yˆÏÓ ò%•z !flõܲyˆÏÓ˚ ã˛y!°ï˛ ܲˆÏÓ˚– xy˛õ!ö •Î˚ï˛ °yí˛zí˛!flõܲyÓ˚ ÓyˆÏ: Óí˛¸ í˛zú˛yÓ˚ (woofer, Ü%˛Ü%˛ˆÏÓ˚Ó˚
í˛yˆÏܲÓ˚ §yò,¢V Ä ˆåÈyê˛ ê%˛•zê˛yÓ˚ (tweeter, ˛õy!áÓ˚ í˛yˆÏܲÓ˚ §ò,¢V !flõܲyÓ˚ ˆòˆÏáˆÏåÈö– ~=!°Ó˚ ≤ÈÏÎ˚yçö#Î˚ï˛y
~ÓyÓ˚ xy˛õ!ö í˛z˛õ°!∏˛ ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏåÈö–
!ã˛e ≠ 13.16
348
~ÓyÓ˚ üy•zˆÏe´yˆÏú˛yö ~ÓÇ °yí˛zí˛!flõܲyÓ˚ §¡∫ˆÏ¶˛ ~ܲ!ê˛ xö%¢#°ö#Ó˚ çÓyÓ !òö–
xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#xö%¢#°ö#ÈÙÈ ÈÙÈ ÈÙÈ ÈÙÈ ÈÙÈ 4 :
(i) ôyÓ˚ܲ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ï%˛ö°yÎ˚ ܲyÓ≈ö üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ ~ܲ!ê˛ §%!Óôy Ä ~ܲ!ê˛ x§%!Óôy !°á%ö–
(ii) í˛z_ü üy•zˆÏe´yˆÏú˛ˆÏöÓ˚ ˆÜ˛yö‰ ˆÜ˛yö‰ =î Ìyܲy í˛z!ã˛ï˛⁄
(iii) °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ ¢∑ÈÙÈí˛zͲõyòö §yí˛¸yñ í˛zͲõߨ ¢ˆÏ∑Ó˚ !ÓÜ,˛!ï˛ Ä !Ó!ܲÓ˚ˆÏîÓ˚ !ò‹Tï˛y ˆÜ˛üö •ÄÎ˚y í˛z!ã˛ï˛
ӈϰ xy˛õ!ö üˆÏö ܲˆÏÓ˚ö⁄
(iv) í˛zû˛yÓ˚ Ä ê%˛•zê˛yÓ˚ !flõܲyˆÏÓ˚Ó˚ ܲyç ܲ#⁄
13.7 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
~•z ~ܲܲ!ê˛ˆÏï˛ ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ≤Ãã˛!°ï˛ ≤ÃÎ%!_´=!°Ó˚ §¡∫ˆÏ¶˛ ~ܲ!ê˛ §Ç!«˛Æ !ÓÓÓ˚î ˆòÄÎ˚y
•ˆÏÎ˚ˆÏåÈ– ≤Ã̈Ïü•z |ö!ÓÇ¢ ¢ï˛y∑#Ó˚ üyé˛yüy!é˛ ˆÌˆÏܲ xyç ˛õÎ≈hsˇ ¢∑ ≤ÃÎ%!_´Ó˚ ˆÎ §Ó Î%àyhsˇÜ˛yÓ˚# xy!Ó‹ÒyÓ˚
Ä í˛zߨÎ˚ö âˆÏê˛ˆÏåÈñ ˆ§=!°Ó˚ ~ܲ!ê˛ Ü˛y°yö%e´!üܲ ï˛y!°Ü˛y ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ– ~Ó˚ ˛õÓ˚ @ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛
x!û˛ˆÏ°áˆÏöÓ˚ Îy!sfܲ ˛õk˛!ï˛ñ ˆã˛Ô¡∫ܲ !ú˛ï˛yÎ˚ x!û˛ˆÏ°áˆÏöÓ˚ ˆã˛Ô¡∫ܲ ˛õk˛!ï˛ ~ÓÇ §Óyܲ ã˛°!Fã˛ˆÏeÓ˚ !ú˛Õ√ Ä
xyˆÏ°yܲ#Î˚ !í˛ˆÏflÒ x!û˛ˆÏ°áˆÏöÓ˚ xyˆÏ°yܲ#Î˚ ˛õk˛!ï˛Ó˚ ˛õ,Ìܲ ˛õ,Ìܲ Óî≈öy ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ– ≤Ã!ï˛!ê˛ ˆ«˛ˆÏe ¢∑
˛õ%öç≈öˆÏöÓ˚ ÓƒÓfliy!ê˛Ä xyüÓ˚y xyˆÏ°yã˛öy ܲˆÏÓ˚!åÈ–
xyˆÏ°yܲ#Î˚ !í˛ˆÏflÒ ˆÎ !í˛!çê˛ƒy° ˛õk˛!ï˛ ÓƒÓ•*ï˛ •Î˚ ï˛y ˆã˛Ô¡∫ܲ !ú˛ï˛y ~ÓÇ ã˛°!Fã˛ˆÏeÓ˚ !ú˛ˆÏÕ√Ä ÓƒÓ•*ï˛
•ˆÏFåÈ– ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ ˆ«˛ˆÏe ~!ê˛•z Óï≈˛üyˆÏö ≤Ãã˛!°ï˛ ˛õk˛!ï˛– §%ï˛Ó˚yÇñ öï%˛ö ~•z ≤ÃÎ%!_´Ó˚
§ˆÏD xy˛õöyˆÏܲ ˛õ!Ó˚!ã˛ï˛ ܲÓ˚ˆÏï˛ ~Ó˚ ü)° ö#!ï˛!ê˛ ˛õ,Ìܲû˛yˆÏÓ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
ˆÎ ˆÜ˛yö ¢∑ ≤ÃÎ%!_´ˆÏï˛•z üy•zˆÏe´yˆÏú˛yö ~ÓÇ °yí˛zí˛!flõܲyÓ˚ xï˛ƒhsˇ =Ó˚&c˛õ)î≈ Ä x˛õ!Ó˚•yÎ≈ xyö%£Ï!Dܲ
Îsf– ï˛y•z ~=!° §¡∫ˆÏ¶˛Ä ˛õ,Ìܲû˛yˆÏÓ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
~•z ~ܲˆÏܲ ÎsfyÓ°#Ó˚ !Ó!û˛ß¨ xLjϢÓ˚ ˛õ%Cyö%˛õ%C Óî≈öyÓ˚ ˆã˛ˆÏÎ˚ ˛õòyÌ≈!ÓòƒyÓ˚ ò,!‹TˆÏܲyî ˆÌˆÏܲ ˆ§=!°Ó˚
ܲyÎ≈ö#!ï˛Ó˚ ÓƒyáƒyÎ˚ ˆÓ!¢ ˆçyÓ˚ ˆòÄÎ˚y •ˆÏÎ˚ˆÏåÈ– ~!ê˛ •Î˚ï˛ xy˛õöyÓ˚ ò,!‹T ~í˛¸yÎ˚!ö–
13.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. @ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛ ¢∑ x!û˛ˆÏ°áö ˛õk˛!ï˛Ó˚ §Ç!«˛Æ Óî≈öy !òö–
2. Îy!sfܲ ˛õ%öç≈öö ÓƒÓfliyÎ˚ !˛õܲÈÙÈxy˛õ !öü≈yˆÏî ˆÎ ≤ÃÎ%!_´ ÓƒÓ•*ï˛ •Î˚ñ üy•zˆÏe´yˆÏú˛yö ÎsfÄ ˆ§•z
≤ÃÎ%!_´=!°Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚•z ܲyç ܲˆÏÓ˚– ÈÙÙÙÈ ~•z í˛z!_´Ó˚ §üÌ≈ˆÏö Î%!_´ ˆòáyö–
3. @ˇÃyüyˆÏú˛yö ˆÓ˚ܲˆÏí≈˛Ó˚ ï%˛°öyÎ˚ ˆã˛Ô¡∫ܲ !ú˛ï˛y ÓƒÓ•yˆÏÓ˚Ó˚ §%!Óôy ܲ# ܲ#⁄
349
4. ˆê˛˛õˆÏÓ˚ܲí≈˛yÓ˚ ΈÏsfÓ˚ ܲyÎ≈≤Ãîy°# §ÇˆÏ«˛ˆÏ˛õ Óî≈öy ܲÓ˚&ö–
5. §Óyܲ ã˛°!Fã˛ˆÏe ܲ#û˛yˆÏÓ ¢ˆÏ∑Ó˚ x!û˛ˆÏ°áö Ä ˛õ%öç≈öö ܲÓ˚y •Î˚⁄
6. ¢∑ x!û˛ˆÏ°áö Ä ˛õ%öç≈öˆÏöÓ˚ !í˛!çê˛ƒy° ˛õk˛!ï˛Ó˚ !ÓÓÓ˚î !òö–
7. ܲü˛õƒyQ !í˛flÒ Óy xyˆÏ°yܲ#Î˚ !í˛ˆÏflÒ !ܲû˛yˆÏÓ ¢∑ x!û˛!°!áï˛ •Î˚ ~ÓÇ ~=!° ˆÌˆÏܲ ܲ#û˛yˆÏÓ ¢ˆÏ∑Ó˚
˛õ%öç≈öö âê˛yˆÏöy •Î˚⁄
8. ≤Ãã˛!°ï˛ ˆÜ˛yöÄ ~ܲ ôÓ˚ˆÏöÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈ö#!ï˛Ó˚ Óî≈öy !òö–
9. §yôyÓ˚î °yí˛zí˛!flõܲyˆÏÓ˚Ó˚ àë˛ö Ä Ü˛yÎ≈˛õk˛!ï˛ Óî≈öy ܲÓ˚&ö–
13.9 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠
1. (i) ˜ÏÓò%ƒ!ï˛Ü˛ !˛õܲÈÙÈxy˛õ ≤Ãôyöï˛ ã˛yÓ˚ ≤ÃܲyˆÏÓ˚Ó˚ •Î˚ ≠ (a) ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ñ (b) ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ Óy
!˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲñ (c) ˆÓ˚yô Ä (d) ôyÓ˚ܲ#Î˚–
(ii) ¢ˆÏ∑Ó˚ ˛õÎ≈yÎ˚ ܲy°
1512
ˆ§ˆÏܲu˛– ~•z §üˆÏÎ˚ ˆÓ˚ܲí≈˛!ê˛
θ
=
331 260512
π ××
= 0·0067 ˆÓ˚!í˛Î˚yö
ˆÜ˛y Ïî â%Ó ÏÓ– §%ï˛ÓyÇñ ¢ Ï∑Ó ~ܲ!ê˛ ˛õ)î≈ ˛õÎ≈yÎñ 0·0067 × 12 xÌ≈yÍ 0·081cm ˜òâ≈ƒ ç% Ïí˛ Ìyܲ ÏÓ–
2. (i) ˆ°ÔˆÏ•Ó˚ x:y•zí˛ Fe0O3 Ä ˆe´y!üÎ˚yü í˛y•zÈÙÈx:y•zí˛–
(ii) ~ !§ ÓyÎ˚y§ x!û˛ˆÏ°áö ˛õk˛!ï˛ ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° §ÇˆÏܲˆÏï˛Ó˚ ï%˛°öyÎ˚ x˛õ¢ˆÏ∑Ó˚ ˛õ!Ó˚üyî xˆÏöܲ
ܲü •Î˚ ~ÓÇ ~Ó˚ ú˛ˆÏ° x!û˛ˆÏ°áˆÏöÓ˚ í˛zÍܲ£Ï≈ ÓyˆÏí˛¸–
(iii) ˆ≤’ÈÙÈÓƒyܲ ˆ•í˛ ˆÎ !Óû˛Ó í˛zͲõߨ ܲˆÏÓ˚ ï˛y §yôyÓ˚îï˛ x!û˛!°!áï˛ ¢ˆÏ∑Ó˚ ܲ¡õyˆÏB˛Ó˚ §üyö%˛õyï˛# •Î˚ñ
ÎyÓ˚ ú˛ˆÏ° í˛zFã˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑=!° ˆÓ!¢ ˆçyˆÏÓ˚ ˆ¢yöy ÎyÎ˚– ~ˆÏï˛ !ü!◊ï˛ Ü˛¡õyˆÏB˛Ó˚ fl∫Ó˚ !ÓÜ,˛ï˛
ˆ¢yöyÎ˚– ~•z !ÓÜ,˛!ï˛ ˆÓ˚yô ܲÓ˚yÓ˚ çöƒ•z §üyö#ܲÓ˚ˆÏîÓ˚ ≤ÈÏÎ˚yçö •Î˚–
(iv) !fiê˛!Ó˚Ä x!û˛ˆÏ°áˆÏö §yôyÓ˚î x!û˛ˆÏ°áˆÏöÓ˚ !m=î §ÇáƒÜ˛ ¢∑!°!˛õÓ˚ ≤ÈÏÎ˚yçö •Î˚ xÌ≈yÍñ
ˆ§yçyü%ˆÏá ò%•z!ê˛ Ä !Ó˛õÓ˚#ï˛ü%ˆÏá ò%•z!ê˛ñ ˆüyê˛ ã˛yÓ˚!ê˛ ¢∑!°!˛õ ÌyˆÏܲ–
3. (i) ˛õ!Ó˚Óï˛#≈ âöc x!û˛ˆÏ°áˆÏö ˆÓ˚áy!ã˛e!ê˛ §)- öy •ˆÏ° ï˛yÓ˚ üôƒ !òˆÏÎ˚ !ú˛ˆÏÕ√Ó˚ í˛z˛õÓ˚ xy˛õ!ï˛ï˛ xyˆÏ°y
xˆÏöܲê˛y ≤Ãfli ç%ˆÏí˛¸ ÌyܲˆÏÓ– ~Ó˚ ú˛ˆÏ° x!û˛ˆÏ°áˆÏöÓ˚ §üÎ˚ ã˛°üyö !ú˛ˆÏÕ√Ó˚ ˆÎ ˆÜ˛yöÄ !Ó®%ˆÏï˛
ˆÓ¢ !ܲå%È«˛î §üÎ˚ ôˆÏÓ˚ xyˆÏ°y ˛õí˛¸ˆÏÓ ~ÓÇ ˙ §üˆÏÎ˚Ó˚ üˆÏôƒ xyˆÏ°yÓ˚ ï˛#Ó ï˛yÎ˚ ¢∑ §ÇˆÏÜ˛ï˛ xö%ÎyÎ˚#
ܲü ˆÓ!¢ âê˛ˆÏ°Ä x!û˛!°!áï˛ ¢∑!°!˛õ!ê˛ˆÏï˛ ˙ §üˆÏÎ˚Ó˚ àí˛¸ ï˛#Ó ï˛y xö%ÎyÎ˚# ܲy!°üyÓ˚ âöc ˆòáy
ÎyˆÏÓ– ~ˆÏï˛ !ӈϢ£Ï ܲˆÏÓ˚ í˛zFã˛ï˛Ó˚ ܲ¡õyB˛=!°Ó˚ x!û˛ˆÏ°áö ÎÌyÌ≈ •ˆÏÓ öy–
(ii) ˛õy¢y˛õy!¢ ò%•z!ê˛ ¢∑!°!˛õÓ˚ ÓƒÓ•yˆÏÓ˚ xÓy!N˛ï˛ x˛õ¢∑ ܲüyˆÏöy ÎyÎ˚– ≤ÈÏÎ˚yçˆÏö ~•z ò%•z!ê˛
¢∑!°!˛õˆÏܲ !fiê˛!Ó˚ĈÏú˛y!öܲ ¢∑ x!û˛ˆÏ°áˆÏöÓ˚ çöƒ ÓƒyÓ•yÓ˚ ܲÓ˚y ÎyÎ˚–
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(iii) ܲü §ÇáƒÜ˛ !ÓˆÏê˛Ó˚ §Çáƒy ÓƒÓ•yÓ˚ ܲÓ˚ˆÏ° ï˛#Ó ï˛y Óƒy!Æ xˆÏ˛õ«˛yÜ,˛ï˛ ܲü ~ÓÇ ¢ˆÏ∑Ó˚ !ÓÜ,˛!ï˛
xˆÏ˛õ«˛yÜ,˛ï˛ ˆÓ!¢ •ˆÏï˛y–
(iv) xyˆÏ°yܲ#Î˚ !í˛ˆÏflÒ ˆáy!òï˛ àï≈˛=!° xï˛ƒhsˇ «%˛o •Î˚– ~ܲüye ˆ°çyÓ˚ Ó˚!Ÿ¬•z àï≈˛=!°Ó˚ xyܲyˆÏÓ˚Ó˚
§ˆÏD ï%˛°ö#Î˚ ≤ÃfliˆÏFåȈÏòÓ˚ •ˆÏï˛ ˛õyˆÏÓ˚–
4. (i) §%!Óôy ≠§%!Óôy ≠§%!Óôy ≠§%!Óôy ≠§%!Óôy ≠ ¢!_´¢y°# ˜Óò%ƒ!ï˛Ü˛ §ÇˆÏÜ˛ï˛–
x§%!Óôy ≠x§%!Óôy ≠x§%!Óôy ≠x§%!Óôy ≠x§%!Óôy ≠ ܲ¡õyB˛ÈÙÙÙȧyí˛¸y x§üñ ú˛ˆÏ° ¢ˆÏ∑Ó˚ x!Óܲ°ï˛y ܲü–
(ii) 13.6.1 xLjϢ xy˛õ!ö ~Ó ˛§¡õ)î≈ í˛z_Ó˚ ˛õyˆÏÓö–
(iii) Ä (iv) 13.6.2 xLjϢ ~ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ˆòˆÏá !öö–
§Ó≈ˆÏ¢£Ï ≤Ãîy°#§Ó≈ˆÏ¢£Ï ≤Ãîy°#§Ó≈ˆÏ¢£Ï ≤Ãîy°#§Ó≈ˆÏ¢£Ï ≤Ãîy°#§Ó≈ˆÏ¢£Ï ≤Ãîy°#
1. 13..3.1 xLjϢ xy˛õ!ö ~ §¡∫ˆÏ¶˛ ˛õˆÏí˛¸ˆÏåÈö–
2. 13.3.2 ~ÓÇ 13.6.1 xÇ¢ ˆòˆÏá !öö– °«˛ƒ ܲÓ˚&ö ˆÎñ !˛õܲÈÙÈxy˛õ ~ÓÇ üy•zˆÏe´yˆÏú˛yöÈÙÙÙÈ í˛zû˛Î˚
ΈÏsfÓ˚ àë˛ˆÏö•z ~ܲ•z ≤ÃÎ%!_´ ÎÌy (i) ã˛y˛õ˜ÏÓò%ƒ!ï˛Ü˛ Óy !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ !e´Î˚yó (ii) ܲyÓ≈ö òyöyÓ˚
í˛z˛õÓ˚ ã˛yˆÏ˛õÓ˚ §ˆÏD ˆÓ˚yˆÏôÓ˚ ˛õ!Ó˚Óï≈˛öó (iii) ôyÓ˚ˆÏܲÓ˚ ò%•z ˛õyˆÏï˛Ó˚ üˆÏôƒ ÓƒÓôyˆÏöÓ˚ ï˛yÓ˚ï˛ˆÏüƒÓ˚ ú˛ˆÏ°
ôyÓ˚ܲˆÏcÓ˚ ˛õ!Ó˚Óï≈˛ö Ä (iv) ï˛!í˛¸Íã%˛¡∫ܲ#Î˚ !e´Î˚y ÓƒÓ•*ï˛ •Î˚–
3 Ä 4. 13.4 xLjϢ ~ !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
5. 13.5.1 xLjϢ xy˛õ!ö ~ !ӣψÏÎ˚ ˛õˆÏí˛¸ˆÏåÈö–
6 Ä 7. 13.5.2 xö%ˆÏFåȈÏò ~ !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
8. 13.6.1 xLjϢ !Ó!û˛ß¨ ôÓ˚ˆÏöÓ˚ üy•zˆÏe´yˆÏú˛yˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈ö#!ï˛ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
9. 13.6.2 xÇ¢!ê˛ ˆòˆÏá !öö–
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~ܲܲ~ܲܲ~ܲܲ~ܲܲ~ܲܲ 14 Óy܉˛ Ä ◊&!ï˛Óy܉˛ Ä ◊&!ï˛Óy܉˛ Ä ◊&!ï˛Óy܉˛ Ä ◊&!ï˛Óy܉˛ Ä ◊&!ï˛ (Speech and Hearing)
àë˛ö
14.1 ≤ÃhflÏyÓöy
í z Ïj¢ƒ
14.2 Óy܉˛ Ä Óyà‰Îsf
14.3 ܲt˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy•
14.4 ܲyö
14.4.1 ܲyˆÏöÓ˚ àë˛ö
14.4.2 ܲyö ܲ#û˛yˆÏÓ Ü˛yç ܲˆÏÓ˚
14.4.3 ܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛y
14.4.4 ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey
14.4.5 ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y
14.4.6 !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑
14.4.7 §%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y–
14.5 fl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫ö
14.6 !mÈÙÈܲî#≈Î˚ !òܲ !öî≈Î˚
14.7 §yÓ˚yÇ¢
14.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
14.9 í˛z_Ó˚üy°y
14.1 ≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy≤ÃhflÏyÓöy
~•z ˛õyë˛e´ˆÏüÓ˚ ˛õ)Ó≈Óï˛#≈ ~ܲܲ=!°ˆÏï˛ xyüÓ˚y ¢∑ï˛Ó˚ˆÏDÓ˚ í˛zͲõ!_ñ ¢ˆÏ∑Ó˚ öyöy!Óô ôü≈ñ ~ܲfliyö ˆÌˆÏܲ
xöƒfliyˆÏö ¢∑ï˛Ó˚ˆÏDÓ˚ §MÈ˛Ó˚îñ ¢ˆÏ∑Ó˚ §ÇÓ˚«˛î Ä ˛õ%öà≈ë˛ö ~ÓÇ xyö%£Ïy!Dܲ ܲˆÏÎ˚ܲ!ê˛ !ӣψÏÎ˚ xyˆÏ°yã˛öy
ܲˆÏÓ˚!åÈ– §%ˆÏÓ˚°y ¢ˆÏ∑Ó˚ ˜Ó!¢‹Tƒ §¡õˆÏÜ≈˛ xy˛õ!ö ˆÎüö !ܲå%Èê˛y çyöˆÏï˛ ˆ˛õˆÏÓ˚ˆÏåÈöñ ˆï˛üö•z §%Ó˚•#ö ¢∑ Óy
x˛õfl∫Ó˚ ܲ#ñ ï˛yÄ xy˛õöyÓ˚ öçˆÏÓ˚ ~ˆÏ§ˆÏåÈ– !ܲv ~=!° åÈyí˛¸yÄ xyÓ˚ ~ܲ ≤ÃܲyˆÏÓ˚Ó˚ ≤ÃyÜ,˛!ï˛Ü˛ ¢∑ xyˆÏåÈ ÎyÓ˚
üyôƒˆÏü xyüÓ˚y ˛õÓ˚flõˆÏÓ˚Ó˚ üˆÏôƒ û˛yÓ !Ó!öüÎ˚ ܲ!Ó˚– xy˛õ!ö •Î˚ï˛ xö%üyö ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏåÈöñ ~áyˆÏö xyüÓ˚y
ˆÜ˛yö ôÓ˚ˆÏöÓ˚ ¢ˆÏ∑Ó˚ ܲÌy Ó°!åÈ– ~!ê˛ •° Óy܉˛ Óy xyüyˆÏòÓ˚ ܲt˛!öà≈ï˛ ¢∑– ~•z ¢∑ í˛zͲõߨ •Î˚ ~ÓÇ ~Ó˚
˜Ó!¢‹Tƒ=!° ܲ#ñ ˆ§ !ӣψÏÎ˚ xy˛õöyÓ˚ !öÿ˛Î˚•z !ܲå%Èê˛y xö%§!¶˛Í§y xyˆÏåÈ–
