acoustic and noise fundamental

8/11/2019 Acoustic and Noise Fundamental

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Chapter 1

Fundamentals

D. Palazzuoli and G. Licitra

The whispering of wind in a wood and the roar of a traffic jam, a mountainwaterfall and a road yard, noise and music: the same physical phenomenonbut 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 inan 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 travelout 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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2 D. Palazzuoli and G. Licitra

In fluids, particles vibrate in the same direction of wave propagation: inthis 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

=

0

0

where k is the adiabatic index (or heat capacity ratio) cp/cv, is the ratiobetween the specific heats at constant pressure (cp) and constant volume(cv),p0and 0are 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 Tcin 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

p

x

p

y

p

z c

p

t

+

+

=

12

2

2

2

2

2 2

2

2 (1.1)

wherep(x,y,z,t) is the instantaneous variation in pressure due to the acousticphenomenon at the point expressed by the coordinates (x,y,z) at the time t.

The general solution1of 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 f1and f2are arbitrary function (derivable in second order) repre-senting a wave travelling in the positive xdirection and in the negative onerespectively with a speed c.

In the case of a source vibrating sinusoidally it can be showed pvariesboth 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= + + + + (1.2)

where k is the acoustic wavenumber (k = /c),p1andp2are the amplitudes

of the positive and negative direction travelling waves, 1and 2are phaseangles, =2f is the angular frequency, and f is the frequency.

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Fundamentals 3

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

c T c

f = =

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 alsoif pressure variation,p, is replaced with the particle displacement, , or theparticle velocity, u.

The (complex) ratio between the value of pressure and particle velocity

defines the specific acoustic impedanceZs[Pa s m1

]. For one-dimensionalpropagation it can be shown that for any plane wave

p

uc=

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

equal to 407 Pa s m1at 22C and 105Pa (density and speed of sound in afluid 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 beevaluated 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 ucos()

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

For a plane wave travelling in the positive xdirection:

I p

c=

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=

2

0

(1.3)

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

rmsT

A t dt A

a

b

1( cos )

2

2= =

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

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

D p

c

rms=

2

0

2

Sound description

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

L p

pp

rms

ref

=

10log dB10

2

2

L I

II

ref

=

10log dB10

L

W

WW ref=

10log dB10

wherepref=20 Pa, Iref=1012W/m2, and Wref=10

12W.




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Fundamentals 5

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

LpLpi

i

n

=

=

10log 10 dB10 10

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 consideredas a composition of different pure tones with specific frequencies, with adiscrete or continuous distribution (spectrum). Fourier theorem3allows 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 definethe energy content in bands at determined frequency intervals. Generallyfrequency analysis is carried out using filters with constant bandwidth orconstant percentage filters. Each band is characterised by lower (f1) andupper (f2) cutoff frequencies, a band centre frequency (fc), and a bandwidthf =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 parametersare defined as

f f= 22 1

f f fc c = =( 2 1 2)

2

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

f f

f fc c= =

2

216

26

f fc ( ) = 2 2

16

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. Thehuman ear, for instance, has a different sensibility to sounds dependingon their frequencies. It is more sensitive in the range of 10004000 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 differentspectral content, are perceived differently by the human ear.

Loudness is therefore a subjective property, unlike pressure levels (indecibels) or frequency spectra, which are objectively measurable physicalquantities. For a given individual, sound loudness can vary not only withpressure 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 canbe 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 linehas a loudness equal to the pressure level of the reference 1000 Hz: if thelevel of a 1000 Hz is 50 dB then all tones on the same line have a loudnessof 50 phons.

The 40 phons iso-loudness contour is chosen as the reference ofA-weighting sound pressure: when a sound is weighted by using a trans-

fer function inverse of that contour, an A-weighted pressure level isobtained.Other commonly used filters are B-, C-, and D-weighting designed for

specific applications. For instance, a D-weighting curve was used for theevaluation 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 Robinsonand Dadson.

Equivalent sound pressure level

In evaluating exposure to noise sources, the most commonly used singleindex 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 levelaveraged over a suitable period, T:

LT

dtT

p tp

dteqL t

T

eff

ref

T

10log 1

10 10log 1 ( )

dB( )

10

0

2

2

0

=

=

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

L T dt T p tp dtAeq

LA tT

Aeffref

T

10log 1 10 10log 1 ( )( )

10

0

2

2

0 =

=

whereLA(t) is the instantaneous sound level A-weighted andpAeff(t) is thesound pressure measured A-weighting frequency filter.

