Proc. iadiau Acad. Sci., Vol. 85 B, No. 5, 1977, pp. 327-339.

W i n g b e a t f r e q u e n c y o f a f l i e r - - M a s s f low t h e o r y

P. G. PURANIK, G. GOPALA KRISHNA AND ADEEL AHMED

Biophysics Laboratory, Physics Department, Nizam College, Hyderabad 500001

AND

N. CHARI

Zoology Department, Postgraduate Centre, Warangal MS received 14 May 1976; in revised form 1 September 1976

ABSTRACT

The paper presents a theoretical approach for computing the wing beat frequency of a flier whether it be an insect, bird or bat in its normal state of hovering on the basis of mass flow of air. The calculated values are found to be in good agreement with the observed values. The paper describes the results of 5 species of insects, 4 species of birds and 3 species of megachiropteran bats totalling 136 samples. The mass of the fliers ranges between 0.07 gins to 744 gms and frequencies range from 125 eps to 4 cps respectively. It is concluded that the wing beat frequency of a flier in general varies with the state of flight.

NOTATION

M~ ----- Mass of the flier

L = Wing span- - the length f rom tip of one wing to the tip of the other wing

1 = Wing length-- the length from the joint to the tip of the wing

B,ff ----- Effective wing breadth = wing area wing length

~rL 2 Sa = Disc area = y

p = Density of the medium in which the flier is hovering

g ----Acceleration due to gravity R = Reacting force v ----- Wing beat frequency of the flier according to mechanical oscil-

lator theory as reported by Greenewalt

327 B 8 ~ May 77

328 P . G . PURANIK et al

Vh

vo K

K'

= Wing beat frequency of the fliel in the state of hovering according to this theory

= Observed wing beat frequency of the flier in its hovering state = Proportionality constant, dimensions of which are L T -~

_ 8g --2086 - - o K - -

1. INTRODUCTION

THE wing beat frequency of the flier is an important aerodynamic para. meter for the understanding of its flight and its energetics. It is rather surprising that ornithological and chiropteran records are not informative regarding the wing beat frequency of different fliers. However, enough data is available in the entomological iitexature.

Vaughan, x Puranik and coworkers, 2 have investigated the flight patterns and aerodynamics of megachiropteran bats in some details. While the flight pattern of insects have received considerable attention at the hands of researchers, the mammalian flight in general seems to have escaped their attention. Recently Pennycuick a has studied the aerodynamics of Rousettus aegyptiacus in a wind tunnel. He has studied the power require- meots for horizontal flight of Columba livia on the basis of mass flow? He has used the high speed photographic technique to determine the wing beat frequency of a pigeon.

In the case of insects, SotavaltaS, 6 has made extensive studies on the wing beat frequencies of various species while Puranik et al. 7 have developed simple experimental techniques for determining the wing beat frequency of insects. Puranik and Adeel Ahmed s have made the Fourier anal:ysis of the flight sound of T. javanica. Puranik and coworkers 9 have also investigated the aerodynamic parameters of C. marginatus and Pteropus giganteus.

Greenewalt 10 has developed an oscillator theory and deduced a relation between the wing length of a flier and it, wing beat frequency, by assuming the flier as an harmonic oscillator. He has set up a differential equation for a damped oscillation, the solution for undamped oscillation is given by

°~ -~ Kbor2/I (1) where o~ is the angtAar frequency, bo is the length of t~e unstrained muscle K a constant related to the elastic modulus of the muscle, r the effective radius of attack aria I the sum of the external and internal moments of inertia. Using Sotavalta's data on the insects and Magnan's n data on

WING BEAT FREQUENCY OF A FLIER 329

birds, Greenewalt has shown the dependence of the wing beat rates on the size of dimensionally similar groups of fliers is consistent with relation (l). Greenewalt then goes on to assume that r oc l and M Qc 13 resulting in I oc 15 and reduces eq. (1) to

vl = constant (2) where

O) ~ ~ °

Gt eenewalt states that the relation (2) holds for insects only and for humming birds the anatomical relationships appear to be quite different. So he proposes that the weight of a bird is proportional to 11"5 and not 13. Taking r oc 1 a n d bo oc M~/3 he deduces the relation

vl 1. ~5 = constant. (3) Then he generalises the frequency wing length expression for insects and birds as

vl n = constant (4)

whore n lies between 1 and 1.25. Probanly on the basis of grapnical rela- tion between the anatomical parameters of fliers, he puts down the relation as

vp.15.-__- 3540 (5) wt, ere l is the wing lengtn, expressed in millimeters.

