Fundamentals - Routledge

1

Chapter 1

Fundamentals

D. Palazzuoli and G. Licitra

The whispering of wind in a wood and the roar of a traffic jam, a mountain waterfall and a road yard, noise and music: the same physical phenomenon but such different effects on human perception and well-being. This chap-ter introduces some fundamental concepts relating to the physics of noise, propagation, attenuation, and the main descriptors.

thE SouNd

The sound phenomenon can be described as a perturbation propagating in an elastic medium, causing a variation in pressure and particles displace-ment from their equilibrium positions. The term “perturbation” is used here because, if energy and information associated with the sound travel out from the source of the perturbation in the form of waves, single par-ticles in the medium remain near their equilibrium positions.

CoNtENtS

The sound ................................................................................................. 1Acoustic energy, levels, and frequency spectrum ........................................ 3

Acoustic energy: Sound intensity and energy density ............................. 3Sound description ................................................................................. 4Frequency and sound spectrum ............................................................. 5

Loudness, frequency weighting, and the equivalent pressure level ............. 6Loudness and frequency weighting ....................................................... 6Equivalent sound pressure level ............................................................ 6

Sources and propagation ........................................................................... 8Directivity ............................................................................................. 8Absorption, reflection, and refraction.................................................... 8Outdoor sound propagation ................................................................. 9

References ............................................................................................... 12

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Copyrighted Material – Taylor & Francis

2 D. Palazzuoli and G. Licitra

In fluids, particles vibrate in the same direction of wave propagation: in this sense, perturbations like sound are defined longitudinal (or compres-sional) waves.

In a perfect gas the speed of sound, c [m/s], is:

c

kp=ρ

0

0

where k is the adiabatic index (or heat capacity ratio) cp/cv, is the ratio between the specific heats at constant pressure (cp) and constant volume (cv), p0 and ρ0 are the equilibrium pressure [Pa] and density [kg m–3].

For the speed of sound in air a useful, approximate formula gives

Tcc 331.6 0.6 m/s+ ⋅=

with Tc in degrees Celsius.In absence of attenuation, the speed of sound is the only parameter that

enters the three-dimensional wave equation, which is given by

px

py

pz c

pt

∂∂

+ ∂∂

+ ∂∂

= ∂∂

12

2

2

2

2

2 2

2

2 (1.1)

where p(x,y,z,t) is the instantaneous variation in pressure due to the acoustic phenomenon at the point expressed by the coordinates (x,y,z) at the time t.

The general solution1 of the wave equation (Equation 1.1) in one dimen-sion in terms of sound pressure, p, is

p x t f ct x f ct x= − + +( , ) ( ) ( )1 2

in which f1 and f2 are arbitrary function (derivable in second order) repre-senting a wave travelling in the positive x direction and in the negative one respectively with a speed c.

In the case of a source vibrating sinusoidally it can be showed p varies both in time, t, and space, x, in a sinusoidal manner:

p x t p t kx p t kx( , ) sin( ) sin( )1 1 2 2= ω − + φ + ω + + φ (1.2)

where k is the acoustic wavenumber (k = ω/c), p1 and p2 are the amplitudes of the positive and negative direction travelling waves, ϕ1 and ϕ2 are phase angles, ω = 2πf is the angular frequency, and f is the frequency.

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Fundamentals 3

The wavelength λ [m] (the distance travelled by the wave in a complete oscillation) is related to the other wave parameters by the relation

c T

cf

λ = ° =

T [s] the period for a complete oscillation and f [Hz] is the frequency 1/T.Having defined a fluid particle as the smaller element of volume that

maintains the bulk properties of the fluid, Equation (1.2) remains valid also if pressure variation, p, is replaced with the particle displacement, ξ, or the particle velocity, u.

The (complex) ratio between the value of pressure and particle velocity defines the specific acoustic impedance Zs [Pa s m–1]. For one-dimensional propagation it can be shown that for any plane wave

pu

c= ρ

if the travelling direction is positive, or –ρc in the negative case.The ratio is the characteristic impedance, Zc, of the fluid. For air it is

equal to 407 Pa s m–1 at 22°C and 105 Pa (density and speed of sound in a fluid are a function of temperature and pressure).

aCouStIC ENErGy, lEvElS, aNd FrEquENCy SPECtrum

acoustic energy: Sound intensity and energy density

If we consider the energy flowing during sound propagation, it can be evaluated with regard to the sound intensity, I: the average rate of the energy that flows through an imaginary surface of a unit area in a direc-tion perpendicular to the surface. The instantaneous acoustic intensity is:

I = p ⋅ u⋅cos(θ)

where θ is the angle between the perpendicular to the unit surface and the propagation direction of the sound wave.

