Chapter 4

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd

Sound PropagationAn Impedance Based Approach

Radiation, Scattering, and Diffraction

Yang-Hann Kim

Chapter 4


Outline

4.1 Introduction/Study Objectives

4.2 Radiation of a Breathing Sphere and a Trembling Sphere

4.3 Radiation from a Baffled Piston

4.4 Radiation from a Finite Vibrating Plate

4.5 Diffraction and Scattering

4.6 Chapter Summary

4.7 Essentials of Radiation, Scattering, and Diffraction

2


4.1 Introduction/Study Objectives

• Scattering and diffraction are physical phenomena, which represent wavesdeflected by characteristics of the discontinuity. Both can therefore beexpressed by solutions of the wave equation which satisfy boundaryconditions.

3

Figure 4.1 (a) Radiation, (b) Scattering, and (c) Diffraction: waves are visualized using a ripple tank. Incidence waves for (b)-(d)are plane waves coming from the left. The depth of the water has to be sufficiently smaller than 1/8th the wavelength to create anon-dispersive wave( has to be smaller than roughly 0.5, where is wave number, and h is water depth)kh k



• To understand these rather complicated phenomena, we need tounderstand sound fields induced by the basic unit sources: the radiation ofa breathing sphere and a trembling sphere.

• The first type of basic radiation unit is a breathing sphere.

4

Figure 4.2 A breathing sphere and its radiation pattern: is the radius, indicates the radial distance, and denotes thevelocity magnitude

a r 0U


• Linear differential equation in polar coordinates can be expressed as

• The solution of Equation 4.1 can be written as

5


2

2 2 2

1 ,r rr c t

,j t krer

A

(4.1)

(4.2)

where is the wave number in the direction.k r

• If the surface of the sphere harmonically vibrates, then the velocity on thesurface ( ) can be written as

0, .j tr a t e u U

( , )r a tu(4.3)

where is the velocity potential.



• The rate of change of the velocity potential with regard to r has to be the velocity at the sphere (4.3). That is,

• The velocity at will then be

6

2

1, .j t krr

jkr t er r r

u A

2

1, .j t kar

jka t ea a

u A

2

0 .1

jkaa ejka

A U

(4.4)

(4.5)

(4.6)

r a

• From Equations 4.3 and 4.5, we can obtain


• The velocity potential, the velocity, and the acoustic pressure cantherefore be written as

• Note that Equations 4.7, 4.8, 4.9 are only valid if r is larger than a.

• Note that the velocity and pressure depend on the relative scales such asthe ratio between the radius of the sphere ( ) and the observation position( ): the sphere’s radius with respect to the wavelength ( ) and theobservation position with regard to the wavelength ( ).

7

2

01, ,

1j t k r aar t e

jka r

U

2

01, ,1

j t k r ar

jkr ar t ejka r

u U

0 0, .

1j t k r ajka ar t c e

jka r

p U

ar ka

kr


(4.7)

(4.8)

(4.9)


• The acoustic impedance of the sphere ( ) can be obtained from Equations4.8 and 4.9, that is,

8


2

0 02 2 2 2

11 .1 11 1 1 1

r

kr kr krc j c jkr kr

kr kr

Z

“Far-field” : when is much larger than 1The radiation from the breathing sphere resembles a one-dimensional

acoustic wave (plane wave), because the acoustic impedance is .

kr

“Near-field” : when is smaller than 1The imaginary part (i.e., the reactive part of the impedance) dominates

the radiation characteristics. Therefore, the radiation is not likely to beeffective.

kr

0c

(4.10)

rZ


• The radiation impedance of a breathing sphere at r=a exhibits how wellthe sphere radiates sound from its surface. This can be obtained fromEquation 4.10, that is,

• The mechanical impedance ( ) can be expressed by multiplying Equation4.11 by the surface area ( ), that is,

9


2

0 02 2 2 2

11 .1 11 1 1 1

r a

ka ka kac j c jka ka

ka ka

Z

22 2

0 02 2 2 2

114 4 .1 11 1 1 1

m

ka ka kaa c j a c jka ka

ka ka

Z

(4.11)