For verifying the semi-empirical relation thus developed Greenewalt has confined his experiment to only one flier ' female ruby throat '. Greene- v, alt's theoretical approach gets vitiated when he starts assuming different proportionalities for bo, r, M and I because of the non-regular surface of the flier both from the point of view of density and regularity of the contour.

Besides the frequency of a flier in genere,1 appears to depend upon the state of flight more so in the case of humming-birds 12 (55 cps-200 cps depend- ing upon the state of fligh0 and as such cannot be considered as constant under different flight conditions. Therefore, the flight appears to be asso- ciated not with a single frequency bt:t with a frequency spectrum.

THEORETICAl, CONSIDERATIONS

The authors have attempted to develop a theory for the evaluation of the vdng beat frequency of a flier in it~ normal state ot hovering on the basis of mass flow of air induced downwards due to wing beat. When the flier is in the state of hovering, the system is said to be in a state of ' dynamical equili~,rium ' which is achieved by the flier by generating a "reacting fo rce"

330 P . G . PURANIK et al

R to just overcome its own weight. Tke reacting force R so due to wing beat is proportional to the following:

l*

2.

3.

4. because

generated

(WingSpan) z The disc area, S a of the fliel, where Sd-----,r --

The effective wing breadth, Bar

The density, p of the medium in which it is hovering

The half of wing beat frequency, vu/2, (the factor half enters in downstroke alone is effective in inducing the air downwards).

TherefoIe, the reacting force R is proportional to the rate of mass flow of air induced downwards. Therefore,

R = K • S a " B ~ • p • vl~/2 (6)

where K is the proportionality constant, the dimensions being L T -x. For the hovering state

R ----- Me • g (7) where M I is the mass ot the flier and g is the acceleration due to gravity From eqs (6) and (7) it follows that '

M ! " g = K " S a " B a , "p • vl,]2. (8)

By putting S a = ~r L~]4 where L is the wing span of the flier in eq. (8) we have

M s • g - - ( K / 8 ) • ~ - L ~ - So, , - p • vh.

Therefore, vh - - ( 8 g / K , ~ p ) • M ~ /Z,~ • S . , , .

--~ K ' • M I / L 2 • Bar.

(9)

(10 a) ( 1 0 b )

The value of K' is computed experimentally by drawing ~ curve between M t and v 0 • L z • Bar, all in CGS units for insects where vo is the ooserved wing beat frequency of the flier in its hovering state and the slope of which gives the value of K' as 2086 (figure 1).

Thus the relation (10) gives the hovering frequency in terms of the mass of the flier, the wing span, the effective wing breadth and the density of the air. This is different from Greenewalt's relation which gives ~¢ frequency in terms of wing length alone.

Hence the relation for the wing beat frequency in the normal l~overing state is

vh ----- (2086)MIlL ~" Bef,. (11)

WING BEAT FREQUENCY OF A FLIER 331

~0

3 e -

3 2 -

2 8 -

24

20 m

¢"..i 16

12

o o o

o,~

O

8 O _

o • O

O

o BOMBUS $p • PERIPLANETA omlricQM o TESSERATOMA jovanko

CHRYSOCORIS purpureu~ APIS $p.

0 0-2 n-t. 0.6 0.8 1.0 1.2 1.4 1.6

Mf Figure 1. The plot between L ~ Bert vo and M t for insects, the slope of which gives the value

of K' of the relation (10 b).

K' = 2086 M -1L s T -1.

Insects have been chosen for computing the value of the constant K' from figure 1 due to the fact that their frequencies can be measured accu- rately by the available experimental techniques. But in the case of other fliers which have low values of hovering frequencies (birds and bats) it is rather difficult if not impossible to determine their hovering frequencies under laboratory conditions. Particularly in such cases eq. (11) is found to be very useful.

Thus eq. (11) enables one for the first time to compute the wing beat frequency of a flier in its normal state of hovering from just the knowledge of the mass of the flier Mr, wing span L and its effective wing breadth, Be~ without taking recourse to high speed photography4,10 and the observer flying along with the flier.