352
x˛õÓ˚˛õˆÏ«˛ñ ˆÎ ˆÜ˛yöÄ í˛zFã˛y!Ó˚ï˛ ¢∑ Óy §D#ˆÏï˛Ó˚ §yÌ≈ܲ ˛õ!Ó˚î!ï˛ ï˛yÓ˚ ◊ÓˆÏî– ˆÜ˛yöÄ ¢∑ x!û˛!°!áï˛
(recorded) •ˆÏ°Ä ï˛y ï˛áö•z §yÌ≈ܲ •Î˚ Îáö ˆ§•z ¢∑ ˛õ%öà≈!ë˛ï˛ •Î˚ ~ÓÇ xyüÓ˚y ˆ§!ê˛ ÷!ö– xyüyˆÏòÓ˚
◊ÓˆÏî!wÎ˚ Óy ܲˆÏî≈Ó˚ àë˛ö Ä Ü˛yÎ≈≤Ãîy°# ~ÓÇ !Ó!û˛ß¨ ܲ¡õyB˛ Ä ï˛#Ó ï˛yÓ˚ ¢ˆÏ∑Ó˚ ≤Ã!ï˛ xyüyˆÏòÓ˚ ◊ÓîÎsf
ܲ#û˛yˆÏÓ §yí˛¸y ˆòÎ˚ñ ˆ§!ê˛Ä xyüyˆÏòÓ˚ ˆÜ˛Ôï%˛•° çy@ˇÃï˛ Ü˛ˆÏÓ˚–
~•z ~ܲܲ!ê˛ˆÏï˛ xyüÓ˚y üö%£ÏƒˆÏòˆÏ•Ó˚ ¢∑§,!‹T Ä ¢∑yö%û)˛!ï˛Ó˚ ç!ê˛° xÌã˛ xï˛ƒhsˇ ܲyÎ≈ܲÓ˚# ò%•z!ê˛
ÎsfÈÙÙÙÈÓyà‰Îsf Ä Ü˛yˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈˛õk˛!ï˛ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó– ~áyˆÏö xy˛õ!ö !ӈϢ£Ï ܲˆÏÓ˚ °«˛ƒ ܲÓ˚ˆÏÓö
ˆÎñ xöƒ ˆÎ ˆÜ˛yöÄ ¢ˆÏ∑Ó˚ üï˛ üyö%ˆÏ£ÏÓ˚ ܲt˛fl∫Ó˚Ä üy•zˆÏe´yˆÏú˛yöñ xƒyü!≤’ú˛yÎ˚yÓ˚ñ x!§ˆÏ°yˆÏflÒy˛õ ≤Ãû,˛!ï˛ ÎˆÏsfÓ˚
§y•yˆÏ΃ §¡õ)î≈ Óƒ!_´ !öÓ˚ˆÏ˛õ«˛û˛yˆÏÓ !ӈϟ’£Ïî ܲÓ˚y §Ω˛Ó– !ܲv ˆÜ˛yö ¢∑ üyö%ˆÏ£ÏÓ˚ ◊ÓˆÏî!wÎ˚ myÓ˚y ◊&ï˛
Ä ˜Óò%ƒ!ï˛Ü˛ §ˆÏB˛ˆÏï˛Ó˚ Ó˚*ˆÏ˛õ öyû≈˛ §ü)ˆÏ•Ó˚ myÓ˚y ü!hflÏˆÏ‹Ò ˆ≤Ã!Ó˚ï˛ •ˆÏÎ˚ ˆÜ˛yö‰ xö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚ñ ï˛yÓ˚ Óî≈öy
Ä !ӈϟ’£Ïî ˆÜ˛Ó°üye ˆ◊yï˛yÓ˚ í˛z˛õ°!∏˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚•z ܲÓ˚y §Ω˛Ó– ~•z Óî≈öy Óy !ӈϟ’£Ïî ~ܲyhsˇ•z
Óƒ!_´!öû≈˛Ó˚ ~ÓÇ ~=!° ~ܲ Óƒ!_´ ˆÌˆÏܲ xöƒ Óƒ!_´Ó˚ ˆ«˛ˆÏe !ܲå%Èê˛y !û˛ß¨ •ÄÎ˚y x§Ω˛Ó öÎ˚–
í˛zˆÏj¢ƒ Éí˛zˆÏj¢ƒ Éí˛zˆÏj¢ƒ Éí˛zˆÏj¢ƒ Éí˛zˆÏj¢ƒ É
~•z ~ܲܲ!ê˛ ˛õyë˛ Ü˛Ó˚ˆÏ°ñ xy¢y ܲÓ˚y ÎyÎ˚ ˆÎ xy˛õ!öÈÙÙÙÈ
üyö%ˆÏ£ÏÓ˚ Óyà‰Îsf ~ÓÇ ◊ÓîÎsf xÌ≈yÍ Ü˛yˆÏöÓ˚ àë˛ö Óî≈öy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ~=!°Ó˚ ≤Ã!ï˛!ê˛ xLjϢÓ˚
ܲyÎ≈ Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
ܲt˛fl∫ˆÏÓ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ í˛z˛õyLjϢÓ˚ í˛z˛õ!fli!ï˛ ~ÓÇ ˆ§=!°Ó˚ üˆÏôƒ «˛üï˛yÓ˚ Órê˛ˆÏöÓ˚ !ÓÓÓ˚î !òˆÏï˛
˛õyÓ˚ˆÏÓö–
!Ó!û˛ß¨ ܲ¡õyˆÏB˛ ˆÜ˛yö‰ !ö¡¨ï˛ü ï˛#Ó ï˛yÓ˚ ¢∑ xyüÓ˚y ÷öˆÏï˛ §«˛ü ••z ~ÓÇ ˆÜ˛yö‰ ï˛#Ó ï˛yÎ˚ ¢∑ xyüyˆÏòÓ˚
ܲˆÏî≈ xfl∫!hflÏÓ˚ §,!‹T Óy «˛!ï˛§yôö ܲˆÏÓ˚ñ ˆ§ !ӣψÏÎ˚ !öˆÏç xÓ!•ï˛ •ˆÏï˛ Ä x˛õÓ˚ˆÏܲ xÓ!•ï˛ ܲÓ˚ˆÏï˛
˛õyÓ˚ˆÏÓö–
¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ §ˆÏD ≤ÃyÓ°ƒüyey Ä ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛yˆÏܲ §¡õ!Ü≈˛ï˛ ܲÓ˚ˆÏï˛ñ ò%!ê˛Ó˚ üˆÏôƒ ˛õyÌ≈ܲƒ
Óƒyáƒy ܲÓ˚ˆÏï˛ ~ÓÇ Úú˛öÛ Ä Úˆ§yöÛ ~ܲܲ=!°Ó˚ §ÇK˛y !òˆÏï˛˛ ˛õyÓ˚ˆÏÓö–
!ü◊ ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ Órê˛ö çyöy ÌyܲˆÏ° ≤Ãï˛ƒy!¢ï˛ ≤ÃÓ°ï˛y àîöy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö–
§%ˆÏÓ˚Ó˚ ܲ¡õyB Ä ï˛#«¯˛ï˛yÓ˚ üˆÏôƒ §¡õÜ≈˛ Óƒyáƒy ܲÓ˚ˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ ï˛#«¯˛ï˛yÓ˚ ~ܲܲ Úˆü°Û ~Ó˚
§ÇK˛y !òˆÏï˛ ˛õyÓ˚ˆÏÓö–
fl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫ˆÏöÓ˚ í˛zͲõ!_Ó˚ ˜ÓK˛y!öܲ Óƒyáƒy !òˆÏï˛ ˛õyÓ˚ˆÏÓö ~ÓÇ
xyüÓ˚y ܲ#û˛yˆÏÓ ò%•z ܲyˆÏöÓ˚ §y•yˆÏ΃ ¢∑ í˛z͈ϧÓ˚ xÓfliyö !öî≈Î˚ ܲ!Ó˚ ï˛y ˆÓyé˛yˆÏï˛ ˛õyÓ˚ˆÏÓö–
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14.2 Óy܉˛ Ä Óyà‰ÎsfÓy܉˛ Ä Óyà‰ÎsfÓy܉˛ Ä Óyà‰ÎsfÓy܉˛ Ä Óyà‰ÎsfÓy܉˛ Ä Óyà‰Îsf
ç!ê˛° ˆò•ÎˆÏsfÓ˚ üˆÏôƒ üyö%ˆÏ£ÏÓ˚ Óyà‰Îsf !öˆÏç•z ~ܲ!ê˛ xï˛ƒhsˇ ܲyÎ≈ܲÓ˚ Îy!sfܲ ÓƒÓfliy– xy˛õ!ö Î!ò ~ܲÓyÓ˚
ˆû˛ˆÏÓ ˆòˆÏáö ˆÎñ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ Ä ï˛#Ó ï˛yÓ˚
Ü˛ï˛ !Ó!ã˛e Ó˚ܲˆÏüÓ˚ ¢∑ ~•z Óyà‰Îsf ˆÌˆÏܲ !öà≈ï˛
•Î˚ ï˛ˆÏÓ xy˛õ!ö ~•z Îsf!ê˛Ó˚ ܲü≈Ü%˛¢°ï˛yÎ˚ xyÿ˛Î≈
öy •ˆÏÎ˚ ˛õyÓ˚ˆÏÓö öy– 14.1 !ã˛ˆÏe üyö%ˆÏ£ÏÓ˚
Óyà‰ÎˆÏsfÓ˚ !Ó!û˛ß¨ xÇ¢=!° ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
¢∑ í˛zͲõyòˆÏö ¢!_´Ó˚ ü)° í˛zͧ xyüyˆÏòÓ˚
Ó%ˆÏܲÓ˚ üyǧˆÏ˛õ!¢– ~•z üyǧˆÏ˛õ!¢Ó˚ §ÇˆÏܲyã˛ˆÏöÓ˚
ú˛ˆÏ° ú%˛§ú%˛§ ˆÌˆÏܲ ÓyÎ˚%ˆÏflÀyï˛ Ÿªy§öy°#Ó˚
(trachea) üôƒ !òˆÏÎ˚ Óyà‰Îsf Óy °ƒy!Ó˚ǧÈÙÈ~
(larynx) ≤ÈÏÓ¢ ܲˆÏÓ˚– ~•z ÓyÎ˚%ˆÏflÀyˆÏï˛Ó˚ ≤ÃÓy•
!öÎ˚sfˆÏîÓ˚ myÓ˚y xÌ≈yÍ ÓyÎ˚%Ó˚ ˆÓà Ä ã˛yˆÏ˛õÓ˚
ï˛yÓ˚ï˛üƒ â!ê˛ˆÏÎ˚•z xyüÓ˚y ¢∑ í˛zͲõyòö ܲ!Ó˚–
ÓyÎ˚%ˆÏflÀyˆÏï˛Ó˚ ~•z !öÎ˚sfî (modulation)
ò%Ûû˛yˆÏÓ âê˛yˆÏöy •Î˚– °ƒy!Ó˚LjϧÓ˚ í˛z˛õˆÏÓ˚Ó˚ ≤Ãyhsˇ
ò%!ê˛ ˛õò≈yÓ˚ üˆÏï˛y fl∫Ó˚ï˛sf# (vocal cord) !òˆÏÎ˚ xyFåÈy!òï˛ ÌyˆÏܲ– fl∫Ó˚ï˛sf# ò%!ê˛Ó˚ üyé˛áyˆÏö ˆÎ ˆÓ˚áy!åÈo (slit)
ÌyˆÏܲñ ˆ§!ê˛Ó˚ ˆáy°y Ä Ó¶˛ •ÄÎ˚yÓ˚ ú˛ˆÏ° °ƒy!Ó˚LjϧÓ˚ üôƒ !òˆÏÎ˚ ≤ÃÓy!•ï˛ ÓyÎ˚%ˆÏflÀyˆÏï˛ ï˛yÓ˚ï˛üƒ âˆÏê˛– xyüyˆÏòÓ˚
§yôyÓ˚î ܲÌyÓyï≈˛yñ Óy àyö àyÄÎ˚yÓ˚ §üÎ˚ ~•z fl∫Ó˚ï˛sf#=!° ܲyç ܲˆÏÓ˚ ~ÓÇ ~•z ¢∑=!°ˆÏܲ xyüÓ˚y ˆây£Ïôù!ö
(voice) Ó!°– fl∫Ó˚ï˛sf#Ó˚ üôƒ !òˆÏÎ˚ !öà≈ï˛ ÓyÎ˚%ˆÏflÀyˆÏï˛Ó˚ ¢∑ã˛yˆÏ˛õÓ˚ xÌ≈yÍ àí˛¸ ÓyÎ˚%ã˛yˆÏ˛õÓ˚ x!ï˛!Ó˚_´ ã˛yˆÏ˛õÓ˚
ܲÓ˚yï˛ÈÙÈòyÑï˛ ï˛Ó˚ˆÏDÓ˚ üˆÏï˛y ï˛yÓ˚ï˛üƒ âˆÏê˛–
§üˆÏÎ˚Ó˚ §ˆÏD ~•z ¢∑ã˛yˆÏ˛õÓ˚ Äë˛yöyüy 14.2(a) ˆ°á!ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ã˛yˆÏ˛õÓ˚ ~•z ï˛yÓ˚ï˛ˆÏüƒÓ˚ ܲ¡õyB˛
fl∫Ó˚ï˛sf#Ó˚ ê˛yö Ä ˜òˆÏâ≈ƒÓ˚ í˛z˛õÓ˚ !öû≈˛Ó˚
ܲˆÏÓ˚– ~ !ӣψÏÎ˚ •Î˚ï˛ ê˛yöy ï˛yˆÏÓ˚Ó˚
§ˆÏD fl∫Ó˚ï˛sf#Ó˚ xyã˛Ó˚ˆÏî xy˛õ!ö !ü°
°«˛ƒ ܲÓ˚ˆÏÓö– fl∫Ó˚ï˛sf#Ó˚ òâ≈ƒ õ%Ó˚&ˆÏ£ÏÓ˚
ˆ«˛ˆÏe ≤ÃyÎ˚ 2.5 cm ~ÓÇ öyÓ˚#Ó˚ «˛ˆÏe
1.5 cm •Î˚– ~•z ˜òâ≈ƒ ܲü •ÄÎ˚yÓ˚
ú˛ˆÏ° öyÓ˚#ܲˆÏt˛Ó˚ ܲ¡õyB˛ fl∫yû˛y!Óܲû˛yˆÏÓ ˛õ%Ó˚&ˆÏ£ÏÓ˚ ï%˛°öyÎ˚ ˆÓ!¢ •Î˚– fl∫Ó˚ï˛sf#Ó˚ ê˛yö Óyí˛¸ˆÏ° í˛zFã˛y!Ó˚ï˛ fl∫ˆÏÓ˚Ó˚
ï˛#«¯˛ï˛y ÓyˆÏí˛¸ ~ÓÇ àyö àyÄÎ˚yÓ˚ §üÎ˚ xyüÓ˚y xû˛ƒy§Ó¢ï˛•z fl∫Ó˚ï˛sf#Ó˚ ê˛yö ܲ!üˆÏÎ˚ Óy!í˛¸ˆÏÎ˚ Ìy!ܲ–
354
fl∫Ó˚ï˛sf# ˆÌˆÏܲ !öà≈ï˛ •ÄÎ˚yÓ˚ ˛õÓ˚ !öÎ˚!sfï˛ ã˛yˆÏ˛õÓ˚ ÓyÎ˚% ܲt˛ (throat) ü%áà•ùÓ˚ Ä öy!§Ü˛y à•ùˆÏÓ˚ ≤ÈÏÓ¢
ܲˆÏÓ˚– ~•z à•ùÓ˚=!°Ó˚ xyܲyÓ˚ Ä xyÜ,˛!ï˛ !ç•ùyñ Ĥ˛ Ä òˆÏhsˇÓ˚ myÓ˚y ˛õ!Ó˚Ó!ï≈˛ï˛ ܲÓ˚y ÎyÎ˚ ~ÓÇ ï˛yÓ˚ myÓ˚y
ÓyÎ˚%≤ÃÓy•ˆÏܲ xyÓ˚Ä !öÎ˚!sfï˛ Ü˛ˆÏÓ˚ xyüÓ˚y öyöy ôÓ˚ˆÏöÓ˚ ˆây£ôù!ö í˛zͲõߨ ܲ!Ó˚– xy˛õ!ö !öˆÏç ܲÌy Ó°yÓ˚
§üÎ˚ !ç•ùyñ Ĥ˛ Ä òˆÏhsˇÓ˚ xÓfliyˆÏöÓ˚ ˛õ!Ó˚Óï≈˛ö °«˛ƒ ܲˆÏÓ˚ ˆòáˆÏï˛ ˛õyˆÏÓ˚ö– ÚxyÛ fl∫Ó˚ôù!ö í˛zFã˛yÓ˚ˆÏîÓ˚ §üÎ˚
¢∑ã˛yˆÏ˛õ ˆÎ ôÓ˚ˆÏöÓ˚ ˛õ!Ó˚Óï≈˛ö âˆÏê˛ ï˛y 14.2(b) !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ–
ˆây£Ïôù!ö åÈyí˛¸yÄ xyüÓ˚y Ÿªy§ôù!ö (breath sound) Óy !ú˛§!ú˛§y!ö !•§yˆÏÓ ˛õ!Ó˚!ã˛ï˛ xyÓ˚ ~ܲ ôˆÏÓ˚öÓ˚
¢∑ ܲˆÏÓ˚ Ìy!ܲ– ~=!°Ó˚
çöƒ fl∫Ó˚ï˛sf#ˆÏܲ ˆÜ˛yöÄ
ܲyç ܲÓ˚ˆÏï˛ •Î˚ öy–
ÓyÎ˚ %ˆ Ïfl À y Ïï˛ §Ó˚y§!Ó ˚
ü%áà•ùÓ˚ ≤Ãû,˛!ï˛Ó˚ üôƒ
!òˆÏÎ˚ !öÎ˚!sfï˛ •ˆÏÎ˚ !öà≈ï˛
•Î˚ ~ÓÇ xˆÏöܲ x“ ï˛#Ó ï˛yÓ˚ ¢∑ í˛zͲõߨ ܲˆÏÓ˚– ÓyÎ˚% ≤ÃÓyˆÏ•Ó˚ â£Ï≈î myÓ˚y (fricative) í˛zͲõߨ ÓƒOöôù!ö
Úú˛Û (f) Ä Ú§Û (s) ~ÓÇ flõ¢≈Óî≈ Ú˛õ‰ñ ê‰˛ Ä Ü‰˛ ÈÙÈ Ä ~•zû˛yˆÏÓ í˛zͲõߨ •Î˚ xÌ≈yÍ ~=!°Ó˚ ˆ«˛ˆÏe fl∫Ó˚ï˛sf#=!°
!ö!‹;˛Î˚ ÌyˆÏܲ–
14.3 ܲt˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy•Ü˛t˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy•Ü˛t˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy•Ü˛t˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy•Ü˛t˛fl∫ˆÏÓ˚ «˛üï˛yÓ˚ Ó!•É≤ÃÓy• (Power output) :
xy˛õ!ö çyˆÏöö ˆÎñ ¢∑ï˛Ó˚D ¢!_´ ˛õ!Ó˚Óy!•ï˛ ܲˆÏÓ˚– ~çöƒ ¢∑ §,!‹T ܲÓ˚yÓ˚ çöƒ Ó_´yˆÏܲ «˛üï˛y ÓƒÎ˚
ܲÓ˚ˆÏï˛ •Î˚– §yôyÓ˚î ܲˆÏÌy˛õÜ˛ÌˆÏöÓ˚ §üÎ˚ ˆÎ «˛üï˛y ÓƒÎ˚ •Î˚ ï˛yÓ˚ üyö x!ï˛ o&ï˛ Äë˛yöyüy ܲˆÏÓ˚– ܲˆÏÎ˚ܲ
ˆ§ˆÏܲu˛ §üˆÏÎ˚Ó˚ çöƒ àí˛¸ !öî≈Î˚ ܲˆÏÓ˚ ˆòáy ÎyÎ˚ ˆÎñ §yôyÓ˚î ܲÌyÓyï≈˛yÎ˚ ≤ÃyÎ˚ 10μ W Süy•zˆÏe´yÄÎ˚yê˛V «˛üï˛y
¢∑ myÓ˚y Óy!•ï˛ •Î˚– fl∫Ó˚ï˛sf#Ó˚ IJõÓ˚ §•ö«˛üï˛yÓ˚ x!ï˛!Ó˚_´ ˛õ#í˛¸ö §,!‹T öy ܲˆÏÓ˚ xyüÓ˚y àˆÏí˛¸ ≤ÃyÎ˚ 200
μ
W«˛üï˛yÓy•# ¢∑ í˛zͲõߨ ܲÓ˚ˆÏï˛ ˛õy!Ó˚– §ˆÏçyˆÏÓ˚ !ã˛ÍܲyÓ˚ ܲÓ˚ˆÏ° ~•z «˛üï˛yÓ˚˛ ˛õ!Ó˚üyî 100
μ
W ~ ˆ˛õÔÑåÈÎ˚–
xÓ˚˛õÓ˚ˆÏ«˛ xyüÓ˚y Îáö !ú˛§!ú˛§ ܲˆÏÓ˚ ܲÌy Ó!° ï˛áö ˆÎ «˛üï˛y Óy!•ï˛ •Î˚ñ ï˛yÓ˚ üyö üye 0·001
μ
W–
Î!ò ò#â≈ §üˆÏÎ˚Ó˚ í˛z˛õÓ˚ àí˛¸ «˛üï˛y öy ôˆÏÓ˚ xy˛õ!ö ~ܲ!ê˛ x«˛Ó˚ (syllable) í˛zFã˛yÓ˚î ܲÓ˚ˆÏï˛ ˆÎê%˛Ü%˛ §üÎ˚ °yˆÏà
xÌ≈yÍ ≤ÃyÎ˚ 0·2 ˆ§ˆÏܲu˛ §üˆÏÎ˚Ó˚ üˆÏôƒ ܲt˛fl∫Ó˚Óy!•ï˛ «˛üï˛yÓ˚ ˛õ!Ó˚üy˛õ ܲˆÏÓ˚öñ ï˛ˆÏÓ ï˛yÓ˚ üyˆÏö xˆÏöܲê˛y
•…y§ÈÙÈÓÈ,!k˛ °«˛ƒ ܲÓ˚ˆÏÓö– í˛zòy•Ó˚îfl∫Ó˚*˛õñ ÚÄÛ fl∫Ó˚Óî≈!ê˛ í˛zFã˛yÓ˚ˆÏîÓ˚ §üÎ˚ «˛üï˛y Î!ò 50
μ
W •Î˚ ï˛ˆÏÓ •zÇÓ˚yç#
‘v’ ÓƒOöÓî≈ í˛zFã˛yÓ˚ˆÏîÓ˚ §üÎ˚ ~•z «˛üï˛y üyeñ 0·03
μ
W - ~ òyÑí˛¸yÎ˚–
Ó‡ !ÓK˛yö# üyö%ˆÏ£ÏÓ˚ ܲt˛fl∫ˆÏÓ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ fl∫ˆÏÓ˚Ó˚ í˛z˛õ!fli!ï˛ Ä ˆ§=!°Ó˚ «˛üï˛yÓ˚ ï%˛°öyü)°Ü˛ üyˆÏöÓ˚
í˛z˛õÓ˚ àˆÏÓ£Ïîy ܲˆÏÓ˚ˆÏåÈö– çyü≈yö !ÓK˛yö# ˆê˛ΔˆÏu˛ˆÏ°öÓ%ˆÏà≈Ó˚ (1924) àˆÏÓ£ÏîyÓ˚ ~ܲ!ê˛ ú˛° xyˆÏ°yã˛öy ܲÓ˚y
Îyܲ– ˛õ%Ó˚&£Ï Ä ü!•°y ܲˆÏt˛ Ú•zÛ Óy ‘t’ fl∫Ó˚Óî≈!ê˛ í˛zFã˛y!Ó˚ï˛ •ˆÏ° ÓyÎ˚%ˆÏï˛ ˆÎ ܲ¡õö §,!‹T •Î˚ ï˛yÓ˚ ˆ°á!ã˛e
14.3(a) Ä (b) !ã˛ˆÏe ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö ˛õ%Ó˚&£ÏܲˆÏt˛Ó˚ ü)° ܲ¡õyB˛ 200Hz ~ÓÇ ü!•°y ܲˆÏt˛Ó˚ ü)°
355
ܲ¡õyB˛ ≤ÃyÎ˚ 250Hz– ˆ°á!ã˛e ò%•z!ê˛Ó˚ ú%˛!Ó˚ˆÏÎ˚ !ӈϟ’£Ïî ܲÓ˚ˆÏ° !Ó!û˛ß¨ ú%˛!Ó˚ˆÏÎ˚ í˛z˛õyLjϢÓ˚ ˆÎ !ÓhflÏyÓ˚ ˛õyÄÎ˚y
ÎyÎ˚ñ ˆ§=!°Ä ˛õyˆÏ¢Ó˚
hflÏΩ˛!ã˛ˆÏe ˆòáyˆÏöy xyˆÏåÈ–
õ%Ó˚&£Ï Ä ü!•°yܲˆÏt˛ ~ܲ•z
fl ∫Ó ˚ í˛ zFã ˛y!Ó ˚ï˛ •ˆ Ï°
ܲ¡õ ÏöÓ˚ Î =îàï˛ ≤à Ïû˛ò
ÌyˆÏܲñ ï˛y !öÿ˛Î˚•z xy˛õöyÓ˚
ò,!‹T ~í˛¸yˆÏÓ öy– !ܲv °«˛ƒ
Ü˛Ó ˚ˆ Ï° ˆòáˆÏÓöñ ò %• z
ˆ«˛ˆÏe•z3000Hz ܲ¡õyˆÏB˛
ˆÓ¢ ¢!_´¢y°# ú%˛!Ó˚ˆÏÎ˚
í˛z˛õyÇ¢ Ó˚ˆÏÎ˚ˆÏåÈñ ˆÎ!ê˛ Ú•zÛ
fl∫Ó˚Óî≈!ê˛Ó˚ ~ܲ!ê˛ ˜Ó!¢‹Tƒ–
xöƒyöƒ àˆ ÏÓ£ÏÜ˛Ó ˚yÄ
ˆòˆ ÏሠÏå Èö ˆÎñ ≤Ã!ï˛
fl∫Ó˚ÓˆÏî≈Ó˚ ˆ«˛ˆÏe•z ü)°
ܲ¡õyB˛ Îy•z ˆ•yܲñ ~ܲ!ê˛
Óy ò%•z!ê˛ í˛zFã˛ï˛Ó˚ ܲ¡õyˆÏB˛ §Ó≈òy•z ¢!_´¢y°# ˛ú%˛!Ó˚ˆÏÎ˚ í˛z˛õyLjÏÏ¢Ó˚ í˛z˛õ!fli!ï˛ °«˛ƒ ܲÓ˚y ÎyÎ˚–
fl∫yû˛y!Óܲ ܲˆÏÌy˛õÜ˛ÌˆÏö ôù!ö «˛üï˛y ܲ¡õyˆÏB˛Ó˚ ~ܲ !Óhfl,Ïï˛ ˛õyÕ‘yÓ˚ üˆÏôƒ Ó!rê˛ï˛ ÌyˆÏܲ– ˛õ%Ó˚&£ÏܲˆÏt˛Ó˚ ˆ«˛ˆÏe
≤ÃyÎ˚ 500Hz ܲ¡õyˆÏB˛ §Ó≈y!ôܲ «˛üï˛y Óy!•ï˛ •ˆÏï˛ ˆòáy ÎyÎ˚– !ö¡¨ï˛Ó˚ ܲ¡õyˆÏB˛ 100Hz ˛õÎ≈hsˇ «˛üï˛y !ӈϢ£Ï