For a plane wave travelling in the positive x direction:

I

pc

=ρ

2

0

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The effective value is simply the root mean square (rms) of the instanta-neous one averaged over a time interval:

I

pc

rms=ρ

2

0

(1.3)

For a sinusoidal signal with amplitude A and period T, the rms value is

rmsT

A t dtA

a

b1

( cos )2

2∫= ° ω =

In three dimensions I is represented by a three-dimensional vector, i.2

It can be useful to define the density of acoustic energy (D) as the energy contained in a unit volume centred in a specified point in a space, by using Equation (1.2):

D

pcrms=

ρ

2

02

Sound description

Audible sounds in air cover a very wide range of both pressure variations, from about 20 μPa to 104 Pa, and acoustic intensity so it is necessary to express these quantities in logarithmic rather than linear scale. Acoustic levels are generally expressed as 10 times the logarithm to the base 10 (decibel) of the ratio relative to a reference level:

L

pp

prms

ref

=

°

10log dB10

2

2

L

II

Iref

=

°

10log dB10

L

WW

Wref

=

°

10log dB10

where pref = 20 μPa, Iref = 10–12 W/m2, and Wref = 10–12 W.

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Fundamentals 5

If two or more sound sources are uncorrelated, the overall pressure (or power) level is determined by the energetic sum (the average squared sound pressure):

Lp

Lpi

i

n

∑=°

=

10log 10 dB1010

1

Frequency and sound spectrum

Acoustics sources in an environment generally do not emit sounds char-acterised by only one frequency (pure tones), but they can be considered as a composition of different pure tones with specific frequencies, with a discrete or continuous distribution (spectrum). Fourier theorem3 allows it to decompose any signal (under weak hypotheses) in a series of sine waves (harmonics) with suitable amplitudes and phases. The amplitudes associ-ated to each frequency represent the sound spectrum.

Frequency analysis can be carried out by using different methods to define the energy content in bands at determined frequency intervals. Generally frequency analysis is carried out using filters with constant bandwidth or constant percentage filters. Each band is characterised by lower (f1) and upper (f2) cutoff frequencies, a band centre frequency (fc), and a bandwidth Δf = f2 – f1. In constant percentage filters the upper and lower frequencies are in a geometric progression:

f fn= ⋅22 1

with

f f fc = ⋅1 2

For n = 1 we obtain the one-octave bands, and the relevant parameters are defined as

f f= ⋅22 1

f f f

cc∆ = − =( 2 1 2)

2

For one-third-octave bands (n = 1/3):

f

ff fc

c= =2

21 6 26

f fc ( )∆ = − −

2 216

16

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Preferred frequencies for the analysis in bands are defined by the techni-cal norm ISO 266-1997 Acoustics.4

loudNESS, FrEquENCy wEIGhtING, aNd thE EquIvalENt PrESSurE lEvEl

loudness and frequency weighting

The concept of loudness is related to the perceived intensity of sound. The human ear, for instance, has a different sensibility to sounds depending on their frequencies. It is more sensitive in the range of 1000–4000 Hz (a range of frequencies corresponding to speech), with a poor response at lower and higher frequencies. Even before being interpreted by the brain, two sounds with the same energetic level (in decibels [dB]), but different spectral content, are perceived differently by the human ear.

Loudness is therefore a “subjective” property, unlike pressure levels (in decibels) or frequency spectra, which are objectively measurable physical quantities. For a given individual, sound loudness can vary not only with pressure level of the sound itself but also with frequency and sound. It is, however, possible to define the correction due to a “standard ear,” that can be applied to more objective measurements (e.g., those taken by a micro-phone) to simulate what a human would “perceive.”

By a procedure of comparison of a reference 1000 Hz pure tone sound with one varying both in frequency and pressure level, standardised equal-loudness contours are designed. A pure tone level in a defined contour line has a loudness equal to the pressure level of the reference 1000 Hz: if the level of a 1000 Hz is 50 dB then all tones on the same line have a loudness of 50 phons.

The 40 phons iso-loudness contour is chosen as the reference of A-weighting sound pressure: when a sound is weighted by using a trans-fer function inverse of that contour, an A-weighted pressure level is obtained.

Other commonly used filters are B-, C-, and D-weighting designed for specific applications. For instance, a D-weighting curve was used for the evaluation of disturbance from aircraft noise.

After the pioneering research done by Fletcher and Munson (1933, Bell Laboratories) the ISO 226:2003 adopted the curves from Robinson and Dadson.