EXPERIMENTAL VERIFICATION

In order to verify the relation (11), the authors have determined the body parameters MI, L and Bg~ of 5 species of insects namely Periplaneta americana, Tesseratoma javanica, Apis sp., Chrysocoris purpureus and Bombus sp. using a sensitive single pan balance and an optical enlarger respec- tively. The wing beat frequencies of the above fliers have also been deter-

332 P .G . PtmANII< et al

mined by recording their flight sound generated in their tethered state by using the tape recorder (Aka;. 1720 L, stereo) and feeding its output to the Oscilloscope (Double beam Philips pH 3230). Figure 2 is the typical oscillo- gram of the flight sound of Bombus sp. and figure 3 is that ofT. javanica, showing the variation in the wing beat frequency due to the variation in body parameters.

A curve is drawn between mass of the flier Mf and vo L~Bof~ which is found to be linear and passes through the origin indicating the validity of the theory. The slope of which gives the value of K' which can be further used for computipg the wing beat frequencies of fliers such as birds and bats whose wing beat frequencies in hovering state is rather difficult to deter- mine under the laboratory conditions.

2. RESULTS AND DISCUSSION

The mass, wing span and the effective wing breadth, the three essential body parameters for the evaluation of the wing beat frequency of different fliers are tabulate6 in tables 1, 2 and 3. Table 1 gives the values of experi- mentally observed frequencies along with the three body parameters for five species of in'..ects, viz., ti) Periplaneta americana (19 to 25 cps), (ii) Tes- seratoma javanica (59 to 77 cps) (figure 3), 0ii) Apis sp. (70 to 106 cps), (iv) Chrysocoris purpureus (82 to 122 cps) and (v)Bombus sp. (125 cps). Table 2 gives the values of frequencies calculated by the mass flow theory along with the three body parameters for four species of birds, viz., O) Corvus splendens (4 to 6 cps), (ii) Psittacula krameri (8 to 10 cps), (iii) Columba livia (9 to 11 cps) and (iv) Passer domesticus (15 to 23 cps). Table 3 gives data for three species of magaechiropteran bats, viz., (j) Pteropus giganteus (8to 10 cps), (ii) Cynopterus marginatus (7 to 10 cps) and (iii) Rousettus sp. (7 to 10 cps)

Figure 4 gives the hovering "frequency characteristics " of 12 fliers. It is a linear plot between the mass of the flier (MI) against the product of the square of the wing span (L ~) and effective wing breadth (B~0. The slope of this curve is proportional to the characteristic hovering frequency of the flier.

It is interesting to note that the hovering frequency for Psittacula krameri and Rousettus sp. is more or less the same (figure 4) even though the wing span, effective wing breadth and mass are different for both the above fliers. Same is the case for Pteropus glganteus and Corvus splendens. Therefore it is seen that the hovering frequency alone is not spec"es specific, but has to be taken along with the three body parameters.

P. G. Puranirc et al. Plate I

Proc. Indian Acad. Sci., Vol. 85 B, No. 5, 1977, pp. 327-339

Figtlre 2. A typical oscillogram of flight sound of Bombus sp. Time mark : 1 Div. = 5 milliseconds Frequency : 125 cps.

( facitlg page 332)

P. G. Puranik et al. Proc. Indian .4cad. Sci., Plate H Vol. 85 B No. 5, 1977 , pp. 3 2 7 - 3 3 9

(a)

(b)

!(c)

(d)

Figure 3 (a-d). Oseillograms of flight sound of T.javanica showing the variation in the frequency due to var ia t ion in body parameters.

Time mark : 1 Div. = 5 milliseconds (a) M t = 0 . 5 7 3 gm L = 4 . 8 5 Beef = 1"02 vo = 59cps (b) M!=l '047gm L = 5 - 7 8 Beef = 1 " 1 8 vo = 6 7 c p s (c) Mt=0.89 gm L = 5 . 6 5 Beee = 1 " 0 6 v o = 7 1 c p s (d) M t = 1" 185 gm L = 5"80 Beer = 1" 11 vo = 77 cps (/'acing page 333)

W I N G BEAT FREQUENCY OF A FLIER 333

Table 1.

SI. Flier (insects) Sex Mt vo L N o .