•…y§ öy ˆ˛õˆÏ°Ä í˛zFã˛ï˛Ó˚ ܲ¡õyˆÏB˛ Óy!•ï˛ «˛üï˛y xˆÏ˛õ«˛yÜ,˛ï˛ o&ï˛ •…y§ ˛õyÎ˚ – ï˛ˆÏÓ «˛üï˛y ܲü •ˆÏ°Ä í˛zFã˛ï˛Ó˚
ܲ¡õyB˛=!°•z í˛zFã˛y!Ó˚ï˛ ¢∑ˆÏܲ §%ˆÏÓyôƒ ܲˆÏÓ˚– fl∫Ó˚ï˛sf#Ó˚ ü)°§%ˆÏÓ˚Ó˚ !ö¡¨Ü˛¡õyB˛=!° ÓÓ˚Ç í˛zFã˛ï˛Ó˚
ܲ¡õyB˛=!°Ó˚ Óy•Ü˛ !•§yˆÏÓ Ü˛yç ܲˆÏÓ˚–
~ ˛õÎ≈hsˇ Îy ˛õí˛¸ˆÏ°ö ï˛yˆÏï˛ •Î˚ï˛ Ó%ˆÏé˛ˆÏåÈö ˆÎñ xyüyˆÏòÓ˚ ܲt˛!öɧ,ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ~ÓÇ
ܲ¡õyB˛ÈÙÙÙÈí˛zû˛ˆÏÎ˚Ó˚•z ˛õyÕ‘y á%Ó Óí˛¸– xyüÓ˚y ˆÎ ◊ÓˆÏî!wÎ˚ !òˆÏÎ˚ ~ï˛ !Ó!ã˛e ¢∑@ˇÃ•î Ä í˛z˛õ°!∏˛ ܲ!Ó˚ñ ï˛yÓ˚
ò«˛ï˛yÄ Óí˛¸ ܲü öÎ˚– xyüÓ˚y ~Ó˚ ˛õˆÏÓ˚Ó˚ xLjϢ ~•z ◊ÓˆÏî!wÎ˚ xÌ≈yÍ Ü˛î≈ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚Ó– ï˛ˆÏÓ
ï˛yÓ˚ xyˆÏà xy˛õöyÓ˚ çöƒ ~ܲ!ê˛ §•ç xö%¢#°ö# Ó˚•z°–
xö%¢#°ö# ÈÙÙÙÈ xö%¢#°ö# ÈÙÙÙÈ xö%¢#°ö# ÈÙÙÙÈ xö%¢#°ö# ÈÙÙÙÈ xö%¢#°ö# ÈÙÙÙÈ 1 :
(i) fl∫Ó˚ï˛sf#ˆÏܲ ÓƒÓ•yÓ˚ öy ܲˆÏÓ˚ xyüÓ˚y ܲ# ܲÌy Ó°ˆÏï˛ ˛õy!Ó˚⁄
(ii) §yôyÓ˚î ܲÌyÓyï≈˛yÎ˚ xyüÓ˚y ܲ# ˛õ!Ó˚üyî «˛üï˛yÓ˚ ¢∑ §,!‹T ܲ!Ó˚⁄
!ã˛e 14.3(a) : ˛õ%Ó˚&£Ï ܲˆÏt˛ Ú•zÛ fl∫Ó˚ÓˆÏî≈Ó˚ ܲ¡õö Ä ú%˛!Ó˚ˆÏÎ˚ í˛z˛õyÇ¢=!°Ó˚ !ÓhflÏyÓ˚–
Sü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ 200Hz)
!ã˛e 14.3(b) : ü!•°yܲˆÏt˛ Ú•zÛ fl∫Ó˚ÓˆÏî≈Ó˚ ܲ¡õö Ä ú%˛!Ó˚ˆÏÎ˚ í˛z˛õyÇ¢=!°Ó˚ !ÓhflÏyÓ˚
Sü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ 250Hz)
356
14.4 ܲyöܲyöܲyöܲyöܲyö
xyüyˆÏòÓ˚ ühflψÏܲÓ˚ ò%•z˛õyˆÏ¢ xÓ!fliï˛ ò%•z!ê˛ Ü˛yö xyüyˆÏòÓ˚ ◊ÓˆÏî!wÎ˚– ܲyö ÓyÎ˚% Óy xöƒ ˆÜ˛yöÄ üyôƒˆÏü
¢∑ï˛Ó˚DˆÏܲ ≤Ã̈Ïü Îy!sfܲ ܲ¡õˆÏö ˛õ!Ó˚îï˛ Ü˛ˆÏÓ˚ ~ÓÇ ï˛yÓ˚˛õÓ˚ ˜Óò%ƒ!ï˛Ü˛ §ˆÏB˛ˆÏï˛ ˛õ!Ó˚îï˛ Ü˛ÛˆÏÓ˚ öyû≈˛ ï˛vÓ˚
myÓ˚y ü!hflÏˆÏ‹Ò ˆ≤ÃÓ˚î ܲˆÏÓ˚– §ühflÏ hflÏöƒ˛õyÎ˚# ç#ˆÏÓÓ˚ ܲyˆÏöÓ˚ àë˛ö ~ܲ•z ôÓ˚ˆÏöÓ˚– xÓ¢ƒ ÓƒyÇ ~ÓÇ ˆÜ˛yöÄ
ˆÜ˛yöÄ !àÓ˚!à!ê˛ çyï˛#Î˚ ≤Ãyî#Ó˚ ܲyö˛õê˛• üyÌyÓ˚ í˛z˛õÓ˚ xöyFåÈy!òï˛ xÓfliyÎ˚ ÌyˆÏܲ– xyÓyÓ˚ üyˆÏåÈÓ˚y ï˛yˆÏòÓ˚
˛õyŸª#≈Î˚ ˆÓ˚áyÓ˚ (interal line) §y•yˆÏ΃ çˆÏ°Ó˚ üˆÏôƒ ¢ˆÏ∑Ó˚ çöƒ ܲ¡õö xö%û˛Ó ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚–
xyüÓ˚y ~áyˆÏö ˆÜ˛Ó° üyö%ˆÏ£ÏÓ˚ ܲyˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈˛õk˛!ï˛ !öˆÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚Ó–
14.4.1 ܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛ö
üyö%ˆÏ£ÏÓ˚ ܲyˆÏöÓ˚ !ï˛ö!ê˛ ≤Ãôyö xÇ¢ xyˆÏåÈ– ~=!° •° Ó!•Éܲî≈ñ üôƒÜ˛î≈ Ä xhsˇÉܲî≈ S!ã˛e 14.4V– ܲyˆÏöÓ˚
˛õeܲ Óy ˛õyï˛y (pinna) Îy Óy•zˆÏÓ˚ ˆÌˆÏܲ ˆòáy ÎyÎ˚ñ ◊ÓîÈÙÈöy!°Ü˛y (auditory canal) Ä Ü˛î≈˛õê˛• Óy ܲyˆÏöÓ˚
˛õò≈y (cardrum), ~=!° !öˆÏÎ˚•z
Ó!•Éܲî≈ à!ë˛ï˛– ܲî≈˛õê˛ˆÏ•Ó˚ xû˛ƒhsˇˆÏÓ˚
üôƒÜ˛ˆÏî≈ xyˆÏåÈ ~ܲ!ê˛ ÓyÎ˚%˛õ)î≈ à•ùÓ˚ñ
ÎyÓ˚ üˆÏôƒ !ï˛ö!ê˛ «%˛o «%˛o x!fli
(ossicle) ˆ˛õ!¢ Ä §!¶˛Ó¶˛ö#Ó˚
(ligament) myÓ˚y §!ߨ!Ó‹T ÌyˆÏܲ–
xyÜ,˛!ï˛ˆÏï˛ !ü° ÌyܲyÓ˚ çöƒ x!fli=!°Ó˚
öyü •yï% !í˛ (malleus), ö•y•z (incus)
Ä ˆÓ˚ܲyÓ (stapes)– •yï%˛!í˛¸Ó˚ §%Ó˚
!òܲ!ê˛ Ü˛yˆÏöÓ˚ ˛õò≈yÓ˚ §ˆÏD Î%_´ ÌyˆÏܲ
~ÓÇ §!¶˛Ó¶˛ö#Ó˚ ê˛yˆÏöÓ˚ ú˛ˆÏ° •yï%˛!í˛¸!ê˛
˛õò≈yˆÏܲ §yüyöƒ ˛õ!Ó˚üyˆÏî !û˛ï˛Ó˚ !òˆÏܲ
ˆê˛ˆÏö Ó˚yˆÏá– xy˛õ!ï˛ï˛ ¢∑ï˛Ó˚ˆÏDÓ˚
ú˛ˆÏ° ˛õò≈yÓ˚ ˆÎ ܲ¡õö •Î˚ñ ï˛yÓ˚ ú˛ˆÏ° •yï%˛!í˛¸ñ ˆö•y•z Ä ˆ¢ˆÏ£Ï ˆÓ˚ܲyÓ!ê˛ Ü˛!¡õï˛ •ˆÏï˛ ÌyˆÏܲ– ˆÓ˚ܲyˆÏÓÓ˚
~ܲ≤ÃyˆÏhsˇ ˛õyòy!öÓ˚ üˆÏï˛y ˆÎ ã˛ƒy˛õê˛y ˛õyï˛ ÌyˆÏܲ ˆ§!ê˛ üôƒÜ˛î≈ Ä xhsˇÉܲˆÏî≈Ó˚ üyé˛áyˆÏö ˆÎ !í˛¡∫yÜ,˛!ï˛ çyöy°y
ÌyˆÏܲ ˆ§!ê˛ˆÏܲ ˆì˛ˆÏܲ Ó˚yˆÏá– xhsˇÉܲî≈!ê˛ ~ܲ≤ÃܲyÓ˚ ï˛Ó˚ˆÏ° ˛õ)î≈ ÌyˆÏܲ ~ÓÇ ˆÓ˚ܲyˆÏÓÓ˚ ܲ¡õö ˙ ï˛Ó˚ˆÏ°
ã˛y˛õï˛Ó˚ˆÏDÓ˚ §,!‹T ܲˆÏÓ˚– üôƒÜ˛ˆÏî≈Ó˚ à•ùÓ˚ Ä Ü˛t˛öy°#ñ •zí˛zˆÏfiê˛!ܲÎ˚ öy°# (Eustachian tube) myÓ˚y Î%_´ ÌyˆÏܲ–
~•z öy°#!ê˛ §yôyÓ˚îû˛yˆÏÓ Ó¶˛ ÌyܲˆÏ°Ä •y•z ˆï˛y°yÓ˚ Óy áyòƒ à°yôÉܲÓ˚ˆÏîÓ˚ §üÎ˚ ˆ§!ê˛ á%ˆÏ° ÎyÎ˚ ~ÓÇ
ˆ§!ê˛Ó˚ üôƒ !òˆÏÎ˚ ÓyÎ˚%ã˛°yã˛° Ü˛ÛˆÏÓ˚ üôƒÜ˛ˆÏî≈Ó˚ !û˛ï˛ˆÏÓ˚ ÓyÎ˚%ã˛y˛õ Óy•zˆÏÓ˚Ó˚ §üyö Ó˚yˆÏá–
xhsˇÉܲˆÏî≈Ó˚ !ï˛ö!ê˛ xÇ¢– ~=!° •°˛ ≤ÈÏÓ¢ ≤ÈÏܲy¤˛ (vestibule), xô≈Ó,_yܲyÓ˚ öy°#ˆÏ◊î# Ä ¢Cöy°#
(cochlea)– xyˆÏà•z ˆçˆÏöˆÏåÈö ˆÎñ xhsˇÉܲˆÏî≈Ó˚ à•ùÓ˚!ê˛ ï˛Ó˚ˆÏ° ˛õ)î≈ ÌyˆÏܲ– üôƒÜ˛î≈ Ä xhsˇÉܲˆÏî≈Ó˚ üyé˛áyˆÏöÓ˚
!ã˛e ≠!ã˛e ≠!ã˛e ≠!ã˛e ≠!ã˛e ≠ 14.4 ܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛öܲyˆÏöÓ˚ àë˛ö
357
ˆòÄÎ˚yˆÏ° !í˛¡∫yÜ,˛!ï˛ çyöy°y åÈyí˛¸yÄ ˛õyï˛°y ˛õò≈y myÓ˚y xyFåÈy!òï˛ ~ܲ!ê˛ ˆày° çyöy°y ÌyˆÏܲñ !ܲv ò%!ê˛
çyöy°y•z Ó¶˛ ÌyܲyÎ˚ ˙ ï˛Ó˚°
üôƒÜ˛ˆÏî≈ ˆÎˆÏï˛ ˛õyˆÏÓ˚ öy– ï˛ˆÏÓ
ˆÓ˚ܲyÓ!ê˛ Îáö ܲ¡õˆÏöÓ˚ ú˛ˆÏ°
!í˛¡∫yÜ,˛!ï˛ çyöy°yÎ˚ ã˛y˛õ òÎ˚ ï˛áö
ˆày° çyöy°yÓ˚ ˛õò≈y!ê˛ üôƒÜ˛ˆÏî≈Ó˚
!ò Ïܲ fl≥˛#ï˛ •Î˚– ~Ó˚ ú˛ Ï° ã˛y˛õï˛Ó˚D
§•ˆÏç•z xhsˇÉܲˆÏî≈Ó˚ ï˛Ó˚ˆÏ°Ó˚ üˆÏôƒ
ã˛°yã˛° ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚– xô≈Ó,_yܲyÓ˚
öy°#ˆÏ◊î# xyüyˆÏòÓ˚ ¢Ó˚#ˆÏÓ˚Ó˚
û˛yÓ˚§yüƒ Ó˚«˛yÎ˚ §•yÎ˚ï˛y ܲˆÏÓ˚–
ï˛ˆÏÓ ◊ÓîܲyˆÏÎ≈ ~=!°Ó˚ ˆÜ˛yöÄ
û)˛!üܲy ÌyˆÏܲ öy– ¢C°yö#!ê˛ ¢ˆÏCÓ˚ üˆÏï˛y ˛õуyã˛ ˆòÄÎ˚y ˆày°yÜ,˛!ï˛ ≤ÃfliˆÏFåȈÏòÓ˚ ~ܲ!ê˛ öy°#– 2
34
˛õуyˆÏã˛Ó˚
~•z öy°#Ó˚ ˆüyê˛ ˜òâ≈ƒ ≤ÃyÎ˚ 3.1 cm– öy°#!ê˛Ó˚ ˜òâ≈ƒ ÓÓ˚yÓÓ˚ ~ܲ!ê˛ !Óû˛yçܲ ˆòÄÎ˚y° (cochlear partition)
öy°#!ê˛ˆÏܲ ˛õy¢y˛õy!¢ ò%•z!ê˛ §%í˛¸ˆÏD !Óû˛_´ ܲˆÏÓ˚– í˛z˛õˆÏÓ˚Ó˚ §%í˛¸D!ê˛ !í˛¡∫yÜ,˛!ï˛ çyöy°yÎ˚ ~ÓÇ !öˆÏã˛Ó˚ §%í˛¸D!ê˛
ˆày° çyöy°yÎ˚ ÷Ó˚& •Î˚ !ܲv §%í˛¸D ò%•z!ê˛ x˛õÓ˚ ≤ÃyˆÏhsˇ ¢C°yö#Ó˚ ¢#ˆÏ£Ï≈ ~ܲ!ê˛ ˆåÈyê˛ !åÈo myÓ˚y §ÇÎ%_´ ÌyˆÏܲ–
!Óû˛yçö ˆòÄÎ˚y°!ê˛Ó˚ àë˛ö xï˛ƒhsˇ ç!ê˛°– ~!ê˛Ó˚ üˆÏôƒ ~ܲy!ôܲ ˛õò≈y ÌyˆÏܲ ÎyÓ˚ üˆÏôƒ ~ܲ!ê˛ •° ˆÓ!§°yÓ˚
˛õò≈y (basilar membrane)– ◊Óî öyû≈˛=!° ~•z ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ üˆÏôƒ ≤ÈÏÓ¢ ܲˆÏÓ˚ ~ÓÇ öyˆÏû≈˛Ó˚ §Ó˚& §Ó˚&
ã%˛ˆÏ°Ó˚ üˆÏï˛y ≤Ãyhsˇ=!° ˛õò≈y!ê˛ ˆÌˆÏܲ ÓyÓ˚ •ˆÏÎ˚ ÌyˆÏܲ–
ܲˆÏî≈Ó˚ àë˛ö!ê˛ Ó%é˛ˆÏï˛ ÎyˆÏï˛ xy˛õöyÓ˚ §%!Óôy •Î˚ñ ˆ§çöƒ 14.5 !ã˛ˆÏe ܲˆÏî≈Ó˚ !Ó!û˛ß¨ xLjϢÓ˚ ~ܲ!ê˛ §°Ó˚#Ü,˛ï˛
öܲ¢y ˆòáyˆÏöy •°– ~ÓyÓ˚ ˆòáy Îyܲ ܲî≈ öyüܲ ~•z ç!ê˛° Îsf!ê˛ Ü˛#û˛yˆÏÓ Ü˛yç ܲˆÏÓ˚–
14.4.2 ܲyö !ܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ܲyö !ܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ܲyö !ܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ܲyö !ܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ܲyö !ܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚
xyüyˆÏòÓ˚ ܲyö ~ܲ!ê˛ xï˛ƒhsˇ §%ò«˛ñ §ÇˆÏÓòö¢#° xÌã˛ ¢_´ˆÏ˛õy_´ Îsf– xï˛ƒhsˇ «˛#î ¢∑Ä xyüyˆÏòÓ˚ ܲˆÏî≈
ôÓ˚y ˛õˆÏí˛¸– ÓyÎ˚%ˆÏï˛ «˛#îï˛ü ˆÎ ¢∑ xyüÓ˚y ÷öˆÏï˛ ˛õy•zñ ï˛yÓ˚ Óy!•ï˛ «˛üï˛y üye 10–12Wm–2 ~ÓÇ ¢∑ã˛y˛õ
2 × 10-5Pa– ~•z ã˛y˛õ ÓyÎ˚%ü[˛ˆÏ°Ó˚ ã˛yˆÏ˛õÓ˚ 5 × 109 û˛yˆÏàÓ˚ ~ܲ û˛yà ~ÓÇ ~•z ¢∑ã˛y˛õ 1000Hz ܲ¡õyˆÏB˛
ܲyˆÏîÓ˚ ˛õò≈yÓ˚ ˆÎ §Ó˚î §,!‹T ܲˆÏÓ˚ ï˛yÓ˚ ˛õ!Ó˚üyî üye 10–9cm– xöƒ!òˆÏܲñ xyô%!öܲ !üí˛z!çܲ !§ˆÏfiê˛ˆÏüÓ˚ ¢∑
Óy ˆÓyüyÈÙÈ˛õê˛Ü˛yÓ˚ ¢ˆÏ∑ xyüyˆÏòÓ˚ ܲî≈ !Óܲ° •ˆÏÎ˚ ÎyÎ˚ öyñ Î!òÄ ~ §Ó ˆ«˛ˆÏe ¢∑ã˛yˆÏ˛õÓ˚ üy˛õ 100Pa åÈy!í˛¸ˆÏÎ˚
ˆÎˆÏï˛ ˛õyˆÏÓ˚– ~åÈyí˛¸y ~!ê˛ ˆÎ ÷ô% 20 ˆÌ Ïܲ 1500Hz ܲ¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏï˛ ˛õyˆÏÓ˚ ï˛y•z öÎ˚– ≤Ãã˛[˛
ˆày°üyˆÏ°Ó˚ üˆÏôƒ ~ܲ!ê˛ !ӈϢ£Ï ܲ¡õyˆÏB˛Ó˚ ¢∑ ~!ê˛ !ã˛ˆÏö !öˆÏï˛ ˛õyˆÏÓ˚– ܲˆÏî≈Ó˚ ~•z ܲü≈«˛üï˛y !öÿ˛Î˚•z
xy˛õöyˆÏܲ !ܲå%Èê˛y xÓyܲ ܲÓ˚ˆÏåÈ–
◊ÓˆÏîÓ˚ ≤Ã!e´Î˚yÓ˚ üˆÏôƒ ܲyˆÏöÓ˚ !ï˛ö!ê˛ xLjϢÓ˚•z !ö!ò≈‹T û)˛!üܲy xyˆÏåÈ– ÓyÎ˚%Óy!•ï˛ ¢∑ï˛Ó˚D ≤Ã̈Ïü•z
358
◊Óîöy!°Ü˛yÓ˚ üˆÏôƒ ≤ÈÏÓ¢ ܲˆÏÓ˚– ◊Óî öy!°Ü˛yÓ˚ Óƒy§ 0·7 cm Ä ˜òâ≈ƒ ≤ÃyÎ˚ 2.5 cm– ~!ê˛Ó˚ !û˛ï˛ˆÏÓ˚Ó˚
≤Ãyhsˇ ܲî≈˛õê˛• myÓ˚y xyFåÈy!òï˛ ÌyܲyÎ˚ ~!ê˛ ~ܲ!ê˛ xyÓk˛ xà≈yö öˆÏ°Ó˚ üˆÏï˛y xyã˛Ó˚î ܲˆÏÓ˚– ÓyÎ˚%ˆÏï˛ ¢ˆÏ∑Ó˚
ˆÓà 340ms–1, ≤Ãyhsˇ#Î˚ e&!ê˛Ó˚ ÷!k˛ ÓƒyˆÏ§Ó˚ 0.3 =î ôˆÏÓ˚ !öˆÏ° ~•z xà≈yö öˆÏ°Ó˚ ≤ÃÌü xö%öyò ܲ¡õyB˛
()34000
42.50.30.7 +×
Óy ≤ÃyÎ˚ 3100 Hz– ˛õÓ˚#«˛y ܲˆÏÓ˚ ˆòáy ˆàˆÏåÈ ˆÎñ ˆüyê˛yü%!ê˛ ~•z ܲ¡õyˆÏB˛ §ï˛ƒ•z xö%öyò
âˆÏê˛ ~ÓÇ Ü˛î≈˛õê˛ˆÏ• ¢∑ã˛y˛õ ◊Óîöy!°Ü˛yÓ˚ ≤ÈÏÓ¢˛õˆÏÌÓ˚ ï%˛°öyÎ˚ 10 ˆÈí˛!§ˆÏÓ° ˆÓ!¢ •Î˚– ~•z xö%öyò xÓ¢ƒ
ï˛#«¯˛ öÎ˚ ~ÓÇ 2000 ˆÌ Ïܲ 6000Hz ܲ¡õyˆÏB˛Ó˚ üˆÏôƒ•z ¢∑ã˛yˆÏ˛õÓ˚ ˆÓ¢ !ܲå%Èê˛y üyˆÏöyߨ!ï˛ (gain) °«˛ƒ
ܲÓ˚y ÎyÎ˚– ~åÈyí˛¸y üyö%ˆÏ£ÏÓ˚ ühflÏܲ myÓ˚y ¢∑ï˛Ó˚ˆÏDÓ˚ ÓƒÓï≈˛ˆÏöÓ˚ (diffraction) ú˛ˆÏ°Ä ü%_´ ÓyÎ˚%Ó˚ ï˛%°öyÎ˚
◊Óîöy!°Ü˛yÓ˚ §yüˆÏö ¢∑ã˛y˛õ !ܲå%Èê˛y ˆÓ!¢ •Î˚–
ܲî≈˛õê˛ˆÏ•Ó˚ ܲ¡õˆÏöÓ˚ §ˆÏD •yï%˛!í˛¸!ê˛Ó˚ ˆÎ ܲ¡õö •Î˚ ï˛y ˆö•y•z ~Ó˚ üyôƒˆÏü ˆÓ˚ܲyˆÏÓ ˆ˛õÔÑåÈyÎ˚– ˆÓ˚ܲyÓ!ê˛Ó˚
x@ˇÃ˛õÿ˛yÍ ˛Ü˛¡õö !í˛¡∫yÜ,˛!ï˛ çyöy°yÓ˚ üôƒ !òˆÏÎ˚ xhsˇÉܲˆÏî≈Ó˚ ï˛Ó˚ˆÏ° ã˛y˛õï˛Ó˚D §,!‹T ܲˆÏÓ˚– ˛õò≈y Ä «%˛o
x!fli=!° fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ܲ¡õyB˛ 800 ˆÌˆÏܲ 1500Hz ~Ó˚ üˆÏôƒ ÌyˆÏܲ– ï˛ˆÏÓ ~•z ܲ¡õö ΈÏÌ‹T
xÓü!®ï˛ (damped) •ÄÎ˚yÎ˚ ܲü≈˛õê˛•!ê˛ !ü!◊ï˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ï˛Ó˚ˆÏD §!ë˛Ü˛û˛yˆÏÓ §yí˛¸y !òˆÏï˛ ˛õyˆÏÓ˚– í˛z˛õÓ˚vñ
˛õê˛ˆÏ•Ó˚ fl∫yû˛y!Óܲ ܲ¡õˆÏöÓ˚ ú˛ˆÏ° ˆÜ˛yö x!ï˛!Ó˚_´ ¢∑yö%û)˛!ï˛Ó˚ §,!‹T •Î˚ öy–
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ˆ¢yöyÓ˚ çöƒ üôƒÜ˛ˆÏî≈Ó˚ ܲ# ≤ÈÏÎ˚yçö⁄ xÌ≈yÍ Ü˛î≈˛õê˛ˆÏ•Ó˚ !ë˛Ü˛ !û˛ï˛ˆÏÓ˚•z
ï˛Ó˚°˛õ)î≈ xhsˇÉܲî≈ ÌyܲˆÏ°Ä ¢∑ ◊Óî §Ω˛Ó •ˆÏï˛y– ~•z ôyÓ˚îy!ê˛ !ë˛Ü˛ öÎ˚– ≤ÃÌüï˛ñ xy˛õ!ï˛ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y
x!ôܲ •ˆÏ° «%˛o x!fli=!°Ó˚ !öÎsfܲ üyǧˆÏ˛õ!¢Ó˚ ê˛yö ~üöû˛yˆÏÓ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚ ˆÎñ •yï%˛!í˛¸Ó˚ à!ï˛ xï˛ƒ!ôܲ
•ˆÏ°Ä ˆÓ˚ܲyˆÏÓÓ˚ à!ï˛ ï˛yÓ˚ §üyö%˛õyï˛# öy •ˆÏÎ˚ ܲü •Î˚ñ ú˛ˆÏ° xhsˇÉܲî≈ ï˛#Ó ¢∑ç!öï˛ «˛!ï˛ ˆÌˆÏܲ Ó˚«˛y
˛õyÎ˚– !mï˛#Î˚ï˛ñ ܲyˆÏöÓ˚ ˛õò≈yÓ˚ ï%˛ö°yÎ˚ !í˛¡∫yÜ,˛!ï˛ çyöy°yÓ˚ ˆ«˛eú˛ˆÏ°Ó˚ xö%˛õyï˛ ≤ÃyÎ˚ 30 : 1 ~ÓÇ !ÓK˛yö#Ó˚y
üˆÏö ܲˆÏÓ˚ö ˆÎñ ˆ«˛eú˛ˆÏ°Ó˚ ~•z xö%˛õyï˛ Ä «%˛ox!fli=!°Ó˚ §Ó˚ˆÏîÓ˚ xö%˛õyï˛ ◊Óîöy!°Ü˛yÓ˚ ÓyÎ˚% Ä xhsˇÉܲˆÏî≈Ó˚
ï˛Ó˚ˆÏ°Ó˚ üˆÏôƒ !ܲå%Èê˛y ≤Ã!ï˛ˆÏÓ˚yôÈÙÈ!ü° (impedance matching) âê˛yÎ˚ñ ÎyˆÏï˛ ˆÓ˚ܲyÓ!ê˛ Îï˛ò)Ó˚ §Ω˛Ó x!ôܲ