(4.12)

r a

mZ24 a

Sound Propagation: An Impedance Based Approach Yang-Hann Kim © 2010 John Wiley & Sons (Asia) Pte Ltd 10

Figure 4.3 Impedances of the breathing sphere: (a) the acoustic impedance and the radiation impedance, and (b) the radiationpower. ( and dominate the characteristics of the impedances; as they become larger, the wave behaves as if it is planar)kr ka



• If we calculate the mean intensity ( ) by using Equations 4.8 and 4.9, then we obtain

11


• The radiation power can be obtained from Equation 4.17 by multiplying the area of interest by

*

22 22 2

0 0 0 02 2

1 Re2

1 1 1 ,2 2 11 1

avgI

kaa ac cr rka

ka

PU

U U

22 2 2 2 2

0 0 0 02 21 1 14 4 4 .2 2 11 1

avg

kaI r c a c a

kaka

U U

(4.17)

(4.18)

avgI

where P and U are the sound pressure magnitude (4.9) and the complex conjugate of the velocity magnitude (4.8).

P *U



• The normalized radiation power ( ) of a circular plate of radius a with a velocity of U0 and frequency w. This is expressed as

22

2 22 2

0 0

4 1 ,1 114 12

avgavg

I r kakac a

ka

U(4.19)

avg a0U

which highlights that the radiation power becomes larger as we increase ka (Figure 4.3(b)).ka


• The second type of basic radiation unit is a trembling sphere (Figure 4.4).

13


Figure 4.4 The trembling sphere. (The direction of vibration is z, the velocity magnitude is , and Ur is the velocity in the rdirection. Other symbols represent coordinates)

z cU rU r


• The acoustic waves generated by this trembling sphere would satisfy theacoustic wave equation in the spherical coordinate, that is,

14


2 2 2

2 2 2 2 2 2 2

1 1 1 1sin .sin sin

rr r r r c t

cos ,r a cr ar

U U

(4.20)

(4.21)

• The boundary condition on the surface of the trembling sphere ( r=a ) canbe written as

r a

where Ur=a denotes the velocity of the trembling sphere in the r direction.r aU r

• The solution satisfying Equation 4.20 is found to be

, , cos .jkrer

r r

A (4.22)



0 0 cos .jkr

j t j tee j et r r

p A

2

2cos .jkr

re

r r r

U = A

2

2cos cos ,jkr

c

r a

er r

U A

• The pressure (p ) can be calculated from Equation 4.22 asp

(4.23)

• The velocity ( ) can be obtained by taking the derivative of the potentialfunction with respect to r as

(4.24)

rU

• Equations 4.21 and 4.24 lead us to write

3

2 .2 2

jkaca eka jka

UA

and we can obtain as

(4.25)

A

(4.26)

r


• The pressure magnitude ( ) can be obtained by using Equations 4.23 and 4.26, that is,

16


• The velocity ( ) can be calculated using Equations 4.24 and 4.26, that is

3

2 2

1 1cos .2 2

jk r a0 c

jkjk c a er rka jka

p = U

3 2

2 3 2

2 2cos .2 2

jk r acr

a jk k er r rka jka

UU

p

rU

(4.28)

(4.27)

• The acoustic impedance ( Z) can be obtained from Equations 4.27 and4.28 as

3

4 3

0 04 4 4 4

1 122 1 ,1 14 4 1 4 1 4

r

kr kr kr kr krc j c jkr kr

kr kr

Z

rZ

(4.29)

which states that the impedance is independent of θ although the pressure and velocity magnitude depend on cosθ.

cos



• The radiation impedance ( ) can be obtained from Equation 4.29, that is,

3

4 3

0 04 4 4 4

1 122 1 .1 14 4 1 4 1 4

r a

ka ka ka ka kac j c jka ka

ka ka

Z

r aZ

(4.30)

• If ka is small (i.e., if the radius of the sphere is small compared to thewavelength of interest), then the reactive term which is the imaginary partof the impedance dominates the radiation characteristics.

• When ka is large, then the resistive term governs the impedance and thetrembling sphere effectively radiates sound waves.

ka

ka



• The normalized radiation power ( ) can be expressed by

(4.31)

42 2

4 41cos cos .