Beff L z Beff v 4 L 2 Beff

1. Periplaneta americana M 0"920 20 7"25 2"00 105 '13 2102"60

2. M 1"124 22 7.00 1"80 188"20 1940"40 3. F 1"287 19 8-15 2-10 139"49 2650'31 4. F 1. 322 20 8" 30 2" 00 137.78 2755" 60

5. Tesseratoma javanica M 0"573 59 4"85 1-02 23.99 1415-41 6. M 0" 655 63 4" 85 0.98 23.05 1452.15 7. M 0"705 71 5"05 0"97 24"71 1756"54 8. M 0.763 67 5"35 1.03 29"48 1975"16 9. M 0" 770 63 5.48 0" 92 27" 62 1740" 06

10. M 0.781 61 5"60 1"03 32.30 1970.30 11. M 0-853 71 5"40 1 "07 31 "20 2215"20 12. M 0" 894 65 5" 75 1.09 36.04 2342" 60 13. F 0"682 69 5"25 1.05 28"94 1996"86 14. F 0"844 71 5"75 1.15 38.02 2699.42 15. F 0" 890 71 5" 65 1.06 33.84 2402" 64 16. F 0"899 73 5"58 1-13 35.18 2568.14 17. F 0"924 68 5"58 0.93 28"96 1969.28 18. F 0"933 67 6"00 1.24 44" 64 2990.88 19. F 1.030 68 5" 80 1" 1 1 37.34 2539.12 20. F 1"047 67 5- 78 1.18 39- 42 2641.14 21. F 1-153 67 5"60 1.14 35"75 2395.25 22. F 1. 185 77 5.80 1"11 37.34 2875.18 23. Apissp. . . 0"073 90 2"20 0.36 1"74 156"60 24. . . 0"075 80 2"25 0-38 1"92 153'60 25. . . 0"076 70 2"40 0.36 2"07 144"90 26. . . 0"079 90 2"20 0"38 1"84 165"60 27. . . 0 '087 90 2"50 0"35 2"07 186.30 28. . . 0.093 80 2-60 0.36 2.43 194-40 29. . . 0-100 100 2-30 0-37 1.96 196.00 30. Chrysocoris purpureus M 0"270 85 3.30 0.62 6"75 573.75 31. M 0-279 100 3 '10 0"68 6-53 653.00 32. M 0"285 120 3-10 0"50 4"81 577"20 33. F 0-264 100 3.30 0"52 5"66 566"00 34. F 0"295 80 3.40 0"65 7"51 600.80 35. F 0"305 120 3"10 0"62 5"96 715"20 36. F 0"360 100 3.35 0.65 7"29 729-00

37. Bombus sp. . . 1.600 125 6"30 0"70 27"78 3472"50

Table 4 gives the comparison of results of the mass flow theory and the harmonic oscillator theory. The values of the frequencies vh and v calcu- lated on the basis of the two theories respectively are compared to the vo, experimentally observed values. It is interesting to note that the values obtained by these are in agreement with the reported values in literature 4 for some fliers.

334

Table 2. i

P. G. PURANIK et al

SI. Flier (birds) Mt L B,rt L 2 Beef Frequency No. (gins) (cm) (em) × 10 ~ K" Mr

(era) v~ = L 2 Bef~ (eps)

1. Corvus splendens 240 81"5 17.0 11"30 4 2. 250 76"5 16"0 8"79 6 3. 255 78"5 16"0 9-25 6 4. 275 81 "5 17.0 11.30 5 5. 275 81"5 18"5 11"97 5

6. 290 83.5 18.5 12"90 5 7. 385 94.0 19"5 17-22 5 8. Psittacula krameri 95 50.0 8"0 2.00 10 9. 100 53-5 9"0 2"58 8

10. 100 49.0 8-5 2"04 10

11. 105 5I "5 8"5 2"25 I0 12. 110 51"0 9"0 3.34 10 13. 125 52-0 9"2 2"49 11 14. Columna livia 170 54.5 10.5 3.12 11 15. 187 61.0 11.5 4.23 11

16. 207 61-0 11.0 4.07 11 17. 229 62-5 12.0 4.67 12 18. 240 61-0 11.3 4.20 12 19. 250 65.5 12.5 5.36 10 20. 260 64.0 12.0 4"92 11

21. 261 66.0 12-6 5"49 10 22. 265 65.0 12.0 5.07 12 23. 276 65.0 12.5 5"28 11 24. 345 70.0 13.5 6" 62 11 25. Passer domesticus 20 23-0 5"5 0-28 15

26. 25 19"5 5"0 0.19 27 27. 25 22" 0 5" 3 O. 25 20 28. 25 23.0 5.5 0.29 18 29. 25 21"5 5-0 0.23 23 30. 25 21" 5 5" 2 O. 24 22

31. 25 21.5 5-0 0.23 23 32. 25 22" 5 5.5 0.28 19 33. 25 23.0 5" 1 0.27 19 34. 25 23.0 5.7 0.31 17

From table 4 it can also be seen that vh/vo--the ratio of the hovering frequency as from 'Mass flow theory ' to that of the observed value is approximately equal to unity for most of the fliers whereas v/vg--the ratio of the wing beat frequency as from mechanical oscillator theory to that of the observed value is significantly different from unity for most of the fliers. This reveals that the mass flow theory explains the phenomenon of hovering flight of insects, birds and bats more adequately than the mecha- nical oscillator theory.