Equivalent sound pressure level

In evaluating exposure to noise sources, the most commonly used single index quantity is the equivalent sound pressure level (Leq). It is useful to

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Fundamentals 7

rating sounds with levels varying in time. Leq is the sound pressure level averaged over a suitable period, T:

LT

dtT

p tp

dteq

L tT

eff

ref

T

10log1

10 10log1 ( ) dB

( )10

0

2

2

0∫ ∫=

°

=°

If pressure levels are weighted by the A-weighting curve it is obtained the equivalent continuous sound pressure level (LAeq):

LT

dtT

p tp

dtAeq

LA tT

Aeff

ref

T

10log1

10 10log1 ( )( )

10

0

2

2

0∫ ∫=

°

=°

where LA(t) is the instantaneous sound level A-weighted and pAeff(t) is the sound pressure measured A-weighting frequency filter.

LAeq is one of the most widely used descriptors in evaluating environmen-tal noise from roads, railways, and industry. In community noise evalua-tion, the evaluation time, T, is a period representing day, night, or evening.

In the framework of noise management, the Directive 2002/49/EC,5 pre-scribed the indicator Lden, which represents the day–evening–night level in dB:

Lden

L L Lday evening night

10log124

12 10 4 10 8 1010

5

10

10

10= ⋅ + ⋅ + ⋅°

+ +

where

• Lday is the A-weighted long-term average sound level as defined in ISO 1996-2:1987, determined over all the day period of a year

• Levening is the A-weighted long-term average sound level as defined in ISO 1996-2:1987, determined over all the evening period of a year

• Lnight is the A-weighted long-term average sound level as defined in ISO 1996-2:1987, determined over all the night period of a year”

(Appendix I, Directive 2002/49/EC)

The day is 12 hours, the evening 4 hours, and the night 8 hours. “The Member States may shorten the evening period by one or two hours and lengthen the day and/or the night period accordingly” and “the start of the day (and consequently the start of the evening and the start of the night) shall be chosen by the Member State (that choice shall be the same

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for noise from all sources); the default values are 07.00 to 19.00, 19.00 to 23.00 and 23.00 to 07.00 local time” (Appendix I, Directive 2002/49/EC).

SourCES aNd ProPaGatIoN

directivity

Most sources are not isotropic, that is, their pattern of emission is not con-stant in all the direction. Directivity is usually a function of frequency; many sources are omnidirectional at low frequency when their dimensions are lower than wavelength of the emitted sound but become directive with increasing frequency.

Directivity factor D(θ,ϕ) is defined as the ratio of mean square sound pressure p2

rms(θ,ϕ) (at angles θ,ϕ and distance r from the source, and the value of p2

rms at the same distance r due to an omnidirectional source of the same emitting power:

D

pp

rms

rms

( , )( , )2

2θ ° = θ °

The directivity index DI(θ,ϕ) is then defined as

DI D

pp

rms

rms

( , ) 10log( ( , )) 10log( , )2

2θ ° = θ ° = θ °

absorption, reflection, and refraction

When an acoustic wave hits a surface, its energy is partly absorbed (Ea) and partly reflected (Er) and transmitted (Et) (see Figure 1.1):

E E E Ei a t r= + +

Ei

Er

EaEt

Figure 1.1 Schematic picture of absorption, reflection, and transmission.

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Fundamentals 9

The adsorption coefficient, α, is defined as

E EEa t

i

α = +

Similarly, the transmission coefficient, τ, is

EEt

i

τ =

The phenomena of reflection and refraction of a sound wave are analo-gous to optical rays. If we consider two different ones, media 1 and 2, with speed sound c1 and c2, respectively (Figure 1.2), the incident angle (θi) is equal to the reflection angle (θr), and for the transmission angle (θt) Snell’s law holds:

cc

i

t

= θθ

sin( )sin( )

1

2

If c2 is smaller than c1, the wave is refracted toward the normal; θt will be smaller than θi.

outdoor sound propagation

A sound propagating without any obstacle or absorption phenomena is defined as the free field propagation condition.

For a point source of power, W, sound intensity, I, at distance, d, in a free field condition is

I

Wd

=π4 2

θi

c1c2 < c1

θr

θt

Figure 1.2 Reflection and refraction at different media boundary.

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and if ρ0c = 407 Pa s m–1:

L L d L dp W w= − °π ° = − −10log(4 ) 20log( ) 112 dB

In the case of a directive source, the pressure level in a point (r, θ, ϕ) is

L L d L d DIp W w10log(4 ) 20log( ) 11 ( , )2= − °π ° = − − + θ φ

Similarly, for an ideal line source on an infinite length pressure level can be calculated as

L L d DIp w 10log( ) 8 ( , )= − − + ° φ

where Lw is the sound power level per unit of length.Sound attenuation from sources to the receivers depends on the following:

• Air absorption• Ground or vegetation absorption• Meteorological conditions• Obstacles along the propagation path

A detailed description of attenuation in atmosphere and a calculation procedure is shown in the technical norms ISO 9613 parts 16 and 2,7 but a brief description of the different terms will be reported here. First, sound energy is dissipated as heat during its propagation in air, but the effect becomes significant only at high frequencies and at a long distance from the source.