14 1 4avg

kaka

ka

avg

• To effectively express this angle dependency of the radiation, we definethe directivity factor as

(4.32),

sphere

IDI

where Ispheis the intensity radiated from the breathing sphere withradiation power equivalent to that of the radiator of interest. denotes theintensity of the radiator whose directivity we wish to characterize.

sphereII

• Note that the pressure and velocity depend on θ.


2cos)( D


Figure 4.5 The directivity factor of a trembling sphere; the circumferential angle is expressed in degrees and the radial distanceis a non-dimensional arbitrary unit



Figure 4.6 Impedances of the trembling sphere: (a) the acoustic impedance and the radiation impedance, and (b) the radiationpower. Note that the radiation impedance and the radiation power are proportional to and , respectively. Thecorresponding characteristics of the breathing sphere are proportional to and . The radiation of the trembling spheretherefore depends on the viewing location and the size of the diameter relative to the wavelength of interest

4( )kr 4( )ka2( )kr 2( )ka



21

0 0 0 0 0 0 0

0 0 0 0 0 0 0

| |

| | .

p

u

S

S

r r r r r r r n dS

r r r r r r n dS

P P G G P

P G G P

Figure 4.7 Volume integral of the Kirchhoff-Helmholtz integral equation. (Sp and Suexpress the boundary surface for thepressure and velocity U. d denotes a surface that is infinitely far from the origin, and and indicate the observationposition and the boundary position vector, respectively)

pS uSP U S r 0r

• We can rewrite the integral equation that emphasizes the individualcontribution of pressure and velocity sources as

(4.33)



• Consider the sound radiation from a baffled piston as illustrated in Figure4.8. Note that we only have the second integral of Equation 4.33 in thiscase.

22

R

r

or

O

oR r r

r

or

O

0S 0

S

n 0U

nU

n 0U

nU

Figure 4.8 Surface integral to calculate the sound radiation from a baffled piston and nomenclature of the coordinate



23

• If we apply Equation 4.33 to this specific case by employing the surface of the integral ( ) (Figure 4.8), then we obtain

0

00.

2

jkR

nS

jk c er dSR

P U (4.34)

0S

• Note that Equation 4.34 states that the radiated sound pressure is inducedby the vibrating surface velocity. Equation 4.34 can also be regarded as anexpression of Huygen’s principle in an integral form.



• We start with the case where Un is constant. In other words, the piston is arigid vibrator (Figure 4.9). By using Equation 4.34, the pressure at anarbitrary position (z) can be expressed by

24Figure 4.9 The radiated sound field from an infinitely baffled circular piston

2 2

02 20

2 .2

jk za

njk c ez d

z

P U (4.35)

nU



• Equation 4.35 can be simplified as

25

00 ,jkRjkz

nz c e e P U

where is the distance from the edge of a circular piston to z(detailed derivation is given in Section 4.7.2).

2 20R z a z

(4.36)

• There are two distinct contributions in the z direction. The first term ( ) isthe pressure coming from the center of the piston, and the second term( ) is from the piston’s rim. The coefficient ( ) is the pressure of theplane wave that is generated by the piston’s motion.

• Note that these two pressure waves interfere with each other. Theinterference sometimes mutually cancels the waves or reinforces them.

• If z is significantly large compared to the radius of the piston, then Equation4.36 becomes

0 nc U

jkze

0jkRe

0 .2

jkzn

jka az c ez

P U (4.37)



• Figure 4.10 depicts the coordinates and nomenclature that we use topredict the radiation from the vibrating piston to an arbitrary position,including the z axis.

• The sound pressure ( ) on the plane is given as

26

z

Figure 4.10 Coordinate set-up and variables for obtaining radiated sound field on x-z plane from an infinitely baffled circularpiston

( , )r P x z

120

sin,

sin

jkr

n

J kaer jk c ar ka

P U (4.38)

where is a Bessel function of the first kind.1J



• It is interesting to look at the sound pressure at , that is,

27

2

0, 0 .2

jkr nkar j c er

UP (4.39)

0

This is identical to Equation 4.37, meaning that Equation 4.38 is a general expression of the radiation sound pressure on the x -z plane.