WING BEAT FREQUENCY OF A FLIER 335 Table 3.

SI. Flier No (Bats)

Sex Mf L Bef f (gins) (el'n) (era)

L 2 B e f f × 104 (cm ~)

Frequency K ' M t

V h - -

L ~ Beff (cps)

1. Pteropus giganteus F 251 105 17.2 19.0 4

2. F 365 116 12"9 17.4 4 3. F 378 107 14.6 16.8 5 4. F 380 103 1~1"4 17.1 5 5. F 384 105 14-0 15" 5 5 6. F 390 103 I4.3 15"2 5 7. F 392 100 14-5 14"5 6 8. F 398 109 14.7 17"5 5 9. F 400 105 16"6 18"4 5

10. F 403 115 14"6 19"4 4 11. F 413 115 14.0 18"5 5 12. F 415 106 15"1 17"0 5 13. F 416 112 14"7 18"5 5 14. F 456 116 18"4 24"8 4 15. F 479 112 13-4 17-1 6 16. F 492 115 18.5 24"5 4 17. F 505 124 15"4 23" 8 4 18. F 543 119 16"9 24.0 5 19. F 600 123 16"2 20.3 6 20. F 606 130 17" 1 28" 4 4 21. F 643 118 13"2 22"1 6 22. F 643 124 15"4 23.9 6 23. F 662 120 16.3 23 "6 6 24. F 734 118 15"7 21 "9 7 25. M 245 97 14"0 13-2 4 26. M 352 112 13.2 20.1 4 27. M 357 111 12"8 15"9 5 28. M 360 115 14"9 19.8 4 29. M 376 100 12" 1 12" 1 7 30. M 379 101 16"0 16"4 5 31. M 387 104 14"2 15"2 5 32. M 394 115 15" 3 20" 3 4 33. M 401 108 14.4 17.2 5 34. M 402 112 13-2 21-0 4 35. M 404 116 14.1 19"0 4 36. M 419 115 15"3 20.3 4 37. M 428 116 16.1 21"7 4 38. M 431 116 16"8 22.g 4 39. M 478 121 16" 9 24" 3 4 40. M 518 ] 14 15"6 20.3 5 41. M 600 116 15-0 24.8 5 42. M 646 119 17.5 25.7 5 43. M 744 125 16" 4 20.3 8 44. C. marginatus M 43 45 6"9 1-4 8 45. M 45 45 6"4 1.3 7 46. M 46 46 6.8 1.4 7 ~7. M 47 45 5.8 1.2 8 48. M 49 47 6.7 1.5 7 49. M 53 48 7.2 1.6 7 50. M 55 49 7.2 1.7 7 51. M 57 48 7.6 1.7 7 52. F 46 48 6.1 1.4 7 53. F 48 48 5-9 1-3 8 54. F 60 48 5.8 1-3 8 55. Rousettus sp. M 49 45 5-8 1"2 10 56. M 62 51 7"2 1"9 7 57, M 66 49 6" 9 1.6 9 58. M 76 $4 7"4 2' 2 7 59. M 91 50 7"6 1"9 10 60. M 98 57 7"8 2"5 8 61. M 101 55 7"4 2"2 9 62. M 104 55 7"5 2" 3 10 63, M 105 55 5"3 2" 5 9

I fll I

B 9~May 77

336 P . G . PURANIK et+al

F,L

1"2

1"0

0"? o CHRY$OCORI~; PURPUREUS

:o ,',, ,'o ,'o ,'o ,,'o ,~ ,'o L 2 I l l /

2S0

2OO

x 150

I00

'JO0

400 !

700

100

i o , "

!