ï˛Ó˚D¢!_´ xhsˇÉܲˆÏî≈Ó˚ ï˛Ó˚ˆÏ° §MÈ˛y!Ó˚ï˛ Ü˛Ó˚ˆÏï˛ ˛õyˆÏÓ˚–
~ÓyÓ˚ ˆòáy Îyܲñ ◊Óî≤Ã!e´Î˚yÓ˚ üˆÏôƒ xhsˇÉܲˆÏî≈Ó˚ û)˛!üܲy ܲ# ¢Cöy°#Ó˚ ˆòÄÎ˚y° ò,ì˛¸ ~ÓÇ xhsˇÉܲˆÏî≈Ó˚
ï˛Ó˚° xöüö#Î˚ (incompressible) •ÄÎ˚yÎ˚ ˆÓ˚ܲyÓ!ê˛ Îáö•z !û˛ï˛Ó˚ !òˆÏܲ ì%˛ˆÏܲ xyˆÏ§ ï˛áö•z ï˛Ó˚°!ê˛ ≤ÃÓy!•ï˛
•Î˚ Ä ˆày° çyöy°yÓ˚ xyFåÈyòö Óy•zˆÏÓ˚Ó˚ !òˆÏܲ §ˆÏÓ˚ ÎyÎ˚– !ö¡¨ ܲ¡õyˆÏB˛ ï˛Ó˚° ¢Cöy°#Ó˚ ¢#ˆÏ£Ï≈ ˆÓ!§°yÓ˚
˛õò≈yÎ˚ ˆÎ !åÈo ÌyˆÏܲ ï˛yÓ˚ üôƒ !òˆÏÎ˚ ã˛°yã˛° ܲˆÏÓ˚– !ܲv x!ôܲ ܲ¡õyˆÏB˛ í˛z˛õˆÏÓ˚Ó˚ §%í˛¸ˆÏDÓ˚ ï˛Ó˚° ˆÓ!§°yÓ˚
˛õò≈yÓ˚ ˆÜ˛yöÄ ~ܲ xÇ¢ˆÏܲ ܲ!¡õï˛ Ü˛ˆÏÓ˚ñ ÎyÓ˚ ú˛ˆÏ° ܲ¡õö §Ó˚y§!Ó˚ !öˆÏã˛Ó˚ §%í˛¸ˆÏDÓ˚ ï˛Ó˚ˆÏ° ˆ˛õÔшÏåÈ ÎyÎ˚–
14.6 !ã˛e ˆòáˆÏ° ≤Ã!e´Î˚y!ê xy˛õöyÓ˚ ܲyˆÏåÈ flõ‹T •ˆÏÓ– ܲ¡õyB˛ Îï˛ ÓyˆÏí˛¸ñ ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ܲ!¡õï˛ xÇ¢!ê˛
ï˛ï˛•z ≤ÈÏÓ¢ ≤ÈÏܲyˆÏ¤˛Ó˚ ܲyåÈyܲy!åÈ ÌyˆÏܲ–
ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ˆÎ ˆÜ˛yöÄ xLjϢÓ˚ öí˛¸yã˛í˛¸y âê˛ˆÏ° ˙ xLjϢÓ˚ ã%˛ˆÏ°Ó˚ üˆÏï˛y öyû≈˛ ˆÜ˛y£Ï=!°Ó˚ ˛õ#í˛¸ö âˆÏê˛–
ˆÜ˛y£Ï=!° ï˛áö !˛õˆÏçy•zˆÏ°Ü˛!ê˛Δܲ ≤Ã!e´Î˚yÎ˚ !Óû˛Ó ≤ÈÏû˛ò §,!‹T ܲˆÏÓ˚– ~•z !Óû˛ÓˆÏܲ xyüÓ˚y ¢Cöy°#ÈÙÈ!Óû˛Ó
359
(cochlear potential) Ó!°– ~•z
!Óû˛Ó•z öyû≈˛ myÓ˚y ü!hflÏˆÏ‹Ò Óy!•ï˛ •Î˚
~ÓÇ ¢ˆÏ∑Ó˚ xö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚–
§ü@ˇÃ ≤Ã!e´Î˚y!ê˛ ˆçˆÏö xy˛õöyÓ˚
!öÿ˛Î˚•z üˆÏö •ˆÏFåÈ ˆÎñ xyüyˆÏòÓ˚
ܲyö ~ܲ!ê˛ § %®Ó˚ ≤ÃyÜ , ˛ !ï˛Ü˛
üy•zˆÏe´yˆÏú˛yö– ï˛ˆÏÓ ~•z ◊ÓîΈÏsfÓ˚Ä
!ܲå%È !ܲå%È fl∫yû˛y!Óܲ §#üyÓk˛ï˛y Ä
!öçfl∫ ˜Ó!¢‹Tƒ xyˆÏåÈ– ~Ó˚˛õÓ˚ xy˛õ!ö ˆ§=!° §¡∫ˆÏ¶˛ çyöˆÏï˛ ˛õyÓ˚ˆÏÓö– ~áö ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òˆÏï˛
ˆã˛‹Ty ܲÓ˚&ö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 2 :
!öˆÏã˛ ˆòÄÎ˚y ܲyˆÏöÓ˚ xÇ¢=!° ¢ˆÏ∑Ó˚ xö%û)˛!ï˛ Ó•ˆÏö ˆÜ˛yöÄ öy ˆÜ˛yöÄ û)˛!üܲy˛ ˛õy°ö ܲˆÏÓ˚– ~=!°ˆÏܲ
ܲyÎ≈ ˛õÓ˚¡õÓ˚y xö%ÎyÎ˚# §y!çˆÏÎ˚ !òö ≠
1. ˆÓ˚ܲyÓñ 2. xhsˇÉܲˆÏî≈Ó˚ ï˛Ó˚°ñ 3. •yï%˛!í˛¸ñ 4. ◊Óî öyû≈˛ñ 4. ◊Óîöy!°Ü˛yñ 6. !í˛¡∫yÜ,˛!ï˛ ˛õò≈yñ 7. ˆö•y•zñ
8. ˆÓ!§°yÓ˚ ˛õò≈yñ 9. ܲyˆÏöÓ˚ ˛õò≈y Óy ܲî≈˛õê˛•–
14.4.3 ܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛yܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛yܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛yܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛yܲyˆÏöÓ˚ ◊Óî¢!_´Ó˚ ˜Ó!¢‹Tƒ Ä §#üyÓk˛ï˛y
xy˛õ!ö •Î˚ï˛ çy Ïöö ˆÎñ üyö% Ï£ÏÓ˚ ܲyö ˆÜ˛Ó°üye !ö!ò≈‹T ~ܲ!ê˛ ˛õÕ‘yÓ˚ ü Ïôƒ Ìyܲy ܲ¡õy ÏB˛Ó˚ ¢∑ ÷ö Ïï˛
˛õyÎ˚– xyÓyÓ˚ ¢ Ï∑Ó˚ ï˛#Ó ï˛y xï˛ƒhsˇ x“ • Ï° ï˛y ˆÎüö ܲy Ïö ôÓ˚y
˛õ Ïí ¸ öyñ ˆï˛üö•z xï% ƒFã˛ ï˛#Ó ï˛yÓ˚ ¢∑ ܲy Ïö ÎsfîyÓ˚ §,!‹T ܲ ÏÓ˚ñ
ܲáöÄ Ü˛áöÄ ï˛y Ó!ôÓ˚ï˛yÄ §,!‹T ܲÓ˚ Ïï˛ ˛õy ÏÓ˚–
§Ó≈!ö¡¨ ˆÜ˛yö‰ ï˛#Ó ï˛yÓ˚ ¢∑ xyüy ÏòÓ˚ ◊Óî§yôƒ •Î˚ ~ÓÇ § ÏÓy≈Fã˛
ˆÜ˛yö‰ ï˛#Ó ï˛y åÈy!í ¸ ÏÎ˚ ˆà Ï° ˆ§•z ¢∑ ܲy Ïö !Ó÷k˛ ¢∑yö%û) !ï˛Ó˚
Óò Ï° xfl∫!hflÏ Óy Ü˛ê‰ Ü˛ê˛y!ö §,!‹T ܲ ÏÓ˚ ï˛y ¢ Ï∑Ó˚ ܲ¡õy ÏB˛Ó˚ í z õÓ˚
!öû≈ Ó˚ ܲ ÏÓ˚– xyüy ÏòÓ˚ ܲyö ≤ÃyÎ˚ 3000Hz ܲ¡õy ÏB˛Ó˚ ¢ Ï∑Ó˚ ˆ«˛ Ïe
§Ó Ïã˛ ÏÎ˚ ˆÓ!¢ §Ç ÏÓòö¢#°– xy˛õ!ö 14.4.2 xÇ Ï¢ •z!ï˛ü Ïôƒ•z
˛õ Ïí ¸ ÏåÈö ˆÎñ ~Ó˚ ܲyÓ˚î ◊Óîöy!°Ü˛yÓ˚ ü Ïôƒ ¢ Ï∑Ó˚ xö%öyò– ≤ÃyÎ˚
1000 Hz ܲ¡õy ÏB˛ §Ó≈!ö¡¨ ˆÎ ï˛#Ó ï˛yÓ˚ ¢∑ xyüÓ˚y ÷ö Ïï˛ §«˛ü ••zñ ˆ§!ê˛ Ïܲ xyüÓ˚y 0 db Sˆí˛!§ ÏÓ°V
ï˛#Ó ï˛y Ó Ï° x!û˛!•ï˛ ܲ!Ó˚– xy˛õ!ö xy Ïà•z ˆç Ïö ÏåÈö ˆÎñ ~•z ï˛#Ó ï˛yÓ˚ üyö 10–12Wm–2– xÓ¢ƒ
◊Óî§yôƒï˛yÓ˚ ~•z !ö¡¨§#üy Óƒ!_´ ˆÌ Ïܲ Óƒ!_´ Ïï˛ !Ó!û˛ß¨ •Î˚ñ xyÓyÓ˚ ܲy ÏöÓ˚ ˛õe ÏܲÓ˚ §yü Ïöñ xÌÓy §¡õ)î≈
360
ˆáy°y çyÎ˚àyÎ˚ ˆÜ˛yÌyÎ˚ ï˛#Ó ï˛y üy˛õy •Î˚ ï˛yÓ˚ í z õÓ˚Ä !öû≈ Ó˚ ܲ ÏÓ˚– ¢∑ ˆÜ˛yö !ò Ïܲ ˆÌ Ïܲ xy§ ÏåÈ ~ÓÇ ˆ◊yï˛y
~ܲ ܲyö !ò ÏÎ˚ ÷ö ÏåÈö öy í zû˛Î˚ ܲyö !ò ÏÎ˚ñ ï˛yÓ˚ í z õ ÏÓ˚Ä ~•z ï˛#Ó ï˛yÓ˚ üyö !öû≈ Ó˚¢#°– 14.7 ˆ°á!ã˛ Ïe
ˆí˛!§ ÏÓ Ï°Ó˚ !•§y ÏÓ Ü˛¡õy ÏB˛Ó˚ § ÏD ◊Óî§yôƒï˛yÓ˚ !ö¡¨§#üy Ä xfl∫!hflÏ xö%û) !ï˛Ó˚ !ö¡¨§#üy ˆòáy Ïöy • ÏÎ˚ ÏåÈ–
~•z ˛õ!Ó˚üy Ï õ !Ó!û˛ß¨ xö%û) !üܲ !òܲ ˆÌ Ïܲ xy§y ¢ Ï∑Ó˚ !ü◊î ÓƒÓ•*ï˛ • ÏÎ˚ ÏåÈ ~ÓÇ ˆ◊yï˛yÓ˚ ò%•z ܲyö•z ˆáy°y
Ó˚yáy • ÏÎ˚ ÏåÈ– ~•z ˆ°á!ã˛e ˆÌ Ïܲ xy˛õ!ö !öÿ˛Î˚•z Ó%é˛ Ïï˛ ˛õyÓ˚ ÏåÈö ˆÎñ ܲ¡õyB˛ 1000 Hz ~Ó˚ !ö Ïã˛ öyü Ï°
~•z ◊Óî§#üy 0 db Ì Ïܲ e´ü¢ Óyí ¸ Ïï˛ Ìy Ïܲ– xÓ¢ƒ 30 Hz ܲ¡õy ÏB˛Ó˚ !ö Ïã ~ ôÓ˚ ÏöÓ˚ õ!Ó˚üy˛õ !öû≈ Ó˚ ÏÎyàƒ
Ìy Ïܲ öy– x˛õÓ˚ õ Ï«˛ñ í zFã˛Ü˛¡õy ÏB˛ ~•z ◊Óî§#üy o&ï˛ Ó,!k˛ ˛õyÎ˚ ~ÓÇ Ü˛¡õy ÏB˛Ó˚ ˆÜ˛yö ~ܲ |ôù≈§#üyÎ˚ ¢∑
xyÓ˚ ◊Óî§yôƒ Ìy Ïܲ öy– ~•z |ôù≈§#üy 30 ÓåÈÓ˚ ÓÎ˚§ ˛õÎ≈hsˇ ˆ◊yï˛yÓ˚ ˆ«˛ Ïe 2000 - 25000 Hz • Ïï˛ ˛õy ÏÓ˚
!ܲv ã˛!Õ‘ ϧyôù≈ ÓÎ˚ ϧÓ˚ ˆ◊yï˛yÓ˚ ˆ«˛ Ïe e´ü¢ ܲ Ïü 15000 ˆÌ Ïܲ 10000 Hz ܲ¡õy ÏB˛ òyÑí ¸yÎ˚– xÓ¢ƒ 1000
Hz ܲ¡õy ÏB˛Ó˚ !ö Ïã˛ ◊Óî§#üy ÓÎ˚ ϧÓ˚ í z õÓ˚ !öû≈ Ó˚ ܲ ÏÓ˚ öy–
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y e´ü¢ Óyí˛¸ˆÏï˛ ÌyܲˆÏ° ˆÎ ï˛#Ó ï˛yÎ˚ ܲˆÏî≈ xfl∫!hflÏÓ˚ §,!‹T •Î˚ñ ï˛yÓ˚ üyö üyé˛yüy!é˛ Ü˛¡õyˆÏB˛
≤ÃyÎ˚ 120 db– ~•z üyö ܲ¡õyˆÏB˛Ó˚ IJõÓ˚ á%Ó ˆÓ!¢ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– 140 db ï˛#Ó ï˛yÎ˚ ܲyˆÏö Îsfîy xö%û)˛ï˛
•Î˚ ~ÓÇ ~Ó˚ ˆÓ!¢ ï˛#Ó ï˛y ˆÓ!¢«˛î fliyÎ˚# •ˆÏ° ◊ÓîΈÏsfÓ˚ fliyÎ˚# «˛!ï˛ •ˆÏï˛ ˛õyˆÏÓ˚– 160 db Óy ï˛ò)ôù≈ ï˛#Ó ï˛yÓ˚
¢∑ ü%•)ˆÏï≈˛Ó˚ üˆÏôƒ ◊Óî¢!_´ ö‹T ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚–
14.4.4 ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey (Loudness level)
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä Ü˛yˆÏöÓ˚ ◊Óî§#üy §¡∫ˆÏ¶˛ xyüÓ˚y xyˆÏàÓ˚ xLjϢ ˆÎ xyˆÏ°yã˛öy ܲÓ˚°yüñ ï˛yÓ˚ ˆÌˆÏܲ
xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎñ ˆÜ˛yö ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö Ü˛ï˛ ˆçyÓ˚ ӈϰ üˆÏö •Î˚ ï˛y ˙ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚
í˛z˛õÓ˚ !öû˛≈Ó˚ ܲˆÏÓ˚– ≤ÃÜ,˛ï˛ âê˛öy ~•z ˆÎñ ~!ê˛ ˙ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä Ü˛¡õyB˛ÈÙÈí˛zû˛ˆÏÎ˚Ó˚ í˛z˛õÓ˚•z !öû≈˛Ó˚ ܲˆÏÓ˚– 14.8
ˆ°á!ã˛e ˆÌˆÏܲ xy˛õ!ö ~ Óƒy˛õyÓ˚!ê˛ !ܲå%Èê˛y Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– ~áyˆÏö !Ó!û˛ß¨ ≤ÃyÓ°ƒüyeyÎ˚ §ü≤ÃyÓ°ƒˆÏÓ˚áy
(Loudness level contour) ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– 20 db ï˛#Ó ï˛yÓ˚ ¢ˆÏ∑Ó˚ ܲÌy•z ôÓ˚&ö– ܲ¡õyB˛ Î!ò 1000,
2000 Óy 5000 Hz •Î˚ ï˛ˆÏÓ ~•z ï˛#Ó ï˛yÓ˚ ¢∑ xyüÓ˚y §•ˆÏç•z ÷öˆÏï˛ ˛õyÓ– !ܲv ˙ ~ܲ•z ï˛#Ó ï˛yÓ˚ ¢∑
100 Hz ܲ¡õyˆÏB˛Ó˚ •ˆÏ° ï˛y xyüyˆÏòÓ˚ ◊Óî§yôƒ •ˆÏÓ öy– §%ï˛Ó˚yÇñ ˆÜ˛yö ¢∑ xyüyˆÏòÓ˚ ܲyˆÏö Ü˛ï˛ ˆÓ!¢
xö%û)˛!ï˛Ó˚ §,!‹T ܲˆÏÓ˚ Óy §•ç û˛y£ÏyÎ˚ Ü˛ï˛ ˆçyÓ˚ ˆ¢yöyÎ˚ ˆ§!ê˛ˆÏܲ xyüÓ˚y ~ܲ!ê˛ !û˛ß¨ öyˆÏü x!û˛!•ï˛ ܲÓ˚Ó–
~!ê˛ •° ¢ˆÏ∑Ó˚ ≤ÃyÓy°ƒüyeyÓ˚ ~ܲܲ ú˛ö (Phon)–
~áö ≤ß¿ñ !ö!ò≈‹T ܲ¡õyˆÏB˛Ó˚ ˆÜ˛yöÄ ¢ˆÏ∑Ó˚˛ ≤ÃyÓ°ƒüyey xyüÓ˚y ܲ#û˛yˆÏÓ !öî≈Î˚ ܲÓ˚Ó⁄ 1000 Hz
ܲ¡õyˆÏB˛Ó˚ ˆÜ˛yö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Îï˛ ˆí˛!§ˆÏÓ°ñ ï˛yÓ˚ ≤ÃyÓ°ƒüyey ï˛ï˛ ú˛ö ӈϰ ôˆÏÓ˚ ˆöÄÎ˚y •Î˚– xÌ≈yÍñ
!Ó÷k˛ 1000 Hz ܲ¡õyˆÏB˛Ó˚ 20db ï˛#Ó ï˛yÓ˚ ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey 20 ú˛öñ 40 db ï˛#Ó ï˛yÓ˚ ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey
40 ú˛ö •zï˛ƒy!ò–
361
~áö Î!ò ¢ˆÏ∑Ó˚
í˛ z͈ϧÓ˚ ܲ¡õyB˛
~ÓÇ ˆ§•z§ˆ ÏD
¢ˆÏ∑Ó˚ ï˛#Ó ï˛y
~üöû˛y ÏÓ ˛õ!ÓÓ!ï≈ ï˛
ܲÓ˚y ÎyÎ˚ñ ÎyˆÏï˛
¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey
~ܲ ÌyˆÏܲ xÌ≈yÍñ
¢ˆ Ï∑Ó˚ ܲ¡õyB˛
˛õ!Ó ˚Ó!ï ≈ ˛ï˛ •ˆ Ï°
ˆ§!ê˛ ˆ◊yï˛yÓ˚ ܲyˆÏö
§üyö ˆçyÓ˚ ӈϰ
üˆÏö •Î˚ñ ï˛ˆÏÓ ˙
!ö!ò≈‹T ≤ÃyÓ°yüyeyÓ˚
çöƒ ܲ¡õyB˛ ï˛#Ó ï˛y
ˆ°á!ã˛e xB˛ö ܲÓ˚y
ÎyˆÏÓ– 14.8 !ã˛ˆÏe
~ ôÓ˚ˆÏöÓ˚ ˆ°á!ã˛e
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~•z !ã˛ˆÏe 40 ú˛ˆÏöÓ˚ ˆÓ˚áy!ê˛ °«˛ƒ ܲÓ˚&ö– ~!ê˛ 1000 Hz ܲ¡õyˆÏB˛ 40 db ï˛#Ó ï˛y !öˆÏò≈¢
ܲˆÏÓ˚ˆÏåÈ (a !Ó®%Vñ Îy xyüÓ˚y ú˛ö ~ܲˆÏܲÓ˚ §ÇK˛y ˆÌˆÏܲ•z xy¢y ܲ!Ó˚– !ܲv ~•z ˆÓ˚ܲy!ê˛100Hz ܲ¡õyˆÏB˛
62 db (b), 500 Hz ܲ¡õyˆÏB˛ 45 db (c) Ä 5000 Hz ܲ¡õyˆÏB˛ 42 db (d) ï˛#Ó ï˛y ˆòáyˆÏFåÈ– xyüyÓ˚
2000 Hz ܲ¡õyˆÏB˛ ï˛#Ó ï˛y 38 db (e), Îy 40 db ~Ó˚ ˆã˛ˆÏÎ˚Ä Ü˛ü– ~Ó˚ ˆÌˆÏܲ xy˛õ!ö Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏåÈö
ˆÎñ ܲ¡õyB˛ ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏ° ≤ÃyÓ°ƒüyey §üyö Ó˚yáˆÏï˛ ï˛#Ó ï˛yÓ˚Ä ˛õ!Ó˚Óï≈˛ö ܲÓ˚ˆÏï˛ •Î˚–
ˆÜ˛yöÄ !Ó÷k˛ ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyeyÓ˚ üyö ¢∑!ê˛ Ü˛ï˛«˛î ôˆÏÓ˚ ◊&ï˛ •Î˚ ï˛yÓ˚ í˛z˛õÓ˚Ä !öû˛≈Ó˚ ܲˆÏÓ˚–
¢∑!ê˛ xyÓ˚Ω˛ •ÄÎ˚yÓ˚ §ˆÏD §ˆÏD ≤ÃyÓ°ƒüyeyÓ˚ üyö o&ï˛ Óyí˛¸ˆÏï˛ ÌyˆÏܲ– ~ܲ ˆ§ˆÏܲˆÏu˛Ó˚ üˆÏï˛y §üˆÏÎ˚
≤ÃyÓ°ƒüyey í˛zFã˛ï˛ü üyˆÏö ˆ˛õÔÑåÈÎ˚ ~ÓÇ ï˛yÓ˚˛õÓ˚ ô#ˆÏÓ˚ ô#ˆÏÓ˚ ܲüˆÏï˛ ÌyˆÏܲ–
≤ÃyÓ°ƒüyeyÓ˚ §ÇK˛y ˆÌˆÏܲ xy˛õöyÓ˚ üˆÏö •ˆÏï˛ ˛õyˆÏÓ˚ ˆÎ ï˛#Ó ï˛y 50 db ˆÌˆÏܲ 60 db •ˆÏ° ˆÎüö ï˛#Ó ï˛yÓ˚
üyö10 =î Ó,!k˛ ˛õyÎ˚ñ ˆï˛üö•z ˆÜ˛yöÄ ~ܲ§%Ó˚ !Ó!¢‹T ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey 50 ú˛ö ˆÌˆÏܲ ˆÓˆÏí˛¸ 60 ú˛ö •ˆÏ°
¢∑!ê˛ ò¢=î ˆçyÓ˚ ˆ¢yöyˆÏÓ– ~•z ôyÓ˚îy !ܲv !ë˛Ü˛ öÎ˚⁄ ú˛ö ~ܲˆÏܲ ≤Ãܲy!¢ï˛ ≤ÃyÓ°ƒüyeyÓ˚ §üyö üyö
362
ò%•z Óy ï˛ˆÏï˛y!ôܲ ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyeyÓ˚ §üï˛y ˆÓyé˛yˆÏï˛ Ü˛yÎ≈ܲÓ˚# •ˆÏ°Ä ò%•z !û˛ß¨ ≤ÃyÓ°ƒüyeyÓ˚ ¢ˆÏ∑Ó˚ ï%˛°öy
ܲÓ˚ˆÏï˛ í˛z˛õˆÏÎyà# öÎ˚– ~ܲ!ê˛ í˛zòy•Ó˚î !òˆÏ° !Ó£ÏÎ˚!ê˛ flõ‹T •ˆÏÓ– ôÓ˚&öñ xy˛õ!ö 1000 Hz ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛
!Ó÷k˛ §%ˆÏÓ˚Ó˚ ï˛#Ó ï˛yÓ˚ üyey 10 db ˆÌˆÏܲ Óy!í˛¸ˆÏÎ˚ 20 db ܲÓ˚ˆÏ°ö– ~ˆÏï˛˛ï˛#Ó ï˛y 10 =î Ó,!k˛ ˛õyˆÏÓ–
≤ÃyÓ°ƒüyey 10 ˆÌˆÏܲ ˆÓˆÏí˛¸ 20 ú˛öÈÙÈ~ òÑyí˛¸yˆÏÓ– !ܲv ~ˆÏï˛ ˆ◊yï˛yÓ˚ xö%û)˛!ï˛ˆÏï˛ ¢ˆÏ∑Ó˚ ˆçyÓ˚yˆÏ°yû˛yÓ Óyí˛¸ˆÏÓ