• Using Equations 4.38 and 4.39, we can obtain an expression whichprovides further significant physical insight, that is,

1 sin, , 0 2 .

sinJ ka

r rka

P P (4.40)

• Also, note that the mean intensity is2

0

.2avgI

c

P(4.41)

x z



• Therefore, the intensity ratio between the time-averaged intensity at anarbitrary position with respect to that of the axis attached to the center ofthe piston can be expressed as

28

21, sin

2, 0 sin

avg

avg

I r J kaI r ka

(4.42)

• Equation 4.42 is often called the directivity or directivity index. Othernames such as spreading and spreading index are also widely used.

• The angular dependency of the radiation significantly decreases as theradiator size decreases relative to the wavelength, or when we have lowerfrequency radiation.

• On the other hand, when ka becomes larger, or the frequency becomeshigher, the radiation strongly depends on the angle.

ka

by using Equations 4.40 and 4.41.



29

8.3ka

4.6ka 20ka

0ka 0.1ka

0.8ka

Figure 4.11 The directivity or directivity index of a baffled circular piston



• Note that the radiation impedance of the plate varies with the position ofthe plate. The velocity is constant (Un) on the surface, but the resultingpressure is not uniform.

• The radiation impedance of this case can be defined as

30

/ ,avgr

n n

S

P FZU U

(4.43)

where F is force acting on the source surface (S), and is the averagepressure on the surface.

S avgPF

2

0 00 0

, ,2 2

jkR jkRan n

S S

jk c jk ce edS d dR R

U UP

nU

• To obtain F, the pressure on the surface of the piston is needed.This can be regarded as the sum of the pressure induced by the otherarea over the entire piston surface, that is,

F ,S P

(4.44)

where Figure 4.12(a) illustrates the variables for the integration.



31

'

2

0 0

2 20

0 0 0 0

,

.2

SSa

S

jkRa an

dS

d d

jk c e d d d dR

F P

P

U

2 2

0 0 0 0

2 2 2 cos

0 0 2 0

2

0 00 0

2

2 1 (2 ) (2 ) ,

jkRa a

jkRa

a

e d d d dR

e RdRd d dR

J k j k d djk

H

• The force acting on the piston surface can be given by

(4.45)

The integration of Equation 4.45 can be written as:

(4.46)

where is the first kind of Bessel function of zero order, andis the zero-order Struve function.0 (2 )k H

0 (2 )J k



32

• The force acting on the piston surface can then be given by

20

0 00 0

2

0 0 00 0

2 1 10

2 1 (2 ) (2 )2

1 (2 ) (2 )

(2 ) (2 )1 ,

an

a

n

n

jk c J k j k d djk

c J k j k d d

J ka kac a jka ka

UF H

U H

HU

(4.47)

where aaaaa is the first kind of Bessel function of the first order, and aaaaaaisthe Struve function of the first order.

1(2 )kaH1(2 )J ka

Figure 4.12 The variables on the surface of the disk for integrating with respect to (a) z and f , and (b) R and y

(b) (a)

R



33

• is then defined as1 1

0(2 ) (2 )1 ,avg n

J ka kac jka ka

HP U

1 10

(2 ) (2 )1 .r

J ka kac jka ka

HZ

avgP

(4.48)

and the radiation impedance of a baffled circular piston with radius a is

(4.49)

Figure 4.13 Radiation impedance, which is normalized to , of the circular baffled piston with respect to . The solid linerepresents the resistance term, and the dashed line the reactance

0c ka



• The radiation due to the plate vibration can be considered to be composedof the radiation of n modes of vibration.

• The radiation due to each mode of vibration can be superimposed by thevibration of many pistons, as illustrated in Figure 4.14(b). We can assumethat each mode of vibration can have equivalent pistons.

34Figure 4.14 (a) Radiation from a finite plate and (b) its possible modeling

0 ( , , 0, )u x y z t

z

y

(b)

(a)



• To understand the radiation from the vibrating plate, we look at basicradiations patterns such as those illustrated in Figure 4.15.