For a plane wave the attenuation of sound in air can be evaluated by

Attair = α ⋅ r dB

where α depends on frequency, humidity, pressure, and temperature; and r is the distance source-receiver.

For low frequencies, air attenuation is smaller than 1 dB/km, whereas for frequencies higher than 12 to 13 kHz the attenuation is very high.

The attenuation due to foliage is generally small. The excess attenuation, when a dense forest is present between the source and the receiver, may be estimated using8

A r ff f= ⋅ ⋅0.01

13

where f is the sound frequency (Hz) and rf is the length (in metres) of the path through the forest. Values of foliage attenuation are generally reported in decibels when sound travel distances rf between 10 and 20 m; and reported in dB/m when rf is between 20 and 200 m.

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Fundamentals 11

The distance of sound in foliage depends on the relative height of the source and receiver, and on the curvature sound path due to meteorological conditions.

The interference between the direct sound from the source and the reflected sound from the ground modify the overall level to the receiver. The ground effect depends on the geometry of source–receiver position and the properties of the ground surface.9 ISO 9613-2:1996 presents a calcula-tion procedure that takes into account the ground effect on sound attenua-tion. It considers the worst case of sound propagation downwind from the source to the receiver. Considering the height of the source and the receiver and their distance, the norm considers three zones: near the source, middle, and near the receiver. The acoustic characteristic of each zone is defined by the parameter, G, varying from 0 (hard ground) to 1 (soft ground). The ground excess attenuation (Ag) is then the sum of the attenuation due to the three zones (source, middle, receiver):

A A A Ag s m r= + +

Meteorological conditions modify sound propagation mainly by the effects of wind gradient and vertical temperature gradient. With tempera-ture inversion phenomena (positive temperature gradient near the ground surface) sound rays are diffracted downward resulting in an increasing sound level. On the other hand a temperature lapse (negative temperature gradient) reduces the sound level on the ground.9

Noise barriers or obstacles, large in comparison with the wavelength of the incident sound, reduce noise level at the receiver (R) in their shadow zone (Figure 1.3) where acoustic rays reach the receiver only for diffrac-tion phenomena.

S

a

c

b

R

Figure 1.3 Sound barrier.

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The sound level in the shadow zone is determined by the fraction of acoustic energy passing through the screen and diffracted by its edge. In order to reduce the fraction of the energy transmitted by the barrier, the surface density has to be >20 kg/m2.

The attenuation [dB], or insertion loss (IL), is used to define the perfor-mance of a barrier:

Att IL L Lp withoutbarrier p withbarrier= = −, ,

Barrier maximum insertion loss is about 20 dB. For a long barrier, in order to ignore the contribution of the diffraction from the lateral edges, the attenuation can be evaluated by using the empirical Maekawa relation10:

Att = 10log(3 + 20N) dB

where N represents the Fresnel number

N = δ

°2

and δ is the difference between the diffracted path and the direct one, δ = a + b − c.

In order to evaluate the noise level reduction from road, the semiempiri-cal formula of Kurze and Anderson11 can also be used:

Att

NN

= ππ

°

20log

2tanh 2

dB + 5 dB10

rEFErENCES

1. Morse P.M., and Ingard K.U., 1968, Theoretical Acoustics (New York: McGraw-Hill).

2. Fahy F.J., 1995, Sound Intensity (London: E&FN Spon, Chapman & Hall). 3. William E., 1999, Fourier Acoustics (London: Academic Press). 4. ISO 266: 1997, Acoustics—Preferred frequencies. 5. Directive 2002/49/EC of the European Parliament and of the Council of

25 June 2002 relating to the assessment and management of environmental noise.

6. ISO 9613-1, 1996, Acoustics—Attenuation of sound during propagation out-doors. Calculation of the absorption of sound in atmosphere.

7. ISO 9613-2, 1996, Acoustics—Attenuation of sound during propagation out-doors. General methods of calculation.

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Fundamentals 13

8. Hoover R.M., 1961, Tree zones as barriers for the control of noise due to air-craft operations. Bolt, Beranek, and Newman, Report 844.

9. Attenborough K., Li K.M., and Horoshenkov K., 2007, Predicting Outdoor Sound (London: Taylor & Francis).

10. Maekawa Z., 1968, Noise reduction by screens. Applied Acoustics 1, 157–173. 11. Kurze U.J., and Anderson G.S., 1971, Sound attenuation by barriers. Applied

Acoustics 4, 35.

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