≤ÃyÎ˚ 6 =îñ 10 =î öÎ˚– xyÓyÓ˚ Î!ò˛ ≤ÃyÓ°ƒüyey 50 ˆÌˆÏܲ Óy!í˛¸ˆÏÎ˚ 60 ú˛ö ܲÓ˚y •ˆÏï˛yñ ï˛ˆÏÓ ˆ◊yï˛yÓ˚
ܲyˆÏåÈ ˙ ˆçyÓ˚yˆÏ°yû˛yÓ Óyí˛¸ï˛ üye 2 =îñ Î!òÄ ï˛#Ó ï˛y Óyí˛¸ï˛ 10 =î– ~Ó˚ ú˛ˆÏ° xyüyˆÏòÓ˚ ~üö ~ܲ!ê˛
~ܲܲ ≤ÈÏÎ˚yçö Îy ˆ◊yï˛y !öû≈˛Ó˚ ≤ÃÓ°ï˛yÓ˚ üy˛õܲy!ë˛ •ˆÏÓ– ˛õÓ˚Óï˛#≈ xLjϢ xyüÓ˚y ~•z ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y
!ӣψÏÎ˚ xyˆÏöyã˛öy ܲÓ˚Ó–
14.4.5 ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛yˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛yˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛yˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛yˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y (Subjective Loudness)
ôÓ˚&öñ xy˛õöyˆÏܲ Äçö üy˛õyÓ˚ çöƒ ~ܲ ˆ§ê˛ Óyê˛áyÓ˚y ˜ï˛!Ó˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– xy˛õ!ö ≤Ã̈Ïü•z ~ܲ!ê˛
Óyê˛áyÓ˚yÓ˚ ÄçöˆÏܲ ~ܲܲ Äçö ӈϰ ôˆÏÓ˚ ˆöˆÏÓö– ~ÓyÓ˚ ï˛yÓ˚ §üyö ÄçˆÏöÓ˚ xyÓ˚ ~ܲ!ê˛ Óyê˛áyÓ˚y !òˆÏÎ˚
ò%!ê˛ Óyê˛áyÓ˚y ~ܲˆÏe Ó˚yáˆÏÓö Ä òyÑ!í˛¸ ˛õyÕ‘yÓ˚ §y•yˆÏ΃ ˆ§ ò%!ê˛Ó˚ §üyö ï,˛ï˛#Î˚ ~ܲ!ê˛ Óyê˛áyÓ˚y ˜ï˛!Ó˚ ܲˆÏÓ˚
ˆ§!ê˛ˆÏܲ ò%•z ~ܲܲ ÄçˆÏöÓ˚ Óyê˛áyÓ˚y Ó°ˆÏÓö– ~•zû˛yˆÏÓ !Ó!û˛ß¨ ÄçˆÏöÓ˚ Óyê˛áyÓ˚y ˜ï˛!Ó˚ ܲˆÏÓ˚ !öˆÏ° ÓƒÓ•yˆÏÓ˚Ó˚
í˛z˛õˆÏÎyà# ˛õ%ˆÏÓ˚y ~ܲˆÏ§ê˛ Óyê˛áyÓ˚y ˛õyÄÎ˚y ÎyˆÏÓ–
~ܲ•z û˛yˆÏÓ Î!ò xy˛õöyˆÏܲ òyÑ!í˛¸ ˛õyÕ‘yÓ˚ ÓòˆÏ° ˆ◊yï˛yÓ˚ ¢∑yö%û)˛!ï˛Ó˚ í˛z˛õÓ˚ !û˛!_ ܲˆÏÓ˚ ˆ◊yï˛y!öû≈˛Ó˚
≤ÃÓ°ï˛yÓ˚ üy˛õܲy!ë˛ ˜ï˛!Ó˚ ܲÓ˚ˆÏï˛ •Î˚ñ ï˛ˆÏÓ ˆ◊yï˛yÓ˚ ܲyˆÏö ˆÜ˛yö ¢∑~ܲ!ê˛ !ö!ò≈‹T ¢ˆÏ∑Ó˚ !m=îñ !ï˛ö=î
•zï˛ƒy!ò ≤ÃÓ°ï˛yÓ˚ ӈϰ üˆÏö •Î˚ ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ •ˆÏÓ– ~!ê˛ Ü˛Ó˚yÓ˚ ò%•z!ê˛ í˛z˛õyÎ˚ xyˆÏåÈ–
(i) ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚ ˆÎñ ˆÜ˛yöÄ ¢∑ ~ܲ ܲyˆÏö ÷öˆÏ° ï˛y ˆ◊yï˛yÓ˚ ܲyˆÏåÈ Îï˛ ˛≤ÃÓ° ӈϰ üˆÏö •Î˚ñ
ò%•z ܲyˆÏö ÷öˆÏ° ï˛yÓ˚ !m=î ≤ÃÓ° üˆÏö •Î˚– §%ï˛Ó˚yÇñ Î!ò ÚÜ˛Û ¢∑ ˆ◊yï˛y ò%•z ܲyˆÏö ÷ˆÏö Îï˛ ≤ÃÓ° ӈϰ
üˆÏö ܲˆÏÓ˚öñ ÚáÛ ¢∑!ê˛ ~ܲ ܲyˆÏö ÷ˆÏö•z ï˛yÑÓ˚ ï˛ï˛ê˛y ≤ÃÓ° üˆÏö •Î˚ñ ï˛ˆÏÓ ÚáÛ ¢ˆÏ∑Ó˚ ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y
ÚÜ˛Û ¢ˆÏ∑Ó˚ !m=î– xÓ¢ƒ ~•z ˛õk˛!ï˛ˆÏï˛ ò%•z!ê˛ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛yÓ˚ 1 : 2 xö%˛õyï˛ §¡∫ˆÏ¶˛•z !ö!ÿ˛ï˛ •ÄÎ˚y
§Ω˛Óñ xöƒ xö%˛õyˆÏï˛Ó˚ ˆ«˛ˆÏe ~•z ˛õk˛!ï˛ áyˆÏê˛ öy–
(ii) xyüÓ˚y xyˆÏà•z ˆòˆÏá!åÈ ˆÎñ ¢ˆÏ∑Ó˚ ~ܲ ~ܲ ܲ¡õyˆÏB˛ ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ~ܲ ~ܲ xÇ¢ ܲ!¡õï˛ •Î˚–
~Ó˚ ú˛ˆÏ° !û˛ß¨ ܲ¡õyˆÏB˛Ó˚ ~ܲ•z ≤ÃÓ°ï˛yÓ˚ ò%•z!ê˛ §%Ó˚ Î!ò ˆ◊yï˛yÓ˚ ܲyˆÏö ≤ÈÏÓ¢ ܲˆÏÓ˚ ï˛ˆÏÓñ ˆ§=!° ˆÓ!§°yÓ˚˛
˛õò≈yÓ˚ ò%•z !û˛ß¨ xLjϢ §yí˛¸y çyàyˆÏÓ ~ÓÇ ˆ◊yï˛yÓ˚ ¢∑yö%û)˛!ï˛ ˆÎ ˆÜ˛yöÄ ~ܲ!ê˛ §%ˆÏÓ˚Ó˚ çöƒ Îï˛ê˛y ≤ÃÓ°
•ï˛ ï˛yÓ˚ !m=î ≤ÃÓ° •ˆÏÓ– ~•z ˛õk˛!ï˛ˆÏï˛ !û˛ß¨ !û˛ß¨ ܲ¡õyˆÏB˛Ó˚ Ä §üyö ≤ÃyÓ°ƒüyeyÓ˚ ò%•zÈÙÈ~Ó˚ x!ôܲ §ÇáƒÜ˛
§%Ó˚ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ ≤ÃÓ°ï˛yÓ˚ xyÓ˚Ä Óí˛¸ xö%˛õyï˛ !ö!ò≈‹T ܲÓ˚y ÎyÎ˚–
~Ó˚ ˛õÓ˚ ˆÎ ܲyç!ê˛ Ü˛Ó˚y ≤ÈÏÎ˚yçö ˆ§!ê˛ •ˆÏ°yñ ≤ÃÓ°ï˛yÓ˚ ˆÜ˛!ê˛ ~ܲܲ !fliÓ˚ ܲÓ˚y– 100 Hz ܲ¡õyˆÏB˛Ó˚
40 db ï˛#Ó ï˛yÓ˚ §%ˆÏÓ˚Ó˚ ≤ÃÓ°ï˛yˆÏܲ ~•z ~ܲܲ !•§yˆÏÓ ôˆÏÓ˚ ˆöÄÎ˚y •Î˚– ~•z ~ܲˆÏܲÓ˚ öyü ˆ§yö (sone)–
363
xy˛õ!ö Î!ò ú˛ö ~ܲˆÏܲÓ˚ §ÇK˛y ˆû˛ˆÏÓ ˆòˆÏáöñ ï˛ˆÏÓ ˆòáˆÏÓö ˆÎñ ~•z §%Ó˚!ê˛Ó˚ ≤ÃÓ°ƒüyey 40 ú˛ö– ~áö
Î!ò 40 ú˛ö xˆÏ˛õ«˛y ܲü Óy ˆÓ!¢ ≤ÃyÓ°ƒüyeyÓ˚ §%ˆÏÓ˚Ó˚ ≤ÃÓ°ï˛yÓ˚ üyö ˆ§yö ~ܲˆÏܲ !öî≈Î˚ ܲÓ˚y ÎyÎ˚ñ ï˛ˆÏÓ
xyüÓ˚y í˛zû˛Î˚ Ó˚y!¢Ó˚ §¡õÜ≈˛ ˆ°á!ã˛ˆÏeÓ˚ §y•yˆÏ΃ ˆòáyˆÏï˛
˛õy!Ó˚– 14.9 !ã˛ˆÏe ~•z ˆ°á!ã˛e!ê˛ ˆòáˆÏï˛ ˛õyˆÏÓö–
ˆ°á!ã˛e!ê˛ û˛y° ܲˆÏÓ˚ °«˛ƒ ܲÓ˚&ö– xy˛õ!ö Ó%é˛ˆÏï˛•z
˛õyÓ˚ˆÏåÈöÈÙÙÙÈ
~•z xö%û)˛!üܲ xˆÏ«˛ ≤ÃyÓ°ƒüyey (LL)
˜Ó˚!áܲû˛yˆÏÓ (linear) ˆòáyˆÏöy •ˆÏ°Ä í˛zÕ‘¡∫ xˆÏ«˛
≤ÃÓ°ï˛y (L) ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ °ày!Ó˚̉!ü܉˛ ˆfl҈ϰ–
40 ú˛ˆÏöÓ˚ x!ôܲ ≤ÃyÓ°ƒüyeyÎ˚ ˆ°á!ê˛ ≤ÃyÎ˚
§Ó˚°˜ÏÓ˚!áܲ– ~•z xÇ¢!ê˛Ó˚ §ü#ܲÓ˚î §•ˆÏç•z !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
ˆ§yˆÏöÓ˚ §ÇK˛y xö%ÎyÎ˚# Îáö LL = 40, ï˛áö L = 1, log
L = 0– Î!ò ôˆÏÓ˚ ˆöÄÎ˚y ÎyÎ˚ Îáö LL = 100, ï˛áö LL
= 100, log L = 2, ï˛ˆÏÓ §ü#ܲÓ˚î!ê˛ •ˆÏÓñ
(log L – 0) = 2 0
100 40−− (LL – 40)
Óyñ logL =
130
(LL – 40) = 0·033LL – 1·33
Óyñ L = 10–1·33·100·033LL = 0·047 100·033LL ...14.1
üˆÏö Ó˚yáˆÏï˛ •ˆÏÓñ 14.1 §ü#ܲÓ˚î!ê˛ 40 ú˛ˆÏöÓ˚ ܲü ≤ÃyÓ°ƒüyeyÎ˚ ÓƒÓ•yÓ˚ ܲÓ˚y ÎyÎ˚ öy–
1000Hz ܲ¡õyˆÏB˛Ó˚ §%ˆÏÓ˚Ó˚ ˆ«˛ˆÏe xyüÓ˚y ≤ÃÓ°ï˛yÓ˚ §ˆÏD ˆí˛!§ˆÏÓ° üyeyÎ˚ ï˛#Ó ï˛yÓ˚ §¡õÜ≈˛ !öî≈Î˚ ܲÓ˚ˆÏï˛
˛õy!Ó˚– ~•z !ӈϢ£Ï ܲ¡õyˆÏB˛
LL = 10log
0I
I⎛⎞ ⎜⎟ ⎝⎠
ˆÎáyˆÏö I = ï˛#Ó ï˛y ~ÓÇ I0= ◊Óî§yôƒï˛yÓ˚ §#üyÎ˚ ï˛#Ó ï˛yñ xÌ≈yÍ 10–12Wm–2
∴
LL = 10(logI – logI0) = 10logI + 120
∴
log L = ·033(10 logI + 120) – 1·33 = 0·33logI + 2·67
!ã˛e ≠ 14.9 ˛≤ÃÓ°ƒüyey Ä ≤ÃÓ°ï˛yÓ˚
ˆ°á!ã˛e
364
xÌ≈yÍñ L = 464I0·33 ...14.2
14.2 §ü#ܲÓ˚î ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎ ≤ÃÓ°ï˛y ˆüyê˛yü%!ê˛û˛yˆÏÓ ï˛#Ó ï˛yÓ˚ âöü)ˆÏ°Ó˚ §üyö%˛õyï˛#–
14.4.6 !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑!ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑!ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑!ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑!ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑
~ ˛õÎ≈hsˇ xyüÓ˚y ≤Ãôyöï˛ ~ܲ!ê˛üye ܲ¡õyˆÏB˛Ó˚ ¢∑ xÌ≈yÍ §%ˆÏÓ˚Ó˚ ≤ÃÓ°ƒüyey ~ ≤ÃÓ°ï˛y §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy
ܲÓ˚°yü– !ܲv ≤ÃÜ,˛!ï˛ˆÏï˛ ˆÜ˛yöÄ ¢∑•z ~ܲ!ê˛üye ܲ¡õyˆÏB˛Ó˚ !Ó÷k˛ §%Ó˚ öÎ˚– xy§ˆÏ° ˆÜ˛y!ܲˆÏ°Ó˚ í˛yܲ•z Ó°%ö
xyÓ˚ •!Ó˚≤çyò ˆã˛ÔÓ˚y!¢Î˚yÓ˚ ÓyÑ!¢•z Ó°%öñ öyöy í˛z˛õ§%Ó˚ !ü!◊ï˛ öy ÌyܲˆÏ° ˆÜ˛yö!ê˛•z ~ï˛ !ü!‹T •ï˛ öy– ~áö
xyüÓ˚y !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y §¡õˆÏÜ≈˛ xyˆÏ°yã˛öy ܲÓ˚Ó–
xyüÓ˚y xyˆÏà•z ˆçˆÏö!åÈñ ˆÎˆÏ•ï%˛ ~ܲ ~ܲ ܲ¡õyˆÏB˛Ó˚ §%Ó˚ ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ~ܲ ~ܲ xÇ¢ˆÏܲ ܲ!¡õï˛ Ü˛ˆÏÓ˚ñ
ˆ§çöƒ ~ˆÏòÓ˚ üˆÏôƒ ˆÜ˛yö Óƒ!ï˛ã˛yÓ˚ (interference) âˆÏê˛ öy ~ÓÇ ~=!°Ó˚ ˆüyê˛ ≤ÃÓ°ï˛y fl∫Ó˚=!°Ó˚ ~ܲܲ
≤ÃÓ°ï˛yÓ˚ ˆÎyàú˛ˆÏ°Ó˚ §üyö •Î˚– ~•z ö#!ï˛!ê˛ xyüyˆÏòÓ˚ ܲyˆÏåÈ xï˛ƒhsˇ =Ó˚&c˛õ)î≈ñ ˆÜ˛ö öyñ ~Ó˚ §y•yˆÏ΃ xyüÓ˚y
!ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤Ãï˛ƒy!¢ï˛ ≤ÃÓ°ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚– ~ܲ!ê˛ í˛zòy•Ó˚î ˆÌˆÏܲ xy˛õ!ö ~Ó˚ ˛õk˛!ï˛!ê˛
Ó%é˛ˆÏï˛ ˛õyÓ˚ˆÏÓö– ôÓ˚y Îyܲñ ˆÜ˛yöÄ !ü◊ ¢∑ 100, 200, 500, 1000, 2000, Ä 5000 Hz ܲ¡õyˆÏB˛ åÈÎ˚!ê˛
§%ˆÏÓ˚Ó˚ §Ç!ü¢ˆÏî à!ë˛ï˛ ~ÓÇ ≤Ã!ï˛!ê˛Ó˚ ï˛#Ó ï˛y 60db– ~áö xyüÓ˚y 14.8 !ã˛ˆÏeÓ˚ §ü≤ÃyÓ°ƒˆÏÓ˚áy=!°Ó˚
§y•yˆÏ΃ ≤Ã!ï˛!ê˛ fl∫ˆÏÓ˚Ó˚ ≤ÃyÓ°ƒüyey !öî≈Î˚ ܲˆÏÓ˚ S~Ó˚ çöƒ !ܲå%Èê˛y ˆã˛yˆÏá ˆòˆÏá xy®yˆÏçÓ˚ ≤ÈÏÎ˚yçö •ˆÏÓV
14.9 ˆ°á!ã˛e ˆÌˆÏܲ ≤Ã!ï˛!ê˛Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚– ~•z ≤ÃÓ°ï˛y=!°Ó˚ ˆÎyàú˛°•z •ˆÏÓ ˆüyê˛
ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y– !öˆÏã˛Ó˚ §yÓ˚!îˆÏï˛ ~ܲ•zû˛yˆÏÓ ˆüyê˛ ≤ÃÓ°ï˛y !öî≈Î˚ ܲÓ˚y •°– xy˛õ!ö !öˆÏç
ˆ°á!ã˛e=!°Ó˚ §y•yˆÏ΃ !öî≈Î˚!ê˛ §!ë˛Ü˛ •ˆÏÎ˚ˆÏåÈ !ܲ öy ˆòˆÏá !öö–
§yÓ˚!î §yÓ˚!î §yÓ˚!î §yÓ˚!î §yÓ˚!î 14.1 : !ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚!ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚!ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚!ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚!ü◊ܲ¡õyˆÏB˛Ó˚ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛y !öî≈Î˚
ܲ¡õyB˛(Hz) ï˛#Ó ï˛y(db) ≤ÃÓ°ƒüyeySú˛öV ≤ÃÓ°ï˛y Sˆ§yöV
100 60 36 0·7200 60 50 2·1500 60 59 4·31000 60 60 4·52000 60 61 4·9
5000 60 58 4·0
ˆüyê˛ ≤ÃÓ°ï˛y 20·5
365
~áyˆÏö ˆüyê˛ ≤ÃÓ°ï˛y 20·5 ˆ§yöñ ÎyÓ˚ çöƒ §yôyÓ˚îû˛yˆÏÓ 83 ú˛ö ≤ÃyÓ°ƒüyeyÓ˚ ≤ÈÏÎ˚yçö– ˆÜ˛Ó°üye
1000 Hz ܲ¡õyˆÏB˛Ó˚ ~ܲ!ê˛ !Ó÷k˛ §%Ó˚ 83 db ï˛#Ó ï˛y!Ó!¢‹T •ˆÏ° ˆ§!ê˛Ó˚ ≤ÃyÓ°ƒüyey 83 ú˛ö •ï˛– !ܲv
~ˆÏ«˛ˆÏe ï˛#Ó ï˛y åÈÎ˚!ê˛ 60db ï˛#Ó ï˛yÓ˚ ˆÎyàú˛° xÌ≈yÍ 60 + 10log 6 Óyñ 68 db– ~Ó˚ ˆÌˆÏܲ xy˛õ!ö !öÿ˛Î˚•z
Ó%é˛ˆÏï˛˛ ˛õyÓ˚ˆÏåÈö ˆÎñ ~ܲ•z ï˛#Ó ï˛y ~ܲ!ê˛ Ü˛¡õyˆÏB˛ ˆÜ˛w#û)˛ï˛ öy •ˆÏÎ˚ !Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ åÈí˛¸yˆÏöy ÌyܲˆÏ°
ˆ◊yï˛yÓ˚ ܲˆÏî≈ ï˛y x!ôܲï˛Ó˚ ≤ÃÓ°ï˛yÓ˚ xö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚– ~ÓyÓ˚ ~ܲê%˛ !ÓÓ˚!ï˛– xyÓ˚Ä ˛õí˛¸yÓ˚ xyˆÏà ~ܲ!ê˛
xö%¢#°ö# xy˛õöyˆÏܲ !Ó£ÏÎ˚!ê˛ Ó%é˛ˆÏï˛ §y•y΃ ܲÓ˚ˆÏÓ–
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# 3 :
~ܲ!ê˛ ¢∑ ˆÓ˚ܲ!í≈˛Ç ÈÙÈ ~Ó˚ fiê%˛!í˛ÄˆÏï˛ 30db ï˛#Ó ï˛yÓ˚ 100 Hz ܲ¡õyˆÏB˛Ó˚ ¢∑ñ ~ܲ!ê˛ Üœ˛y§âˆÏÓ˚ 70 db
ï˛#Ó ï˛yÓ˚ !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑ ~ÓÇ ~ܲ!ê˛ Ü˛yÓ˚áyöyÎ˚ 130 db ï˛#Ó ï˛yÓ˚ !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑ ˜ï˛!Ó˚ •Î˚– ~•z
!ï˛ö!ê˛ ˆ«˛ˆÏe ~ܲçö ˆ◊yï˛yÓ˚ ܲˆÏî≈ ˆÜ˛üö xö%û)˛!ï˛ •ˆÏÓ⁄
xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# xö%¢#°ö# 4 :
!öˆÏã˛Ó˚ í˛z!_´=!°Ó˚ üˆÏôƒ ˆÎ=!° !ë˛Ü˛ ˆ§=!°Ó˚ ˛õyˆÏ¢ !ê˛Ü‰˛ ã˛•´ !òö–
(a) °yí˛zí˛!flõܲyÓ˚ Óy ˛õê˛Ü˛yÓ˚ ¢∑ xyüyˆÏòÓ˚ ◊ÓîΈÏsf ˆÜ˛yöÄ «˛!ï˛ Ü˛ˆÏÓ˚ öy–
(b) 60 db ï˛#Ó ï˛yÓ˚ ¢ˆÏ∑Ó˚ ܲ¡õyB˛ 1000 Hz •ˆÏ° ï˛yÓ˚ ≤ÃyÓ°ƒüyey 60 ú˛öñ !ܲv ܲ¡õyB˛ ܲüˆÏ°
≤ÃyÓ°ƒüyeyÄ Ü˛ü •ˆÏÓ–
(c) ï˛#Ó ï˛y 10 db Óyí˛¸ˆÏ° ï˛yÓ˚ üyö 10 =î •Î˚ñ §%ï˛Ó˚yÇ ≤ÃyÓ°ƒüyey 10 ú˛ö ˛õ!Ó˚üyˆÏî Óyí˛¸ˆÏ° ˆ◊yï˛yÓ˚
ܲˆÏî≈ ï˛y 10 =î ˆçyÓ˚yˆÏ°y ӈϰ üˆÏö •ˆÏÓ–
(d) ¢ˆÏ∑Ó˚ ˆ◊yï˛y!öû≈˛Ó˚ ≤ÃÓ°ï˛y ˆüyê˛yü%!ê˛û˛yˆÏÓ ï˛#Ó ï˛yÓ˚ âöü)ˆÏ°Ó˚ §üyö%˛õyï˛#–
(e) ~ܲ•z ï˛#Ó ï˛y ~ܲ!ê˛üye ܲ¡õyˆÏB˛Ó˚ ˛õ!Ó˚ÓˆÏï≈˛ !Ó!û˛ß¨ ܲ¡õyˆÏB˛ !ÓöƒhflÏ •ˆÏ° ˆ◊yï˛yÓ˚ ܲyˆÏåÈ ¢∑!ê˛