I +I ° I 2e

L ~ lillff x 10 t"

I I I 41o lO 20 30 L2 B e f f . I 10 6

Figare 4, The hovering flight characteristics of insects, birds and bats,

WING BEAT FREQUENCY OF A FLIER

Table 4. Compar ison* o f results for insects, birds and bats

337

S1. Flier vo vn vh/vo v V/Vo No. (cps) (cps) (cps)

Insects

1. Periplaneta americana

2. Tesseratoma javanica

3. Apis sp.

4. Chrysocoris purpureus

5. Bombus sp.

Birds

6. Corvus splendens

7 Psittacula krameri

8. Columba livia

20 21 1"05 25 1"25

68 66 0"97 88 1"30

87 85 0"98 276 3"20

95 100 1"04 189 2"00

125 124 1"00 87 0"69

5 5 1 "00 6 1 "20

9 10 1 .10 11 1"22

11 11 1"00 9 0"82

9. Passer domestieus 17 20 1.18 29 1"70

Bats

10. Pteropus giganteus 4 5 1 "25 4 1.00

11. C. marginatus 6 8 1.33 12 2.00

12. Rousettus sp. 7 9 1"29 11 1 '57 i w

* Mean values.

From figure 5 it can be observed that whatever be the flier insect, bird or bat, the plot between the mass of the flier against the product of observed frequency v 0, effective wing breadth Beff and the square of the wing span L 2 for 12 species of fliers is linear thereby indicating the validity of Uae relation (11). It may be noted that the eq. (11) arrived at by 'mass flow theory' characterises the hovering state as a particular state of flight and ascribes a particular frequency (vh) known as 'hovering frequency' to that state. The mechanical oscillator theory, however considers the wing beat frequency of a flier to be the same under a variety of flight conaitions-- hovering flight, flight at different linear velocities, take-offs from a perch, and escape routines when the birds became trightened.

The mass flow theory further gains support from the observations made on humming birds 1~ where the frequencies are found to range from 55 to 200 cps under different states of flight, and on pigeons 4 where a similar variation has been reported,

338 P . G . PURANIK et al

6.4 PTEROPUS giganteus ~ d ~

" CORVUS $ p l e n d e n s ~ - S-6 -- COLUMBA l i v i a ' - " ~ ~

_ ROUSETTUS s p . - ~ CYNOPTERUS m a r g i n a t u ~ /

/ "8 - - PASSER do m e s t i c w ~ / / /

: : 4 0 -- ~r PSITTACULA

T E SSI~RATp M A - ~ Q ~ ~- PERIPLANETA 3.2 - javamc6 ,~ ~ amer icana

. CHRVSO C O R I S ~ ' ~ " - purpureus_ /

-- 2-~ ApE; s p . y

0-~

0 , 1 , I t I I I , I 0 0.0 1.6 2. / . 3.2 /.. 0 /..8 5.6

log Mf

Figure 5. A plot between M t and v0 L s Ber~ drawn on a logarithmic scale showing the validity of the relation (11) for insects, birds and bats.

From this it may be concluded that the wing beat frequency of a flier in general varies with the state of flight and as such cannot be considered as constant under variety of flight conditions.

ACI~NOWLEDGEMENT

Thanks are due to Sri A. S. N. Murthy, Department of Physics, Nizam College and Sri R. B. Ramchander, Department of Physics, Engineering College, Osmaoia University, for many useful discussions. One of the authors (A.A.) acknowledges with thanks the award of a fellowship by the CSIR.

REFERENCES

1. Vaughan, Tery, A., Biology o f Bats, Vol. I, ed. William A. Wimsatt, Academic Press, New York and London, Chap. 5 (1970).

2. Puranik, P. G., Gopala Krishna, 13. and Chari, N., Prec. Indian Acad. Sci. 83 B 160 (1976).

3. Pennycuick, C. J., J. Exp. BioL, 55 833 (1971).

4. Ponnycuick, C. J., J. Exp. Biol., 49 527 (1968).

5. Sotavalta, O., Ann. Zoo. See. Vanamo., 15 (1952).

6. Sotavalta, O., Ann. Entomologica Fennici 20 93 (1951)

WING BEAT FREQUENCY OF A FLIER 339

7. Puranik, P. G., Chari, N. and Sukhdev, R. D., Indian J. Expt. Biol. 11 579 (1973).

8. Puranik, P. G. and Adeel Alined, Indian J. Expt. Biol. 14 279 (1976).

9. Paranik, P. G., Gopala Krishna, G. and Chaxi, N., All-India Symp. on Comparative Animal Physiology, Marathwada University, Aurangabad, 1975.

10. Greenewalt, C. H., Proc. Amer. Phil. Soc. 104 605 (1960).

11. Magnan, A., Ann. Sci. Nat. 5 125 (1922).

12. Oliver, L Austin, Jr., Birds of the World, Paul Hamlyn, London, pp. 169 (1966).

B 10--May 77