LAeqis 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,5pre-

scribed the indicator Lden, which represents the dayeveningnight level in dB:

Lden

L L Lday evening night

10log 1

2412 10 4 10 8 1010

5

10

10

10= + +

+ +

where

Lday

is the A-weighted long-term average sound level as defined in ISO1996-2:1987, determined over all the day period of a year

Leveningis the A-weighted long-term average sound level as defined inISO 1996-2:1987, determined over all the evening period of a year

Lnight is the A-weighted long-term average sound level as defined inISO 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. TheMember States may shorten the evening period by one or two hours and

lengthen the day and/or the night period accordingly and the start ofthe day (and consequently the start of the evening and the start of thenight) 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 to23.00 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 dimensionsare lower than wavelength of the emitted sound but become directive withincreasing frequency.

Directivity factor D(,) is defined as the ratio of mean square soundpressure p2rms(,) (at angles ,and distance r from the source, and thevalue ofp2rmsat the same distance rdue to an omnidirectional source of thesame emitting power:

D p

p

rms

rms

( , ) ( , )2

2 =

The directivity indexDI(,) is then defined as

DI D p

p

rms

rms

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

2 = =

Absorption, reflection, and refraction

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

E E E Ei a t r= + +

Ei

Er

Ea

Et

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




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Fundamentals 9

The adsorption coefficient, , is defined as

E E

E

a t

i

=+

Similarly, the transmission coefficient,, is

E

E

t

i

=

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 c1and c2, respectively (Figure 1.2), the incident angle(i) is equal to the reflection angle (r), and for the transmission angle (t)Snells law holds:

c

c

i

t

=

sin( )

sin( )

1

2

If c2is smaller than c1, the wave is refracted toward the normal; twill besmaller than i.

Outdoor sound propagation

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

For a point source of power, W, sound intensity, I, at distance, d,in a

free field conditionis

I

W

d=

4 2

i

c1

c2 < c1

r

t

Figure 1.2 Reflection and refraction at different media boundary.




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and if 0c=407 Pa s m1:

L L d L d p W w= = 10log(4 ) 20log( ) 11

2 dB

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

L L d L d DI p W w10log(4 ) 20log( ) 11 ( , )

2= = +

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

L L d DI p w 10log( ) 8 ( , )= +

where Lwis 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 calculationprocedure is shown in the technical norms ISO 9613 parts 16and 2,7but abrief description of the different terms will be reported here. First, soundenergy is dissipated as heat during its propagation in air, but the effectbecomes significant only at high frequencies and at a long distance fromthe source.

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

Attair= rdB

where 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 forfrequencies 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 beestimated using8

A r ff f= 0.01

1

3

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




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Fundamentals 11

The distance of sound in foliage depends on the relative height of the sourceand 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 sourcereceiver position andthe properties of the ground surface.9ISO 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 thesource to the receiver. Considering the height of the source and the receiverand their distance, the norm considers three zones: near the source, middle,and near the receiver. The acoustic characteristic of each zone is definedby the parameter, G, varying from 0 (hard ground) to 1 (soft ground). Theground excess attenuation (A

g) 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 theeffects of wind gradient and vertical temperature gradient. With tempera-ture inversion phenomena (positive temperature gradient near the groundsurface) sound rays are diffracted downward resulting in an increasing

sound level. On the other hand a temperature lapse (negative temperaturegradient) reduces the sound level on the ground.9

Noise barriers or obstacles, large in comparison with the wavelength ofthe incident sound, reduce noise level at the receiver (R) in their shadowzone (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 ofacoustic energy passing through the screen and diffracted by its edge. Inorder 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, inorder to ignore the contribution of the diffraction from the lateral edges, theattenuation can be evaluated by using the empirical Maekawa relation10:

Att = 10log(3 + 20N) dB

where Nrepresents the Fresnel number

N=

2

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

In order to evaluate the noise level reduction from road, the semiempiri-cal formula of Kurze and Anderson11can also be used:

Att N

N=

20log

2

tanh 2dB + 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, AcousticsPreferred 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 environmentalnoise.

6. ISO 9613-1, 1996, AcousticsAttenuation of sound during propagation out-

doors. Calculation of the absorption of sound in atmosphere. 7. ISO 9613-2, 1996, AcousticsAttenuation 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 OutdoorSound(London: Taylor & Francis).

10. Maekawa Z., 1968, Noise reduction by screens. Applied Acoustics1, 157173. 11. Kurze U.J., and Anderson G.S., 1971, Sound attenuation by barriers. Applied

Acoustics4, 35.




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