≤ÃÓ°ï˛Ó˚ ӈϰ üˆÏö •Î˚–
14.4.7 §%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y§%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y§%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y§%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y§%Ó˚ Ä fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y (pitch)
àyö ÷öˆÏï˛ xy˛õ!ö !öÿ˛Î˚•z û˛yˆÏ°yÓyˆÏ§ö– àyö ˆ¢yöyÓ˚ §üÎ˚ xy˛õ!ö !öÿ˛Î˚•z ˆáÎ˚y° ܲˆÏÓ˚ö ˆÎñ àyÎ˚ܲ
Óy ày!Î˚ܲyÓ˚ ܲt˛ ~ܲÓyÓ˚ áyˆÏò ˆöˆÏü xyˆÏ§ xyÓyÓ˚ ã˛í˛¸yÎ˚ ĈÏ벖 ~ˆÏ«˛ˆÏe xy˛õöyÓ˚ ܲyˆÏö ~ܲê˛yöy ¢ˆÏ∑Ó˚
366
ˆÎ =î!ê˛ ôÓ˚y ˛õˆÏí˛¸ ˆ§!ê˛ •ˆÏ°y ï˛yÓ˚ ï˛#«¯˛ï˛y (pitch)– xÌ≈yÍñ ˆÜ˛yöÄ §%Ó˚ Óy §%fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y Ó°ˆÏï˛ xyüÓ˚y
Ó%!é˛ ˆ◊yï˛yÓ˚ xö%û)˛!ï˛ˆÏï˛ ˆ§!ê˛ Ü˛ï˛ê˛y ã˛í˛¸y ӈϰ üˆÏö •Î˚–
ˆÜ˛yöÄ §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y ≤Ãôyöï˛ ï˛yÓ˚ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– ï˛#«¯˛ï˛y Ä Ü˛¡õyˆÏB˛Ó˚ §¡õÜ≈˛ !öî≈Î˚
ܲÓ˚ˆÏï˛ ˆ◊yï˛yÓ˚ Óƒ!_´!öû≈˛Ó˚ (subjective) x!û˛üï˛ @ˇÃ•î ܲÓ˚y ≤ÈÏÎ˚yçöñ ˆÜ˛ööyñ ≤ÃyÓ°ƒüyeyÓ˚ üˆÏï˛y ï˛#«¯˛ï˛yÄ
§¡õ)î≈Ó˚*ˆÏ˛õ ˆ◊yï˛yÓ˚ xö%û)˛!ï˛Ó˚ myÓ˚y•z §Ç!K˛ï˛ •Î˚– ~•z §¡õÜ≈˛ ò%•zû˛yˆÏÓ !öî≈Î˚ ܲÓ˚y ÎyÎ˚–
(i) ˆ◊yï˛y §üyö ≤ÃyÓ°ƒüyeyÓ˚ (40 Óy 60 ú˛öV ò%•z!ê˛ !Ó÷k˛ §%Ó˚ ~ܲÓyÓ˚ ~!ê˛ñ xyÓ˚ ~ܲÓyÓ˚ xöƒ!ê˛ñ
~•zû˛yˆÏÓ ˆ¢yˆÏöö ~ÓÇ ~ܲ!ê˛Ó˚ ܲ¡õyB˛ !fliÓ˚ ˆÓ˚ˆÏá xöƒ!ê˛Ó˚ ܲ¡õyB˛ ~üöû˛yˆÏÓ í˛z˛õˆÏÎyçö ܲˆÏÓ˚ö ÎyˆÏï˛ ˆ§!ê˛Ó˚
ï˛#«¯˛ï˛y ≤ÃÌü!ê˛Ó˚ ï%˛°öyÎ˚ !ë˛Ü˛ xˆÏô≈ܲñ xÌ≈yÍ ~ܲ §Æܲ !öˆÏã˛ ÓˆÏ° üˆÏö •Î˚– ~•z ≤Ã!e´Î˚y!ê˛ ÓyÓ˚ ÓyÓ˚ ≤ÈÏÎ˚yà
ܲˆÏÓ˚ §ühflÏ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚ çöƒ ܲ¡õyˆÏB˛Ó˚ !•§yˆÏÓ ï˛#«¯˛ï˛yÓ˚ üy˛õܲy!ë˛ ˜ï˛!Ó˚ ܲÓ˚y ÎyÎ˚– ~•z ˛õk˛!ï˛ˆÏï˛
§yôyÓ˚îï˛ 1000Hz ܲ¡õyˆÏB˛Ó˚ !Ó÷k˛ §%Ó˚ˆÏܲ üyöܲ !•§yˆÏÓ ôÓ˚y •Î˚ ~ÓÇ ~!ê˛Ó˚ ï˛#«¯˛ï˛yˆÏܲ 1000 ˆü°
(•zÇÓ˚yç# melody ¢∑ ˆÌˆÏܲ í˛zq(ï˛V Ó°y •Î˚– ˆòáy ÎyÎ˚ ˆÎñ 60 ú˛ö ≤ÃyÓ°ƒüyeyÎ˚ ˆÎ ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛y ~Ó˚
xˆÏô≈ܲ xÌ≈yÍ 500 ˆü° ӈϰ üˆÏö •Î˚ñ ï˛yÓ˚ ܲ¡õyB˛ üye 400Hz– xyÓyÓ˚ ï˛#«¯˛ï˛y ~Ó˚ !m=î xÌ≈yÍ 2000
ˆü° •ˆÏï˛ •ˆÏ° ï˛yÓ˚ ܲ¡õyB˛ •Î˚ ≤ÃyÎ˚ 3100 Hz– ~Ó˚ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ ˆÎ ï˛#«¯˛ï˛y Ä Ü˛¡õyB˛ §üyö%˛õy!ï˛Ü˛
öÎ˚–
(ii) !mï˛#Î˚ ~ܲ!ê˛ ˛õk˛!ï˛ˆÏï˛ ˆ◊yï˛y §%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ §Ó≈!ö¡¨ ˆÎ ˛õ!Ó˚Óï≈˛ö
f Δ
ôÓ˚ˆÏï˛ ˛õyˆÏÓ˚ö ˆ§!ê˛ !öî≈Î˚
ܲÓ˚y •Î˚– àí˛¸ ܲ¡õyB˛ f Îáö 500Hz-~Ó˚ !öˆÏã˛ ï˛áö
f Δ
~Ó˚ üyö 60 ú˛ö ≤ÃÓ°ƒüyeyÓ˚ ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe
≤ÃyÎ˚ 3 Hz •ˆÏï˛ ˆòáy ÎyÎ˚– í˛zFã˛ï˛Ó˚ ܲ¡õyˆÏB˛
ffΔ
Ó˚y!¢!ê˛ ≤ÃyÎ˚ x˛õ!Ó˚Ó!ï≈˛ï˛ ÌyˆÏܲ ~ÓÇ ~Ó˚ üyö •Î˚ 0·002
ˆÌˆÏܲ 0·003– ôˆÏÓ˚ ˆöÄÎ˚y •Î˚ ˆÎñ ܲ¡õyB˛
f Δ
˛õ!Ó˚üyî ˛õ!Ó˚Ó!ï≈˛ï˛ •ˆÏ° ï˛#«¯˛ï˛y §Ó ܲ¡õyˆÏB˛•z ~ܲ•z ˛õ!Ó˚üyî
ܲˆÏü Óy ÓyˆÏí˛¸– §ü@ˇÃ ◊Óî˛õyÕ‘y ç%ˆÏí˛¸ ˛õÓ˚˛õÓ˚
f Δ
~Ó˚ üyö !öî≈Î˚ ܲˆÏÓ˚ ≤Ã!ï˛!ê˛ §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y Úˆü°Û ÈÙÈ ~Ó˚
!•§yˆÏÓ !öî≈Î˚ ܲÓ˚y •Î˚– ˆÜ˛yöÄ §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ í˛z˛õˆÏÓ˚Ä !ܲå%Èê˛y !öû≈˛Ó˚ ܲˆÏÓ˚– ܲ¡õyB˛ !fliÓ˚
ˆÓ˚ˆÏá ï˛#Ó ï˛y Óyí˛¸yˆÏ° ï˛#«¯˛ï˛y !ܲå%Èê˛y ܲü ӈϰ xö%û)˛ï˛ •Î˚– ï˛ˆÏÓ 100Hz-~Ó˚ ܲyåÈyܲy!åÈ Ü˛¡õyˆÏB˛•z ï˛#Ó ï˛yÓ˚
§ˆÏD ï˛#«¯˛ï˛yÓ˚ ˛õ!Ó˚Óï≈˛ö §Ó≈y!ôܲ •Î˚ñ 1000 HzÈÙÈ~Ó˚ x!ôܲ ܲ¡õyˆÏB˛ ï˛#«¯˛ï˛y ï˛#Ó ï˛yÓ˚ í˛z˛õÓ˚ !ӈϢ£Ï !öû≈˛Ó˚
ܲˆÏÓ˚ öy– xyÓyÓ˚ x!ï˛ í˛zFã˛ Ü˛¡õyˆÏB˛ (> 3000 Hz) ï˛#Ó ï˛y Óyí˛¸ˆÏ° ï˛#«¯˛ï˛yÄ ÓyˆÏí˛¸ñ ~üöÄ ˆòáy ˆàˆÏåÈ–
367
60 ú˛ö ˛≤ÃyÓ°ƒüyeyÎ˚ §%ˆÏÓ˚Ó˚ ܲ¡õyB˛ Ä ï˛#«¯˛ï˛yÓ˚ !öî#≈ï˛ §¡õÜ≈˛ 14.10 !ã˛ˆÏe ˆ°ˆÏáÓ˚ §y•yˆÏ΃
ˆòáyˆÏöy •ˆÏÎ˚ˆÏåÈ– ~áyˆÏö °«˛ƒ ܲÓ˚yÓ˚ !Ó£ÏÎ˚ ~•z ˆÎñ 1000 Hz ܲ¡õyˆÏB˛Ó˚ !öˆÏã˛ ˆü°ÈÙÈ~Ó˚ !•§yˆÏÓ ï˛#«¯˛ï˛y
§Ó≈òy•z ܲ¡õyˆÏB˛Ó˚ ˆã˛ˆÏÎ˚ ˆÓ!¢– !ܲv 1000 Hz ܲ¡õyˆÏB˛Ó˚ |ˆÏôù≈ ï˛#«¯˛ï˛y ܲ¡õyˆÏB˛Ó˚ ï%˛°öyÎ˚ e´ü¢ ܲü
•ˆÏï˛ ÌyˆÏܲ–
~ ˛õÎ≈hsˇ xyüÓ˚y ˆÎ xyˆÏ°yã˛öy ܲÓ˚°yü ï˛y ÷ô% !Ó÷k˛ §%ˆÏÓ˚Ó˚ ˆ«˛ˆÏe•z áyˆÏê˛– ~ܲy!ôܲ §%ˆÏÓ˚Ó˚ §üß∫ˆÏÎ˚
ˆÎ fl∫Ó˚ à!ë˛ï˛ •Î˚ ï˛yÓ˚ ï˛#«¯˛ï˛y Ü˛ï˛ •ˆÏÓ⁄ ~•z ≤Èϟ¿Ó˚ í˛z_Ó˚ §Ó˚y§!Ó˚ ˆòÄÎ˚y ÎyÎ˚ öy– §yôyÓ˚îû˛yˆÏÓ ˆÎ ¢ˆÏ∑
~ܲ!ê˛ ü)°§%Ó˚ Ä ï˛yÓ˚ í˛z˛õ§%Ó˚=!° !ü!◊ï˛ xyˆÏåÈñ ï˛yÓ˚ ï˛#«¯˛ï˛y ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– !ܲv
ܲ¡õyB˛ (Hz)
ï˛#«
¯˛ï˛yÓ
˚ Sˆü
°V
368
xöƒˆÏ«˛ˆÏe xÓfliyÓ˚ Óƒ!ï˛e´ü âˆÏê˛– ˛õÓ˚#«˛y ܲˆÏÓ˚ ˆòáy ˆàˆÏåÈñ 100, 200, 300,400 Ä 500 Hz ܲ¡õyˆÏB˛Ó˚
§üyö ï˛#Ó ï˛yÓ˚ §%ˆÏÓ˚Ó˚ §üß∫ˆÏÎ˚ ˆÎ fl∫Ó˚ à!ë˛ï˛ •Î˚ ï˛yÓ˚ ï˛#«¯˛ï˛y ≤ÃyÎ˚ 160 ˆü°– ~áö Î!ò 100 Hz ÈÙÈ ~Ó˚
§%Ó˚!ê˛ Ü˛!üˆÏÎ˚ ˆòÄÎ˚y ÎyÎ˚ ï˛ˆÏÓ xy˛õ!ö !öÿ˛Î˚•z xy¢y ܲÓ˚ˆÏÓö ˆÎñ ï˛#«¯˛ï˛yÓ˚ Ó,!k˛ âê˛ˆÏÓ– !ܲv ÓyhflψÏÓ ˆòáy
ÎyÎ˚ ˆÎñ ï˛#«¯˛ï˛y ~ܲ•z xÌ≈yÍ 160 ˆü° ÌyˆÏܲ– xyÓyÓ˚ 400, 500, 600, 700, 800, 900 Ä 1000 Hz
ܲ¡õyˆÏB˛Ó˚ §üyö ï˛#Ó ï˛yÓ˚ §yï˛!ê˛ §%ˆÏÓ˚Ó˚ !ü◊ˆÏîÓ˚ ï˛#«¯˛ï˛yÄ ≤ÃyÎ˚ 160 ˆü° •Î˚– !ܲv 500, 700 Ä 900
Hz ܲ¡õyˆÏB˛Ó˚ §%Ó˚=!° Óyò !òˆÏ° ï˛#«¯˛ï˛y ˆÓˆÏí˛¸ ≤ÃyÎ˚ 300 ˆü°ÈÙÈ~ òyÑí˛¸yÎ˚– ~ ˆÌˆÏܲ ~ê˛y•z ˆÓyé˛y ÎyÎ˚ ˆÎñ
ï˛#«¯˛ï˛y ˆÜ˛Ó°üye §Ó≈!ö¡¨ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– ÓÓ˚Ç Ü˛¡õyB˛=!°Ó˚ ÓƒÓôyˆÏöÓ˚ §ˆÏDÄ ï˛#«¯˛ï˛yÓ˚
â!ö¤˛ §¡õÜ≈˛ Ó˚ˆÏÎ˚ˆÏåÈ– ~•z âê˛öyÓ˚ ܲyÓ˚î xyüyˆÏòÓ˚ ܲî≈˛õê˛ˆÏ• ã˛y˛õ Ä ◊ÓîÈÙÈöyˆÏû≈˛Ó˚ í˛zj#˛õöyÓ˚ üˆÏôƒ x˜ÏÓ˚!áܲ
(non-linear) §¡õÜ≈˛– ~ §¡∫ˆÏ¶˛ xy˛õ!ö ˛õÓ˚Óï˛#≈ xLjϢ xyÓ˚Ä çyöˆÏï˛ ˛õyÓ˚ˆÏÓö–
ˆÜ˛yöÄ !ü◊ fl∫ˆÏÓ˚Ó˚ üˆÏôƒ Îáö Ó‡§ÇáƒÜ˛ ܲ¡õyB˛ á%Ó Ü˛yåÈyܲy!åÈ ÌyˆÏܲñ ï˛áö ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ xˆÏöܲê˛y
xMÈ˛° ~ܲˆÏe í˛zj#!˛õï˛ •Î˚ ~ÓÇ ˆÜ˛yöÄ !ö!ò≈‹T ï˛#«¯˛ï˛y•z ˆ◊yï˛y xö%û˛Ó ܲˆÏÓ˚ö öy– ~ çyï˛#Î˚ ¢∑ˆÏܲ xyüÓ˚y
x˛õfl∫Ó˚ Óy ˆÜ˛y°y•° (noise) ӈϰ Ìy!ܲ– x˛õÓ˚˛õˆÏ«˛ñ !ö!ò≈‹T ï˛#«¯˛ï˛yÓ˚ xö%û)˛!ï˛ çyàyˆÏöy ¢∑ ◊Óî§%áܲÓ˚
~ÓÇ ~ˆÏܲ•z xyüÓ˚y §%ˆÏÓ˚°y ¢∑ (musical sound) Ó!°–
§%ˆÏÓ˚°y ¢ˆÏ∑Ó˚ xyÓ˚ ~ܲ!ê˛ ôˆÏü≈Ó˚ í˛zˆÏÕ‘á ܲˆÏÓ˚ xyüÓ˚y ~•z xÇ¢!ê˛ ˆ¢£Ï ܲÓ˚Ó– ~!ê˛ •° §%ˆÏÓ˚°y ¢ˆÏ∑Ó˚
çy!ï˛ (timbre)– ¢ˆÏ∑Ó˚ ï˛Ó˚DyÜ,˛!ï˛Ó˚ §ˆÏD §Ç!Ÿ’‹T ~•z ˜Ó!¢‹Tƒ!ê˛Ó˚ çöƒ•z xyüÓ˚y ≤ÃÓ°ï˛y Ä ï˛#«¯˛ï˛y ~ܲ
•ÄÎ˚y §ˆÏ_¥Ä ÓyÑ!¢ Ä ˆÓ•y°yÓ˚ ¢ˆÏ∑Ó˚ üˆÏôƒ ˛õyÌ≈ܲƒ ôÓ˚ˆÏï˛ ˛õy!Ó˚– ï˛ˆÏÓ ~ê˛yÄ üˆÏö Ó˚yáˆÏï˛ •ˆÏÓ ˆÎñ §%ˆÏÓ˚°y
¢ˆÏ∑Ó˚ çy!ï˛ ˆÜ˛Ó°üye ï˛Ó˚DyÜ,˛!ï˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ öy– ~ܲ•z ¢∑ ˆÓ˚ܲ!í≈˛ÇÈÙÈ~Ó˚ ˛õÓ˚ x!ôܲ ≤ÃyÓ°ƒüyeyÎ˚
ÓyçyˆÏöy •ˆÏ° xÌÓy ÓyçyˆÏöyÓ˚ §üÎ˚ !í˛flÒ Óy ˆã˛Ô¡∫ܲ ˆê˛ˆÏ˛õÓ˚ à!ï˛ Óy!í˛¸ˆÏÎ˚ ܲ¡õyˆÏB˛Ó˚ Ó,!k˛ âê˛yˆÏ° ˛õ%öà≈!ë˛ï˛
¢∑ ܲyˆÏö ~ܲ•z çy!ï˛Ó˚ ӈϰ üˆÏö •Î˚ öy– çy!ï˛Ó˚ ˆÜ˛yöÄ ˛õ!Ó˚üyîàï˛ ˛õ!Ó˚üy˛õ §Ω˛Ó öÎ˚ ~ÓÇ ~!ê˛ §¡õ)î≈
Óƒ!_´!öû≈˛Ó˚ Ä =îàï˛ xö%û)˛!ï˛ !•§yˆÏÓ•z Ó!î≈ï˛ •Î˚–
˛õˆÏÓ˚Ó˚ xLjϢ ÎyÄÎ˚yÓ˚ xyˆÏà ~ܲ!ê˛ xö%¢#°ö#Ó˚ í˛z_Ó˚ !òö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 5 :
!öˆÏã˛Ó˚ Óyܲƒ=!°ˆÏï˛ ¢)öƒfliyö=!° í˛z˛õÎ%_´ ¢∑ !òˆÏÎ˚ ˛õ)î≈ ܲÓ˚&ö ≠ÈÙÙÙÈ
ˆÜ˛yöÄ §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y ≤Ãôyöï˛ ï˛yÓ˚..................ÈÙÈ~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲÓ˚ˆÏ°Ä..................§ˆÏDÄ ï˛#«¯˛ï˛yÓ˚
!ܲå%Èê˛y ˛õ!Ó˚Óï≈˛ö âê˛ˆÏï˛ ˛õyˆÏÓ˚–..................~ܲܲ ˆü°– 100 Hz ܲ¡õyˆÏB˛Ó˚ !Ó÷k˛ fl∫ˆÏÓ˚Ó˚
ï˛#«¯˛ï˛yˆÏܲ..................ˆü° ôÓ˚y •Î˚– §üyö ÓƒÓôyˆÏö Ìyܲy ܲ¡õyB˛ !Ó!¢‹T ܲˆÏÎ˚ܲ!ê˛ §üï˛#Ó ï˛yÓ˚ §%Ó˚ ~ܲˆÏe
ôù!öï˛ •ˆÏ° ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛y §%Ó˚=!°Ó˚..................í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚– xï˛ƒhsˇ ܲyåÈyܲy!åÈ Ìyܲy Ó‡§ÇáƒÜ˛
§%Ó˚ .................. §,!‹T ܲˆÏÓ˚– §%ˆÏÓ˚°y ¢ˆÏ∑Ó˚ ˆÎ ôˆÏü≈Ó˚ çöƒ xyüÓ˚y §yöy•z Ä ÓyÑ!¢Ó˚ ¢ˆÏ∑Ó˚ ˛õyÌ≈ܲƒ Ó%é˛ˆÏï˛
˛õy!Ó˚ñ ˆ§!ê˛ •° ï˛yÓ˚..................– ~•z ôü≈!ê˛ ü)°ï˛..................ÈÙÈ~Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚¢#° •ˆÏ°Ä..................Ä
ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚Ä !ܲå%Èê˛y !öû˛≈Ó˚ ܲˆÏÓ˚–
369
14.5 fl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫öfl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫öfl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫öfl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫öfl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫ö (Beats, aural harmonics and
combination tones)
xyüyˆÏòÓ˚ ◊ÓîÎsf!ê˛Ó˚ ܲyÎ≈ܲy!Ó˚ï˛y Ä !öçfl∫ ˜Ó!¢ˆÏ‹TƒÓ˚ ˆÓ¢ !ܲå%È ˛õ!Ó˚ã˛Î˚ xy˛õ!ö •z!ï˛üˆÏôƒ•z ˆ˛õˆÏÎ˚ˆÏåÈö–
~•z xLjϢ xy˛õ!ö ˆòáˆÏÓö ˆÎñ xyüyˆÏòÓ˚ ܲyˆÏö ~üö ܲ¡õyˆÏB˛Ó˚ xö%û)˛!ï˛ §,!‹T •ˆÏï˛ ˛õyˆÏÓ˚ Îy í˛zj#˛õܲ
ã˛y˛õï˛Ó˚ˆÏDÓ˚ Óy ã˛y˛õï˛Ó˚D§ü)ˆÏ•Ó˚ ˆÜ˛yöÄ!ê˛Ó˚ ܲ¡õyˆÏB˛Ó˚ §üyö öÎ˚– öï%˛ö ~•z ܲ¡õyB˛=!°Ó˚ §,!‹T ܲyˆÏöÓ˚
Îy!sfܲ ÓƒÓfliyÓ˚ üˆÏôƒ•z âˆÏê˛ ÌyˆÏܲ ~ÓÇ ¢∑ @ˇÃy•ˆÏܲÓ˚ à!ï˛≤ÃÜ,˛!ï˛Ó˚ ày!î!ï˛Ü˛ !ӈϟ’£Ïî myÓ˚y ~=!°Ó˚ Óƒyáƒy
ˆòÄÎ˚y §Ω˛Ó– fl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫ö ~•z çyï˛#Î˚ ܲ¡õyˆÏB˛Ó˚ í˛zòy•Ó˚î– ~áö ˆòáy Îyܲñ ܲ#û˛yˆÏÓ
~=!° í˛zͲõߨ •Î˚–
fl∫Ó˚ܲ¡õ fl∫Ó˚ܲ¡õ fl∫Ó˚ܲ¡õ fl∫Ó˚ܲ¡õ fl∫Ó˚ܲ¡õ ≠ ò%•z!ê˛ §üˆÏÓ˚á §Ó˚° ˆòy°à!ï˛ Îáö ≤ÃyÎ˚ §üyö ܲ¡õyˆÏB˛Ó˚ •Î˚ ï˛áö ï˛yˆÏòÓ˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚
ú˛ˆÏ° ~üö ~ܲ!ê˛ ˆòy°à!ï˛ í˛zͲõߨ •Î˚ ÎyÓ˚ ܲ¡õyB˛ §Ó˚° ˆòy°à!ï˛ ò%•z!ê˛Ó˚ ܲ¡õyˆÏB˛Ó˚ àí˛¸ ~ÓÇ !ÓhflÏyÓ˚
ܲ¡õyB˛ ò%•z!ê˛Ó˚ xhsˇˆÏÓ˚Ó˚ §üyö ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •ˆÏï˛ ÌyˆÏܲ– EPH 03 ˆÜ˛yˆÏ§≈ xy˛õ!ö
~ !ӣψÏÎ˚ !Ó¢òû˛yˆÏÓ ˛õˆÏí˛¸ˆÏåÈö– ≤ÃyÎ˚ §üyö ~ÓÇ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ §%Ó˚ Îáö ܲˆÏî≈ ◊&ï˛ •Î˚ñ ï˛áö ˆ§=!°
ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ~ܲ•z xLjϢ í˛zj#˛õöy §,!‹T ܲˆÏÓ˚– ï˛ˆÏÓ §%Ó˚ ò%•z!ê˛Ó˚ ò¢y e´üyß∫ˆÏÎ˚ §ü Ä !Ó˛õÓ˚#ï˛ •ÄÎ˚yÓ˚
˛ú˛ˆÏ° ˆÓ!§°yÓ˚ ˛õò≈yÓ˚ ܲ¡õˆÏöÓ˚ !ÓhflÏyÓ˚ Äë˛yöyüy ܲÓ˚ˆÏï˛ ÌyˆÏܲñ ú˛ˆÏ° í˛zj#˛õöyÓ˚ üyˆÏöÄ xö%Ó˚*˛õ ï˛yÓ˚ï˛üƒ
âˆÏê˛– ~Ó˚ ú˛ˆÏ° ˆ◊yï˛y üyé˛yüy!é˛ Ü˛¡õyˆÏB˛Ó˚ ¢∑ ÷öˆÏï˛ ˛õyö !ܲv ï˛yÑÓ˚ xö%û)˛ï˛ ≤ÃyÓ°ƒüyey ܲ!¡õï˛ •ˆÏï˛
ÌyˆÏܲ– ~ˆÏܲ•z xyüÓ˚y fl∫Ó˚ܲ¡õ Ó!°–
ò%•z!ê˛ §%ˆÏÓ˚Ó˚ ܲ¡õyˆÏB˛Ó˚ üˆÏôƒ §üï˛y xyöyÓ˚ çöƒ ~ܲ!ê˛Ó˚ ܲ¡õyB˛ !fliÓ˚ ˆÓ˚ˆÏá xöƒ!ê˛ í˛z˛õˆÏÎy!çï˛ (adjust)
ܲÓ˚y •Î˚ ÎyˆÏï˛ fl∫Ó˚ܲˆÏ¡õÓ˚ ܲ¡õyB˛ e´ü¢ ܲˆÏü ¢)öƒ •Î˚– ÓyòƒÎˆÏsfÓ˚ §%Ó˚ ÓyÑôyÓ˚ çöƒ ~!ê˛ ~ܲ!ê˛ ≤Ãã˛!°ï˛
í˛z˛õyÎ˚– •yˆÏüy≈!öÎ˚yü ΈÏsfÓ˚ ~ˆÏܲÓyˆÏÓ˚ áyˆÏòÓ˚ xÌ≈yÍ Óyü!òˆÏܲÓ˚ ˛õÓ˚˛õÓ˚ ò%!ê˛ !Ó˚í˛ ~ܲ§ˆÏD !ê˛ˆÏ˛õ Îsf!ê˛ ÓyçyˆÏ°
xy˛õ!ö fl∫Ó˚ܲ¡õ ÷öˆÏï˛ ˛õyˆÏÓö–
◊&!ï˛ í˛z˛õ§%Ó˚ ≠ ◊&!ï˛ í˛z˛õ§%Ó˚ ≠ ◊&!ï˛ í˛z˛õ§%Ó˚ ≠ ◊&!ï˛ í˛z˛õ§%Ó˚ ≠ ◊&!ï˛ í˛z˛õ§%Ó˚ ≠ fl∫Ó˚ܲ¡õ í˛zͲõyòˆÏö xyüyˆÏòÓ˚ ◊ÓîÎsf ˜Ó!áܲû˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ xÌ≈yÍ ˆÓ!§°yÓ˚ ˛õò≈yÓ˚
í˛zj#˛õöy ◊ÓîÈÙÈöyˆÏû≈˛ ˆÎ §yí˛¸y (response) ˜ï˛!Ó˚ ܲˆÏÓ˚ ï˛y í˛zj#˛õöyÓ˚ §ˆÏD ˜Ó˚!áܲû˛yˆÏÓ ˛õ!Ó˚Ó!ï≈˛ï˛ •Î˚– ¢ˆÏ∑Ó˚
ï˛#Ó ï˛y x“ •ˆÏ° í˛z!j˛õܲ ¢∑ã˛y˛õ Ä öyˆÏû˛≈Ó˚ §yí˛¸yˆÏܲ §üyö%˛õyï˛# ӈϰ ôÓ˚y ÎyÎ˚– !ܲv ï˛#Ó ï˛y Ó,!k˛ ˆ˛õˆÏ°
§yí˛¸yÓ˚ ~•z ˜Ó˚!áܲï˛y xyÓ˚ ÓçyÎ˚ ÌyˆÏܲ öy– ~•z xÓfliyÎ˚ ~ܲ!ê˛ !Ó÷k˛ §%Ó˚ x!ôܲ ï˛#Ó ï˛yÓ˚ ôù!öï˛ •ˆÏ°
ˆ◊yï˛y !Ó÷k˛ §%Ó˚!ê˛Ó˚ ܲ¡õyˆÏB˛Ó˚ =!îï˛Ü˛ ܲ¡õyˆÏB˛Ó˚ §%Ó˚Ä ÷öˆÏï˛˛ ˛õyö– ~•z =!îï˛Ü˛ ܲ¡õyˆÏB˛Ó˚ §%Ó˚=!°ˆÏܲ•z
◊&!ï˛ í˛z˛õ§%Ó˚ Ó°y •Î˚–
◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ í˛zͲõ!_ ày!î!ï˛Ü˛û˛yˆÏÓ Óƒyáƒy ܲÓ˚y ÎyÎ˚– ôÓ˚y Îyܲ ¢∑ ã˛y˛õ P Ä öyˆÏû˛≈Ó˚ §yí˛¸y R-~Ó˚
üˆÏôƒ §¡õÜ≈˛ ≠
R = ap + bp2 + cp3 ...14.3
370
bp2 Ä cp3 Ó˚y!¢mÎ˚ ~áyˆÏö í˛zû˛ˆÏÎ˚Ó˚ §¡õˆÏÜ≈˛Ó˚ x˜ÏÓ˚!áܲï˛y (non-linearity) !öˆÏò≈¢ ܲˆÏÓ˚ˆÏåÈ– ~áö ôÓ˚y
Îyܲ ¢∑ã˛y˛õ n ܲ¡õyˆÏB˛ §Ó˚° ˆòy°à!ï˛ˆÏï˛ Ü˛!¡õï˛ •Î˚– p = P cos 2
π
nt !°áˆÏ°ñ
R = aPcos2
π
nt + bP2cos22
π
nt + cP3cos32
π
nt
= aPcos2
π
nt +
12
bP2(cos 4
π
nt + 1) +
14
cP3 (cos 6
π
nt + 3cos 2
π
nt )
Óyñ R =
3 34
aPcP ⎛⎞ + ⎜⎟ ⎝⎠
cos2
π
nt +
12
bP2cos4
π
nt +
14
cP3cos 6
π
nt +
12
bP2 ...14.4
14.4 §ü#ܲÓ˚ˆÏîÓ˚ í˛yö!òˆÏܲÓ˚ ≤ÃÌü !ï˛ö!ê˛ Ó˚y!¢ ÎÌye´ˆÏü n, 2n Ä 3n ܲ¡õyˆÏB˛Ó˚ xyˆÏ®y°ö §)!ã˛ï˛ ܲÓ˚ˆÏåÈ–
~ ˆÌˆÏܲ ≤Ãüy!îï˛ •Î˚ ˆÎñ ¢∑ã˛yˆÏ˛õÓ˚ ˛õ!Ó˚Óï≈˛ö ˆÜ˛Ó°üye n ܲ¡õyˆÏB˛ •ˆÏ°Ä ◊ÓîΈÏsfÓ˚ x˜ÏÓ˚!áܲï˛yÓ˚
ú˛ˆÏ° §yí˛¸yÓ˚ üˆÏôƒ í˛zFã˛ï˛Ó˚ í˛z˛õ§%Ó˚Ä Óï≈˛üyö ÌyܲˆÏÓ– ~=!° ◊&!ï˛ í˛z˛õ§%Ó˚–
◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ í˛z˛õ!fli!ï˛ á%Ó §•ˆÏç•z ˆòáyˆÏöy •Î˚– ôÓ˚y Îyܲñ ˆÜ˛yöÄ ˆ◊yï˛yˆÏܲ 200 Hz ܲ¡õyˆÏB˛Ó˚
~ܲ!ê˛ !Ó÷k˛ §%Ó˚ ˆÓ¢ í˛zFã˛ ï˛#Ó ï˛yÎ˚ ˆ¢yöyˆÏöy •ˆÏ°– ˆ◊yï˛y 200 Hz ܲ¡õyˆÏB˛Ó˚ §%Ó˚!ê˛Ó˚ §ˆÏD 400 Hz,
600 Hz ≤Ãû,˛!ï˛ Ü˛¡õyˆÏB˛Ó˚ §%ˆÏÓ˚Ó˚ xö%û)˛!ï˛Ä °yû˛ ܲÓ˚ˆÏÓö– ~áö Î!ò 403 Hz ܲ¡õyˆÏB˛Ó˚ xyÓ˚Ä ~ܲ!ê˛
!Ó÷k˛ §%Ó˚ ï˛yˆÏܲ ˆ¢yöyˆÏöy •Î˚ ï˛ˆÏÓ !ï˛!ö 400 Hz ܲ¡õyˆÏB˛Ó˚ í˛z˛õ§%ˆÏÓ˚Ó˚ §ˆÏD ~•z §%Ó˚!ê˛Ó˚ í˛z˛õ!Ó˚˛õyï˛ˆÏöÓ˚
ú˛ˆÏ° 3Hz ܲ¡õyˆÏB˛Ó˚ fl∫Ó˚Ü˛Ï¡õÄ ÷öˆÏï˛ ˛õyˆÏÓö– ~ˆÏï˛ 400 Hz í˛z˛õ§%ˆÏÓ˚Ó˚ í˛z˛õ!fli!ï˛ !ö!ÿ˛ï˛û˛yˆÏÓ ≤Ãüy!îï˛
•Î˚– 100 Hz-~Ó˚ ܲyåÈyܲy!åÈ Ü˛¡õyˆÏB˛Ó˚ !Ó÷k˛ §%Ó˚ xˆÏ˛õ«˛yÜ,˛ï˛ §•ˆÏç•z ◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ §,!‹T ܲÓ˚ˆÏï˛ ˛õyˆÏÓ˚–
≤ÃyÓ°ƒüyey 20 ú˛ö ˆÌˆÏܲ Óyí˛¸ˆÏï˛ ÌyܲˆÏ° e´ü¢ !mï˛#Î˚ñ ï,˛ï˛#Î˚ ≤Ãû,˛!ï˛ í˛z˛õ§%Ó˚à!° ˆ¢yöy ÎyÎ˚–
Î%_´fl∫ö ≠ Î%_´fl∫ö ≠ Î%_´fl∫ö ≠ Î%_´fl∫ö ≠ Î%_´fl∫ö ≠ ≤ÃyÎ˚ §üyö ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ !Ó÷k˛ §%Ó˚ Î!ò ~ܲ•z §ˆÏD ◊&ï˛ •Î˚ñ ï˛ˆÏÓ ï˛yÓ˚ ú˛° xy˛õ!ö
•z!ï˛üˆÏôƒ•z ˆçˆÏöˆÏåÈö– !ܲv §%Ó˚ ò%•z!ê˛Ó˚ ܲ¡õyB˛ Î!ò ~ï˛ê˛y•z ˛õ,Ìܲ •Î˚ ˆÎñ ï˛yˆÏòÓ˚ ÓƒÓôyöÄ ◊yÓƒ ܲ¡õyˆÏB˛Ó˚
üˆÏôƒ ˛õˆÏí˛¸ñ ï˛áö ܲ¡õyB˛mˆÏÎ˚Ó˚ !ÓˆÏÎ˚yàú˛° ~ÓÇ ˆÎyàú˛ˆÏ°Ó˚ §üyö xÌ≈yÍ §ü!‹T Ä xyhsˇÓ˚ ܲ¡õyˆÏB˛Ó˚ §%Ó˚Ä
ˆ¢yöy ÎyÎ˚– ~•z §%Ó˚=!°ˆÏܲ xyüÓ˚y Î%_´fl∫ö Ó!°–
◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ üˆÏï˛y Î%_´fl∫ˆÏöÓ˚ í˛zͲõ!_Ä ◊ÓîΈÏsfÓ˚ x˜ÏÓ˚!áܲï˛yÎ˚ çöƒ•z âˆÏê˛– 14.3 §ü#ܲÓ˚ˆÏîÓ˚
xˆÏ˛õ«˛yÜ,˛ï˛ «%˛o Ó˚y!¢ cp3-ˆÜ˛ í˛zˆÏ˛õ«˛y ܲˆÏÓ˚ xyüÓ˚y !°áˆÏï˛ ˛õy!Ó˚–
R = ap + bp2 (R = öyˆÏû≈˛Ó˚ §yí˛¸yñ p = ¢∑ã˛y˛õV
ôˆÏÓ˚ !ööñ ¢∑ã˛y˛õ n Ä m ò%•z ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ ˛õ,Ìܲ !Ó÷k˛ §%ˆÏÓ˚Ó˚ myÓ˚y í˛zͲõߨ xÌ≈yÍñ
p = P1 cos 2
π
nt + P2cos 2
π
mt
¢∑ã˛yˆÏ˛õÓ˚ Ä•z Ó˚y!¢ ÓƒÓ•yÓ˚ ܲˆÏÓ˚ñ
371
R = a (P1cos 2
π
nt + P2cos 2
π
mt) + b(P12cos22
π
mt + P22 cos22
π
mt + 2P1P
2 cos2
π
ntcos2
π
mt)
= a (P1 cos2
π
nt + P2 cos2
π
mt) +
12
bP12 (cos 4
π
nt + 1) +
12
bP2
2 (cos 4
π
mt + 1)
+ bP1P
2[cos 2
π
(n + m)t + cos 2
π
(n – m)t] ...14.5
°«˛ƒ ܲˆÏÓ˚ ˆòá%ö R - ~Ó˚ Ó˚y!¢!ê˛Ó˚ üˆÏôƒ n, m, 2n, 2m åÈyí˛¸yÄ n + m Ä n – m ܲ¡õyˆÏB˛Ó˚ §ü!‹T
Ä xyhsˇÓ˚ í˛z˛õyÇ¢ Ó˚ˆÏÎ˚ˆÏåÈ– ~Ó˚ ˆÌˆÏܲ ˆÓyé˛y ÎyÎ˚ñÎ%_´fl∫ö=!° ◊ÓîΈÏsfÓ˚ x˜ÏÓ˚!áܲï˛yÓ˚ ú˛ˆÏ°•z í˛zͲõߨ •Î˚–
!Óí˛¸yˆÏ°Ó˚ ܲyˆÏöÓ˚ ¢Cöy°#ÈÙÈ!Óû˛ˆÏÓÓ˚ ˛õ!Ó˚üy˛õ ܲˆÏÓ˚ Î%_´fl∫ˆÏöÓ˚ x!hflÏc ˆòáy ˆàˆÏåÈ– 700 Hz Ä 1200
Hz ܲ¡õyˆÏB˛Ó˚ ò%•z!ê˛ !Ó÷k˛ §%Ó˚ ܲˆÏî≈ ≤ÈÏÓ¢ ܲ!Ó˚ˆÏÎ˚ ¢Cöy°#ÈÙÈ!Óû˛ˆÏÓ 500 Hz, 1900 Hz, 1700 Hz,
Ä 3100 Hz ܲ¡õyˆÏB˛Ó˚ §yí˛¸yÓ˚ §¶˛yö ˛õyÄÎ˚y ÎyÎ˚– °«˛ƒ ܲÓ˚&öñ ~•z ܲ¡õyB˛=!° ÎÌye´ˆÏü 1200 –700,
1200 + 700, 2 × 1200 – 700 Ä 2 × 1200 + 700 Hz - ~Ó˚ §üyö–
◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ ˆ«˛ˆÏe Ó!î≈ï˛ ˛õk˛!ï˛ˆÏï˛Ä Î%_´fl∫ö=!°Ó˚ §¶˛yö ܲÓ˚y ÎyÎ˚– ~ˆÏ«˛ˆÏe 700 Ä 1200 Hz
ܲ¡õyˆÏB˛Ó˚ ò%!ê˛ !Ó÷k˛ §%Ó˚ ܲyˆÏö ≤ÈÏÓ¢ ܲ!Ó˚ˆÏÎ˚ ¢Cöy°# !Óû˛ˆÏÓ 500 Hz, 1900 Hz, 1700 Hz Ä 3100
Hz ܲ¡õyˆÏB˛Ó˚ §yí˛¸yÓ˚ §¶˛yö ˛õyÄÎ˚y ÎyÎ˚– °«˛ƒ ܲÓ˚&öñ ~•z ܲ¡õyB˛=!° ÎÌye´ˆÏü 1200 – 700, 1200
+ 700, 2 × 1200 + 700 Hz Ä 2 × 1200 + 700 Hz ~Ó˚ §üyö–
◊&!ï˛ í˛z˛õ§%ˆÏÓ˚Ó˚ ˆ«˛ˆÏe Ó!î≈ï˛ ˛õk˛!ï˛ˆÏï˛Ä Î%_´fl∫ö=!°Ó˚ §¶˛yö ܲÓ˚y ÎyÎ˚– ~ˆÏ«˛ˆÏe 700 Hz Ä 1200
Hz ܲ¡õyˆÏB˛Ó˚ ò%!ê˛ !Ó÷k˛ §%Ó˚ ˆüyê˛yü%!ê˛ í˛zFã˛ ï˛#Ó ï˛yÎ˚ ôù!öï˛ Ü˛ˆÏÓ˚ ~=!°Ó˚ §ˆÏD ï,˛ï˛#Î˚ ~ܲ!ê˛
í˛z˛õˆÏÎyçöˆÏÎyàƒ §¶˛yö# §%Ó˚ ˆ◊yï˛yˆÏܲ ˆ¢yöyˆÏöy •Î˚– §¶˛yö# §%Ó˚!ê˛Ó˚ ܲ¡õyB˛ Îáö 500 Hz, 1900 Hz
≤Ãû,˛!ï˛Ó˚ ܲyåÈyܲy!åÈ •Î˚ñ ï˛áö ˆ◊yï˛y fl∫Ó˚ܲ¡õ ÷öˆÏï˛ ˛õyö ~ÓÇ ~Ó˚ myÓ˚y !Ó!û˛ß¨ §ü!‹T Ä xyhsˇÓ˚ ܲ¡õyˆÏB˛Ó˚
í˛z˛õ!fli!ï˛ §)!ã˛ï˛ •Î˚–
~•z ≤çˆÏD ~ܲ!ê˛ Ü˛Ìy Ó°y ˆÎˆÏï˛ ˛õyˆÏÓ˚– xyç ˆÌˆÏܲ ˆòí˛¸¢ ÓåȈÏÓ˚Ó˚Ä xyˆÏà !ÓK˛yö# Äü‰ (G.S. Ohm,
1787-1854) ~•z !§k˛yˆÏhsˇ ~ˆÏ§!åȈϰö ˆÎñ xyüyˆÏòÓ˚ ܲyˆÏö !Ó÷k˛ §Ó˚° ˆòy°à!ï˛Ó˚ ܲ¡õö ~ܲ!ê˛üye §%ˆÏÓ˚Ó˚
xö%û)˛!ï˛ §,!‹T ܲˆÏÓ˚ñ !ܲv xöƒ ôÓ˚ˆÏöÓ˚ ˛õÎ≈yÓ,_ ܲ¡õö ܲyˆÏö !ӈϟ’!£Ïï˛ •Î˚ ~ÓÇ ÎˆÏÌ‹T ï˛#Ó ï˛y ÌyܲˆÏ° ≤Ã!ï˛!ê˛
í˛z˛õ§%Ó˚ ˛õ,Ìܲû˛yˆÏÓ xö%û)˛ï˛ •Î˚– !ܲv ~•z xLjϢ xyüÓ˚y °«˛ƒ ܲÓ˚°yü ˆÎñ ܲyˆÏö ~üö ܲ¡õyˆÏB˛Ó˚ xö%û)˛!ï˛
§,!‹T •ˆÏï˛ ˛õyˆÏÓ˚ Îy ◊&ï˛ ¢ˆÏ∑ §¡õ)î≈ xö%˛õ!fliï˛– §%ï˛Ó˚yÇñ Äüñ ÈÙÈ ~Ó˚ !§k˛yhsˇ §¡õ)î≈ §ï˛ƒ öÎ˚–
14.6 !mܲî#≈Î˚ !òܲ !öî≈Î˚!mܲî#≈Î˚ !òܲ !öî≈Î˚!mܲî#≈Î˚ !òܲ !öî≈Î˚!mܲî#≈Î˚ !òܲ !öî≈Î˚!mܲî#≈Î˚ !òܲ !öî≈Î˚ (Binaural localization)
xy˛õöyÓ˚ •Î˚ï˛ üˆÏö •ˆÏÎ˚ˆÏåÈ ˆÎñ ¢ˆÏ∑Ó˚ ≤ÃÓ°ï˛yñ ï˛#«¯˛ï˛y Óy çy!ï˛ xö%û˛Ó ܲÓ˚ˆÏï˛ ~ܲ!ê˛ Ü˛yö•z ΈÏÌ‹T–
§%ï˛Ó˚yÇñ üyö%£Ï Óy xöƒyöƒ hflÏöƒ˛õyÎ˚# ç#ˆÏÓÓ˚ ò%•z!ê˛ Ü˛yˆÏöÓ˚ !ܲ ≤ÈÏÎ˚yçö⁄ ~Ó˚ í˛z_ˆÏÓ˚ Ó°y ÎyÎ˚ ˆÎñ xyüÓ˚y
ˆÎ ¢∑ñ ÷!ö ï˛yÓ˚ í˛zͧ!ê˛ ˆÜ˛yÌyÎ˚ ˆ§ê˛yÄ xyüyˆÏòÓ˚ çyöyÓ˚ ≤ÈÏÎ˚yçö •Î˚– üˆÏö ܲÓ˚&öñ !¢«˛y §•yÎ˚ܲ xö%¤˛yˆÏö
372
ˆÜ˛yö ~ܲçö !¢«˛yÌ#≈ ~ܲ!ê˛ ≤ß¿ ܲÓ˚ˆÏ°ö– !¢«˛Ü˛ ü•y¢ˆÏÎ˚Ó˚ ÷ô% ≤ß¿!ê˛ ÷öˆÏ°•z ã˛°ˆÏÓ öyñ ï˛yшÏܲ çyöˆÏï˛
•ˆÏÓñ ˆÜ˛ ≤ß¿!ê˛ Ü˛Ó˚ˆÏ°ö xÌ≈yÍ ˆÜ˛yö !òˆÏܲ ˆÌˆÏܲ ≤Èϟ¿Ó˚ ¢∑!ê˛ ~°– ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ !òܲ !öî≈Î˚ ܲÓ˚yÓ˚ çöƒ
ò%•z ܲyˆÏö ¢∑!ê˛ ˆ¢yöy ~ܲyhsˇ•z ≤ÈÏÎ˚yçö–
ò%•z ܲyˆÏö ¢∑ ˆ¢yöyÓ˚ ú˛ˆÏ° ò%!ê˛ ˛õ,Ìܲ í˛z˛õyˆÏÎ˚ xyüÓ˚y ¢ˆÏ∑Ó˚ !òܲ !öî≈Î˚ ܲ!Ó˚– ~=!° •ˆÏ°y–
ò%•z ܲyˆÏö xö%û)˛ï˛ ≤ÃÓ°ï˛yÓ˚ !û˛!_ˆÏï˛ ï˛#Ó ï˛yÓ˚ ï%˛°öyü)°Ü˛ !Óã˛yÓ˚
«˛îfliyÎ˚# ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe ò%•z ܲyˆÏö ¢∑ ˆ˛õÔÑåȈÏöyÓ˚ §üˆÏÎ˚Ó˚ ÓƒÓôyöñ xÌÓy ò#â≈fliyÎ˚# ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe ◊&ï˛
¢ˆÏ∑Ó˚ ò¢yÓ˚ ≤ÈÏû˛ò–
~•z ò%•z!ê˛ í˛z˛õyÎ˚ §¡∫ˆÏ¶˛ xyüÓ˚y˛ ˛õ,Ìܲû˛yˆÏÓ xyˆÏöyã˛öy ܲÓ˚Ó–
ò%•z ܲyˆÏö ◊&ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÓ˚ ≤ÈÏû˛ò •ÄÎ˚yÓ˚ ܲyÓ˚î ˆ◊yï˛yÓ˚ ühflÏܲ Ä Ü˛yö ˛õeˆÏܲÓ˚ ¢∑åÈyÎ˚y– ~ˆÏ«˛ˆÏe
ˆÎ §yôyÓ˚î ö#!ï ≤ÈÏÎyçƒñ ˆ§!ê˛ •°ñ xöFåÈ Ó›Ó˚ xyܲyÓ˚ Îáö ï˛Ó˚ˆÏDÓ˚ ˜òˆÏâ≈ƒÓ˚ ï%˛°öyÎ˚ «%˛o •Î˚ ï˛áö
ˆ§!ê˛ ˙ ï˛Ó˚ˆÏDÓ˚ çöƒ ˆÜ˛yöÄ åÈyÎ˚yMÈ˛° §,!‹T ܲˆÏÓ˚ öy– ¢∑ï˛Ó˚ˆÏB˛Ó˚ ܲ¡õyB˛ 1000 Hz-~Ó˚ ܲü xÌ≈≈yÍ
ï˛Ó˚D˜Ïòâ≈ƒ 35 cm xˆÏ˛õ«˛y ˆÓ!¢ •ˆÏ° ˆ◊yï˛yÓ˚ ühflÏܲñ ˆÎ!ê˛Ó˚ Óƒy§ ≤ÃyÎ˚ 20 cmñ ˆÜ˛yöÄ åÈyÎ˚yMÈ˛° §,!‹T
ܲˆÏÓ˚ öy ~ÓÇ ò%•z ܲyˆÏö ◊&ï˛ ¢ˆÏ∑Ó˚ ï˛#Ó ï˛yÎ˚ !ӈϢ£Ï ˛õyÌ≈ܲƒ ÌyˆÏܲ öy– 1000 Hz-~Ó˚ x!ôܲ ܲ¡õyˆÏB˛ ühflψÏܲÓ˚
¢∑åÈyÎ˚yÎ˚ xÓ!fliï˛ •ÄÎ˚yÓ˚ ú˛ˆÏ° ¢∑ í˛zͧ x!û˛ü%á# ܲyˆÏöÓ˚ ï%˛°öyÎ˚ ò)Ó˚Óï˛#≈ ܲyˆÏö ˆ˛õÔÑåȈÏöy ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y
ܲü •Î˚– ˆ◊yï˛yÓ˚ ò%•z ܲyˆÏö ˆê˛!°ˆÏú˛yö !Ó˚!§û˛yˆÏÓ˚Ó˚ §y•yˆÏ΃ §yüyöƒ ܲü ˆÓ¢# ï˛#Ó ï˛yÓ˚ ¢∑ ˆ˛õÔшÏåÈ !òˆÏÎ˚
ˆòáy ˆàˆÏåÈ 1000 Hz ~Ó˚ x!ôܲ ܲ¡õyˆÏB˛Ó˚ !Ó÷k˛ §%ˆÏÓ˚Ó˚ ˆ«˛ˆÏe ˆ◊yï˛y ˆÎ !òˆÏܲÓ˚ ܲyˆÏö ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y
ˆÓ!¢ñ ¢ˆÏ∑Ó˚ í˛zͧ ˆ§•z xö%ÎyÎ˚# Óyü Óy í˛yö˛õyˆÏ¢ ӈϰ üˆÏö ܲˆÏÓ˚ö– 3000 Hz-~Ó˚ x!ôܲ ܲ¡õyˆÏB˛ ܲî≈˛õeܲ
ò%!ê˛Ä ◊Óîöy!°Ü˛yÓ˚ §yüˆÏö åÈyÎ˚yÓ˚ §,!‹T ܲˆÏÓ˚ñ ú˛ˆÏ° ˆ◊yï˛y ¢ˆÏ∑Ó˚ í˛zͧ ï˛yÓ˚ §yüˆÏö öy !˛õåȈÏö xyˆÏåÈ ï˛yÄ
Ó%é˛ˆÏï˛ ˛õyˆÏÓ˚ö–
x˛õÓ˚˛õˆÏ«˛ñ ò%•z ܲyˆÏö ˆ˛õÔÑåÈyˆÏöy !Ó÷k˛ ~ܲ §%ˆÏÓ˚Ó˚ ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe ò¢yÓ˚ ≤ÈÏû˛ò ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ ~ÓÇ
¢ˆÏ∑Ó˚ ˆ◊yï˛yÓ˚ ˆ§yçy§%!ç §yüˆÏöÓ˚ !òˆÏܲ ˆÌˆÏܲ Óyü Óy í˛yö!òˆÏܲ ܲï˛ê˛y ˆÜ˛yˆÏî §ˆÏÓ˚ xyˆÏåÈñ ï˛yÓ˚ í˛z˛õÓ˚
!öû≈˛Ó˚ ܲˆÏÓ˚– í˛zͧ!ê˛ §¡õ)î≈ ~ܲ˛õyˆÏ¢ ÌyܲˆÏ° ò¢yÓ˚ ≤ÈÏû˛ò •ˆÏÓ
2dπλ
, ˆÎáyˆÏö d = ühflψÏܲÓ˚ Óƒy§ Ä
λ
= ¢ˆÏ∑Ó˚ ï˛Ó˚D˜Ïòâ≈ƒ– 150 Hz-~Ó˚ üˆÏï˛y x!ï˛ !ö¡¨ ܲ¡õyˆÏB˛
dλ
~Ó˚ üyö ≤ÃyÎ˚ 0.03, §%ï˛Ó˚yÇñ ò¢yÓ˚ ≤ÈÏû˛ò
xï˛ƒhsˇ ܲü– 800 = 900 Hz ܲ¡õyˆÏB˛ ò¢yÓ˚ ≤ÈÏû˛ò ≤ÃyÎ˚ 180° •Î˚– !ܲv ~Ó˚ ˆÓ!¢ ܲ¡õyˆÏB˛ ò¢yÓ˚ ≤ÈÏû˛ò
180° ÈÙÈ ~Ó˚ ˆã˛ˆÏÎ˚ ˆÓ!¢ •Î˚ ~ÓÇ ~Ó˚ ú˛ˆÏ° í˛z͈ϧÓ˚ !öܲê˛Óï˛#≈≈ ܲyˆÏö ˆÎ ¢∑ï˛Ó˚D ˆ˛õÔÑåÈÎ˚ñ ˆ§!ê˛ ò¢yˆÏܲyˆÏî
~!àˆÏÎ˚ öy ˆÌˆÏܲ !˛õ!åȈÏÎ˚ xyˆÏåÈ ÓˆÏ° üˆÏö •Î˚– §%ï˛Ó˚yÇñ 1000 Hz Óy ï˛ò)ôù≈ ܲ¡õyˆÏB˛ ò¢yÓ˚ ≤ÈÏû˛ò ¢∑
í˛z͈ϧÓ˚ !òܲ !öî≈ˆÏÎ˚ §y•y΃ ܲˆÏÓ˚ öy– 200 ˆÌˆÏܲ 800 Hz ܲ¡õyˆÏB˛Ó˚ ˆ«˛ˆÏe ˆÎ ~!ê˛ Ü˛yÎ≈ܲÓ˚# ÌyˆÏܲ ï˛y
˛õÓ˚#«˛y myÓ˚y ≤Ãüy!îï˛ •ˆÏÎ˚ˆÏåÈ–˛˛õ)ˆÏÓ≈Ó˚ ˛õÓ˚#«˛yÓ˚ üˆÏï˛y ˆ◊yï˛yÓ˚ ò%•z ܲyˆÏö ˆê˛!°ˆÏú˛yö !Ó˚!§û˛yˆÏÓ˚Ó˚ §y•yˆÏ΃ !û˛ß¨
ò¢yÎ˚ !Ó÷k˛ §)Ó˚ ◊Óî ܲ!Ó˚ˆÏÎ˚ ˆòáy ˆàˆÏåÈñ ˆ◊yï˛y ˆÎ!òˆÏܲÓ˚ ܲyˆÏö ò¢yÎ˚ ~!àˆÏÎ˚ Ìyܲy ¢∑ ˆ¢yˆÏööñ ¢ˆÏ∑Ó˚
í˛zͧ ˆ§!òˆÏܲ•z xyˆÏåÈ ÓˆÏ° üˆÏö ܲˆÏÓ˚ö–
373
ÓyhflψÏÓ ◊&ï˛ ¢ˆÏ∑Ó˚ üˆÏôƒ §yôyÓ˚îï˛ í˛zFã˛ Ä ö#ã˛ÈÙÙÙÈ§Ó Ü˛¡õyB˛•z í˛z˛õ!fliï˛ ÌyˆÏܲ– í˛z˛õÓ˚v ˆ◊yï˛y •zFåÈyüï˛
üyï˛y ˆâyÓ˚yˆÏï˛Ä ˛õyˆÏÓ˚ö– ~Ó˚ ú˛ˆÏ° xyüÓ˚y ï˛#Ó ï˛yÓ˚ ≤ÈÏû˛ò Ä ò¢yÓ˚ ≤ÈÏû˛òÈÙÙÙÈò%!ê˛ˆÏܲ•z ܲyˆÏç °yày•z ~ÓÇ
ˆüyê˛yü%!ê˛ §!ë˛Ü˛û˛yˆÏÓ ¢ˆÏ∑Ó˚ í˛z͈ϧÓ˚ !òܲ !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚–
◊&!ï˛ §¡∫¶˛#Î˚ xyˆÏ°yã˛öy ~áyˆÏö•z ˆ¢£Ï •°– ~•z xyˆÏ°yã˛öy ˆÎˆÏ•ï%˛ üyö%ˆÏ£ÏÓ˚•z ◊ÓˆÏî!wÎ˚ Ä ◊Óî «˛üï˛yÓ˚
§¡∫ˆÏ¶˛ñ ~!ê˛ §Ω˛Óï˛ xy˛õöyÓ˚ ܲyˆÏåÈ ˆÜ˛Ôò%•ˆÏ°yj#˛õܲ ӈϰ üˆÏö •ˆÏÎ˚ˆÏåÈ– ~ÓyÓ˚ xy˛õ!ö ~ܲ!ê˛ xö%¢#°ö#
í˛z_Ó˚ ˆòÄÎ˚yÓ˚ ˆã˛‹Ty ܲÓ˚&ö–
xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ xö%¢#°ö# ÈÙÈ 6 :
(a) ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲü ÌyܲˆÏ° ◊&!ï˛ í˛z˛õ§%Ó˚ ˆ¢yöy ÎyÎ˚ öy ˆÜ˛ö⁄
(b) 600 Hz Ä 1000 Hz ܲ¡õyˆÏB˛Ó˚ ò%!ê˛ !Ó÷k˛ §%Ó˚ ~ܲ§ˆÏD ôù!öï˛ •°– ~üö ã˛yÓ˚!ê˛ Ü˛¡õyˆÏB˛Ó˚ í˛zˆÏÕ‘á
ܲÓ˚&ö ˆÎ=!°ˆÏï˛ Î%_´fl∫ö ˆ¢yöy ÎyˆÏÓ–
(c) 100 Hz ܲ¡õyˆÏB˛Ó˚ !Ó÷k˛ ¢∑ ò%•z ܲyˆÏö §üyö ≤ÃÓï˛°yÎ˚ ◊&ï˛ •Î˚– ~Ó˚ ܲyÓ˚î ܲ#⁄
14.7 §yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢§yÓ˚yÇ¢
Óï≈˛üyˆÏö ~ܲܲ!ê˛Ó˚ í˛z˛õç#Óƒ !Ó£ÏÎ˚ üyö%ˆÏ£ÏÓ˚ ¢Ó˚#ˆÏÓ˚ ≤ÃÜ,˛!ï˛ò_ ¢∑ í˛zͲõyòܲ Ä ¢∑@ˇÃy•Ü˛ ò%!ê˛ ÎsfÈÙÙÙÈÓyà‰Îsf
Ä Ü˛yö– ~ܲܲ!ê˛Ó˚ ò%•z ü)° xLjϢÓ˚ ≤ÃÌü!ê˛ˆÏï˛ Óyà≈ΈÏsfÓ˚ àë˛ö Ä Ü˛yÎ≈≤Ãîy°# xyˆÏ°y!ã˛ï˛ •ˆÏÎ˚ˆÏåÈñ ÎyÓ˚ üˆÏôƒ
xyˆÏåÈ ˆây£Ïôù!ö Ä Ÿªy§ôù!öñ í˛zû˛Î˚ ≤ÃܲyÓ˚ ܲt˛fl∫ˆÏÓ˚Ó˚ í˛zͲõyòö– ܲt˛fl∫ˆÏÓ˚ ~ «˛üï˛y Óy!•ï˛ •Î˚ ï˛yÓ˚ ˛õ!Ó˚üyî
xï˛ƒhsˇ x“– ~•z «˛üï˛y ˛õ%Ó˚&£Ï Ä ü!•°yÓ˚ ܲt˛fl∫ˆÏÓ˚Ó˚ !Ó!û˛ß¨ ú%˛!Ó˚ˆÏÎ˚ í˛z˛õyLjϢ ܲ#û˛yˆÏÓ Ó!rê˛ï˛ ÌyˆÏܲ ï˛yÓ˚ §Ç!«˛Æ
í˛zˆÏÕ‘á ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
!mï˛#Î˚ xÇ¢!ê˛ ◊&!ï˛ §¡õܲ#≈Î˚– ܲyˆÏöÓ˚ àë˛ö Ä Ü˛yÎ≈≤Ãîy°# åÈyí˛¸yÄ Ü˛yˆÏöÓ˚ ◊Óî§#üy xÌ≈yÍ ◊Óî§yôƒ
¢ˆÏ∑Ó˚ §Ó≈!ö¡¨ ï˛#Ó ï˛y Ä §•ö#Î˚ §ˆÏÓ≈yFã˛ ï˛#Ó ï˛yÓ˚ í˛zˆÏÕ‘á ܲÓ˚y •ˆÏÎ˚ˆÏåÈ– ï˛ˆÏÓ ˜öÓ≈ƒ!_´Ü˛û˛yˆÏÓ ˛õ!Ó˚üy˛õˆÏÎyàƒ
ï˛#Ó ï˛y åÈyí˛¸yÄ ˆ◊yï˛y !öû≈˛Ó˚ ò%•z!ê˛ Ó˚y!¢ ÓƒÓ•yÓ˚ ܲÓ˚yÓ˚ ≤ÈÏÎ˚yçö •Î˚– ~Ó˚ ~ܲ!ê˛ ≤ÃyÓ°ƒüyeñ ˆÎ!ê˛ ˆÎ ¢∑=!°
!Ó!û˛ß¨ ܲ¡õyˆÏB˛Ó˚ xÌã˛ ˆ◊yï˛yÓ˚ ܲyˆÏåÈ §üyö ≤ÃyӈϰƒÓ˚ ӈϰ üˆÏö •Î˚ ˆ§=!°ˆÏܲ §¡õ!Ü≈˛ï˛ ܲˆÏÓ˚– xöƒ!ê˛
ˆ◊yï˛y !öû≈˛Ó˚ ≤ÃÓ°ï˛yñ ˆÎ!ê˛Ó˚ §y•yˆÏ΃ ò%•z!ê˛ ¢ˆÏ∑Ó˚ ~ܲ!ê˛ ˆ◊yï˛yÓ˚ ܲyˆÏö xöƒ!ê˛Ó˚ ï%˛°öyÎ˚ ܲï˛=î ˆçyÓ˚yˆÏ°y
ӈϰ üˆÏö •Î˚ ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ §y•y΃ ܲˆÏÓ˚– ~•z ò%•z Ó˚y!¢Ó˚ !ӣψÏÎ˚ xyˆÏ°yã˛öy åÈyí˛¸y ~=!°Ó˚ ~ܲܲñ ÎÌye´ˆÏü
ú˛ö Ä ˆ§yö ~áyˆÏö §ÇK˛yï˛ Ä ÓƒÓ•*ï˛ •ˆÏÎ˚ˆÏåÈ–
˛õ!Ó˚ˆÏ¢ˆÏ£Ïñ ò%•z!ê˛ Ü˛yˆÏö ◊ÓˆÏîÓ˚ §y•yˆÏ΃ xyüÓ˚y ܲ#û˛yˆÏÓ ¢∑ í˛z͈ϧÓ˚ !òܲ !öî≈Î˚ ܲ!Ó˚ ˆ§ §¡∫ˆÏ¶˛ ~•z
~ܲˆÏܲ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
374
14.8 §Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ܲt˛fl∫Ó˚ í˛zͲõyòˆÏö (a) ú%˛§‰ú%˛§‰ñ (b) fl∫Ó˚ï˛sf#ñ (c) !ç•ùyñ Ĥ˛ Ä òˆÏhsˇÓ˚ û)˛!üܲy Óƒyáƒy ܲÓ˚&ö–
2. üyö%ˆÏ£ÏÓ˚ ܲt˛fl∫ˆÏÓ˚ ܲ# ˛õ!Ó˚üyˆÏî !Ó!û˛ß¨ ܲ¡õyB˛ í˛z˛õ!fliï˛ ÌyˆÏܲ⁄
3. üyö%ˆÏ£ÏÓ˚ ܲyˆÏöÓ˚ àë˛ö Óî≈öy ܲÓ˚&ö ~ÓÇ Ü˛yˆÏöÓ˚ !Ó!û˛ß¨ xLjϢÓ˚ ܲyÎ≈ Óƒyáƒy ܲÓ˚&ö–
4. ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y Ä ≤ÃyÓ°ƒüyeyÓ˚ §¡õÜ≈˛ Óî≈öy ܲÓ˚&ö–
5. ¢ˆÏ∑Ó˚ ≤ÃyÓ°ƒüyey Ä ≤ÃÓ°ï˛yÓ˚ ≤ÈÏû˛ò ܲ#⁄ ≤ÃÓ°ï˛y Ó˚y!¢!ê˛Ó˚ K˛yö xyüyˆÏòÓ˚ ܲyˆÏç °yˆÏà ~üö
ò%•z!ê˛ ˆ«˛ˆÏeÓ˚ í˛zˆÏÕ‘á ܲÓ˚&ö–
6. ï˛#«¯˛ï˛y ܲ# ~ÓÇ ~!ê˛ ˆÜ˛yö‰ ˆÜ˛yö‰ Ó˚y!¢Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚⁄
7. fl∫Ó˚ܲ¡õñ ◊&!ï˛ í˛z˛õ§%Ó˚ Ä Î%_´fl∫ö ܲ#û˛yˆÏÓ í˛zͲõߨ •Î˚ ï˛y Óƒyáƒy ܲÓ˚&ö–
8. ¢∑ í˛z͈ϧÓ˚ !òܲ !öî≈ˆÏÎ˚ ò%•z!ê˛ Ü˛yö ܲ#û˛yˆÏÓ Ü˛yç ܲˆÏÓ˚ñ Óƒyáƒy ܲÓ˚&ö–
14.9 í˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°yí˛z_Ó˚üy°y
xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠xö%¢#°ö# ≠
1. (i) •ƒyÑñ Ÿªy§ôù!öˆÏï˛ xÌ≈yÍ !ú˛§!ú˛§ ܲˆÏÓ˚ ܲÌy Ó°ˆÏï˛ fl∫Ó˚ï˛sf#Ó˚ ÓƒÓ•yÓ˚ •Î˚ öyñ ï˛ˆÏÓ ÓyÎ˚%ˆÏflÀyï˛
ü%áà•ùÓ˚ñ öy!§Ü˛yà•ùÓ˚ ≤Ãû,˛!ï˛Ó˚ üˆÏôƒ !öÎ˚!sfï˛ •ˆÏÎ˚ xÌ≈˛õ)î≈ ¢∑ §,!‹T ܲˆÏÓ˚–
(ii) §yôyÓ˚î ܲÌyÓyï≈˛yÎ˚ ≤ÃyÎ˚ 10
μ
W «˛üï˛yÓ˚ ¢∑ §,‹T •Î˚ñ ï˛ˆÏÓ ˆçyˆÏÓ˚ ܲÌy Ó°ˆÏ° ~!ê˛ 200
μ
W
˛õÎ≈hsˇ ˆÎˆÏï˛ ˛õyˆÏÓ˚–
2. ◊Óîöy!°Ü˛yÈÙÙÙÈܲî≈ õê˛•ÈÙÙÙÈ•yï% !í ¸ÈÙÙÙȈö•y•zÈÙÙÙȈÓ˚ܲyÓÈÙÙÙÈ!í˛¡∫yÜ, !ï˛ õò≈yÈÙÙÙÈxhsˇÉܲ Ïî≈Ó˚ ï˛Ó˚°ÈÙÙÙȈÓ!§°yÓ˚
˛õò≈yÙÙÙÈ◊ÓîÙÙÙÈöyû≈ –
3. 30 db ï˛#Ó ï˛yÓ˚ 100 Hz ܲ¡õyˆÏB˛Ó˚ ¢∑ÈÙÙÙÈ≤ÃyÓ°ƒüyey 0 ú˛ˆÏöÓ˚ !öˆÏã˛– §%ï˛Ó˚yÇ ◊Óî§yôƒ öÎ˚–
70 db ï˛#Ó ï˛yÓ˚ !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑ÈÙÙÙÈˆÜ˛yöÄ xfl∫!hflÏ Óƒï˛#ï˛ ˆ¢yöy ÎyˆÏÓ–
130 db ï˛#Ó ï˛yÓ˚ !ü◊ ܲ¡õyˆÏB˛Ó˚ ¢∑ÈÙÙÙÈܲyˆÏö xfl∫!hflÏÓ˚ §,!‹T ܲÓ˚ˆÏÓ–
4. (b), (d), (c) ÈÙÙÙÈ !ë˛Ü˛–
5. ܲ¡õyB˛ñ ï˛#Ó ï˛yÓ˚ñ ï˛#«¯˛ï˛yÓ˚ñ 1000, ÓƒÓôyˆÏöÓ˚ñ x˛õfl∫Ó˚ñ çy!ï˛ñ ï˛Ó˚DyÜ,˛!ï˛ñ ≤ÃyÓ°ƒüyey–
375
6. (a) ¢ˆÏ∑Ó˚ ï˛#Ó ï˛y ܲü ÌyܲˆÏ° 14.3 §ü#ܲÓ˚ˆÏî p-~Ó˚ üyö ˆåÈyê˛ ôÓ˚ˆÏï˛ •ˆÏÓ– ˆ§ˆÏ«˛ˆÏe bp2 Ä cp3
Ó˚y!¢=!° í˛zˆÏ˛õ«˛î#Î˚ •ˆÏÓñ xÌ≈yÍ R Ä p ~Ó˚ §¡õÜ≈˛ ˜Ó˚!áܲ •ˆÏÓ– 14.4 §ü#ܲÓ˚ˆÏî R ~Ó˚
Ó˚y!¢üy°yÎ˚ ü%°§%Ó˚ Óƒï˛#ï˛ xöƒyöƒ í˛z˛õ§%Ó˚=!° ÌyܲˆÏÓ öy– ~Ó˚ ú˛ˆÏ° ◊&!ï˛ í˛z˛õ§%Ó˚ ˆ¢yöy ÎyˆÏÓ
öy–
(b) 400 Hz, 1600 Hz, 1400(1000 × 2 – 600) Hz, 2600 (1000 × 2 + 600) Hz
(c) 100 Hz ܲ¡õyˆÏB˛ ï˛Ó˚D˜Ïòâ≈ƒ ≤ÃyÎ˚ 3.5 m, Îy ühflψÏܲÓ˚ ÓƒyˆÏ§Ó˚ ï%˛ö°yÎ˚ xˆÏöܲ Óí˛¸– ú˛ˆÏ° ˆ◊yï˛yÓ˚
ühflÏܲ ~•z ܲ¡õyˆÏB˛ ˆÜ˛yöÄ åÈyÎ˚yMÈ˛° §,!‹T ܲˆÏÓ˚ öy ~ÓÇ ¢∑ í˛zû˛Î˚ ܲyˆÏö ≤ÃyÎ˚ §üyö ï˛#Ó ï˛yÎ˚
ˆ˛õÔÑåÈyÎ˚–
§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#§Ó≈ˆÏ¢£Ï ≤ß¿yÓ°#
1. ~Ó˚ í˛z_Ó˚ xy˛õ!ö 14.2 xLjϢ ˛õyˆÏÓö–
2. 14.3 xLjϢ ~ !ӣψÏÎ˚ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
3. 14.4.1 xLjϢ ܲyˆÏöÓ˚ àë˛ö Ä 14.4.2 xLjϢ !Ó!û˛ß¨ xLjϢÓ˚ ܲyÎ≈≤Ãîy°# Óƒyáƒy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
4. 14.4.4 xLjϢ xy˛õ!ö ≤ß¿!ê˛Ó˚ í˛z_Ó˚ ˛õyˆÏÓö–
5. ú˛ö ~ܲˆÏܲ §ü≤ÃyÓ°ƒüyeyÓ˚ ¢∑ ܲyˆÏö §üyö ˆçyÓ˚yˆÏ°y ˆ¢yöyÎ˚ !ܲv ≤ÃyÓ°ƒüyeyÓ˚ üyˆÏöÓ˚ Ó,!k˛Ó˚
§ˆÏD ¢∑ ܲï˛ê˛y ˆçyÓ˚yˆÏ°y ˆ¢yöyˆÏÓ ï˛yÓ˚ !ö!ò≈‹T §¡õÜ≈˛ ˆö•z– Î!ò ~ܲ!ê˛ ¢∑ í˛z͈ϧÓ˚ §ˆÏD xö%Ó˚*˛õ
xyÓ˚ ~ܲ!ê˛ ¢∑ í˛zͧ ÓƒÓ•yÓ˚ ܲÓ˚y •Î˚ ï˛ˆÏÓ ï˛#Ó ï˛y !m=î •ˆÏ°Ä (3db Ó,!k˛V ≤ÃyÓ°ƒüyey
§yôyÓ˚îû˛yˆÏÓ 3 ú˛ö Ó,!k˛ ˛õyˆÏÓ öy– x˛õÓ˚˛õˆÏ«˛ñ ܲyˆÏö ¢∑ Ü˛ï˛ ˆçyÓ˚ ˆ¢yöy ÎyˆÏÓ ï˛y ˆ§yö ~ܲˆÏܲ
≤ÃÓ°ï˛y !òˆÏÎ˚ !öˆÏò≈¢ ܲÓ˚y ÎyÎ˚– ˆçyÓ˚yˆÏ°yû˛yÓ !m=î Ó,!k˛ ˆ˛õˆÏ° ≤ÃÓ°ï˛yÄ !m=î •ˆÏÓ– !Ó!û˛ß¨
í˛zͧ ˆÌˆÏܲ xy§y ¢ˆÏ∑Ó˚ çöƒ ˆüyê˛ ≤ÃÓ°ï˛y ~ܲܲ ≤ÃÓ°ï˛y=!°Ó˚ ˆÎyàú˛° ~ÓÇ ˆ◊yï˛yÓ˚ ܲyˆÏåÈ
!ü!°ï˛ ¢∑ Ü˛ï˛ ˆçyÓ˚ ˆ¢yöyˆÏÓ ï˛y !öˆÏò≈¢ ܲˆÏÓ˚–
≤ÃÓ°ï˛y Ó˚y!¢!ê˛ çyöy ÌyܲˆÏ° xyüÓ˚y (i) !Ó!û˛ß¨ í˛zͧ ˆÌˆÏܲ xy§y ¢∑ !ü!°ï˛û˛yˆÏÓ ˆ◊yï˛yÓ˚ ܲyˆÏö
ܲï˛ê˛y ˆçyÓ˚yˆÏ°y ˆ¢yöyˆÏÓ ï˛y !öî≈Î˚ ܲÓ˚ˆÏï˛ ˛õy!Ó˚ Ä (ii) ˆÎ ¢ˆÏ∑ !Ó!û˛ß¨ ï˛#Ó ï˛yÓ˚ ~ܲy!ôܲ
ܲ¡õyˆÏB˛Ó˚ í˛z˛õyÇ¢ xyˆÏåÈ ï˛y ˆ◊yï˛yÓ˚ ܲyˆÏåÈ Ü˛ï˛ê˛y ˆçyÓ˚ üˆÏö •ˆÏÓ ï˛yÓ˚ àîöy ܲÓ˚ˆÏï˛ ˛õy!Ó˚–
6. ˆ◊yï˛yÓ˚ xö%û)˛!ï˛ˆÏï˛ Ü˛yöÄ §%Ó˚ xÌÓy §%fl∫Ó˚ ܲï˛ê˛y ã˛í˛¸y üˆÏö •Î˚ñ ï˛yˆÏܲ•z §%Ó˚ Óy §%fl∫ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y
Ó°y •Î˚– ˆÜ˛yöÄ §%ˆÏÓ˚Ó˚ ï˛#«¯˛ï˛y ≤Ãôyöï˛ ï˛yÓ˚ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚ñ ï˛ˆÏÓ ï˛#Ó ï˛yÓ˚ í˛z˛õˆÏÓ˚Ä
~!ê˛ !ܲå%Èê˛y !öû≈˛Ó˚¢#°– ~ܲ!ê˛ ü)°§%Ó˚ Ä ï˛yÓ˚ !ܲå%È í˛z˛õ§%ˆÏÓ˚Ó˚ §üß∫ˆÏÎ˚ à!ë˛ï˛ ¢ˆÏ∑Ó˚ ï˛#«¯˛ï˛y
ü)°§%ˆÏÓ˚Ó˚ ܲ¡õyB˛ myÓ˚y•z !öô≈y!Ó˚ï˛ •Î˚– !ܲv ü)°§%Ó˚ Ä ï˛yÓ˚ í˛z˛õ§%Ó˚ öÎ˚ ~üö ܲˆÏÎ˚ܲ!ê˛ §%ˆÏÓ˚Ó˚
376
!ü◊ˆÏî à!ë˛ï˛ ¢ˆÏ∑Ó˚ ˆ«˛ˆÏe ï˛#«¯˛ï˛y §Ó≈!ö¡¨ ܲ¡õyˆÏB˛Ó˚ í˛z˛õÓ˚ !öû≈˛Ó˚ öy ܲˆÏÓ˚ ܲ¡õy=!°Ó˚ ÓƒÓôyˆÏöÓ˚
í˛z˛õÓ˚ !öû≈˛Ó˚ ܲˆÏÓ˚–
7. 14.5 xLjϢ ~ §¡∫ˆÏ¶˛ xyˆÏ°yã˛öy ܲÓ˚y •ˆÏÎ˚ˆÏåÈ–
8. 14.6 xLjϢ ~•z ≤Èϟ¿Ó˚ í˛z_Ó˚ ˛õyˆÏÓö–
~•z ~ܲܲ!ê˛Ó˚ Ó˚ã˛öyÎ˚ !öˆÏß√y_´ ˛õ%hflÏܲ=!°Ó˚ §•yÎ˚ï˛y ˆöÄÎ˚y •ˆÏÎ˚ˆÏåÈ– xy˛õ!ö •zFåÈy ܲÓ˚ˆÏ° ~=!° ˛õˆÏí˛¸
ˆòáˆÏï˛ ˛õyˆÏÓ˚ö–
1. Fundamentals of Acoustics — L.E. Kinsler & A.R. Frey, Wiley Eastern Ltd.
2. Acoustics — A. Wood, Blackie & Son Ltd.
3. í˛zFã˛ï˛Ó˚ fl∫ö!ÓòƒyÈÙÙÙÈÎ%à°!ܲˆÏ¢yÓ˚ ü%ˆÏáy˛õyôƒyÎ˚ñ ˛õ!ÿ˛üÓD Ó˚yçƒ ˛õ%hflÏܲ ˛õ£Ï≈ò–
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