• The radiated fields are obtained using the Rayleigh integral equation.• The radiator’s typical dimensions are larger than the wavelengths that are

generated; the radiation efficiencies are therefore fairly good.• The sound pressures at the distance from the radiators ( ) become

negligible when the distance is long compared to the wavelength, exceptfor case (a).

• This is because the far field sound propagation tends to a plane wave,resulting in a perfect cancellation for the cases of (b) and (c) in Figure 4.15.

• This kind of cancellation becomes more and more significant when wehave higher order modes.

35

lz



36

Figure 4.15 Basic examples of plate vibration and radiation (A indicates the observation position, and can be expressed as(0,0,zl) in the Cartesian coordinate)



• To look more specifically at the radiation from a plate, we describe thewaves in space using rectangular coordinates for convenience.

• The wave is assumed to be harmonic in space and time, without loss ofgenerality. The acoustic wave can now be written as

37

, , , .x y zj k x k y k zjk r j t j tx y z t e e e e

p P P

• Equation 4.50 must satisfy the linear acoustic wave equation; the followingequality must therefore hold, that is,

2 2 2 2 ,x y zk k k k

,xx

mkL

,yy

nkL

(4.50)

(4.51)

(4.52)

(4.53)

where and are the length of the plate in the x and y directions, and mand n are integers.

xL yL

/k cwhere , which is a dispersion relation.The wave numbers in the x and y directions ( and ) can be written asxk yk



• Figure 4.16(a), Equations 4.51, 4.52, and 4.53 tell us that the wavenumber in the z direction ( ), which describes how the wave in the zdirection propagates, could be real or imaginary.

38

Figure 4.16 Radiation from a finite plate: (a) corner mode, edge modes, and radiation circle in wave number domain and (b)edge (corner) mode dominates the radiation in the left- (right-) hand image

z zzk



• The development of either exponentially decaying propagation(evanescent wave) in the z direction or a continuous phase changing waveis determined by the location (ox, ky) in reference to the radiation circle.

39

z( , )x yk k

2 2 2.x yk k k

2 2 2.x yk k k

(4.54)

(4.55)

Case 2) When is located outside a radiation circle,( , )x yk k

• has to be imaginary and the wave therefore decays exponentially.• The wave is less likely to propagate in the z direction with a larger or , in

other words, as the wavelength becomes increasingly smaller.

zkxk yk

Case 1) When is located inside a radiation circle,( , )x yk k

• The propagator can be mathematically written as . zjk ze• The wave in the z direction continuously changes its phase as it propagates.



• In acoustic holography, the spatial distribution of a sound wave isexpressed in terms of the wave number domain of interest, and thenpropagated to a plane that is not measured by using a propagator.

• Figures 4.17 and 4.18 illustrate the basic procedures of acousticholography.

40

x

x

y

y

xk

yk

( ) Hx, y , z zP ( )x yk ,kP

( )( ) z Hjk z zx yk ,k e P

)( Hz zzjke

transformFourier 2D

transformFourier

inverse 2D

k

circle radiation

domain Space domainnumber Wave

wavegPropagatin wave

Evanescent

Figure 4.17 Conceptual diagram of acoustic holography in rectangular coordinate



41

Figure 4.18 Illustration of acoustic holography and its fundamental procedure



• Figure 4.19 illustrates a typical diffraction and scattering phenomena. Thediffraction tends to be stronger as the wavelength becomes larger. In otherwords, we have more diffraction at the back of the wall as the wavelengthincreases.

• Diffraction is generally used to describe the physical circumstances underwhich we can hear sound but cannot see the sound source.

• Scattering describes waves that induced due to an abrupt impedancechange in space, especially when the waves spread out in space.

42

Figure 4.19 Diffraction around a straight barrier (note that the diffraction strongly depends on the wavelength)



• Beginning with the simplest case which considers all the necessaryfundamentals, suppose that we have a plane wave impinging on anarbitrary scatterer, as illustrated in Figure 4.20.

43

.t i sc P P Pthe complex amplitude of the incident wavethe complex amplitude of the scattering wave

i

sc

PP

(4.58)

• The total sound pressure can be expressed as

scP

a

rkji e

BP

0S

n

iPx

0S

xjkrkji

xee BBP

y

bAfA

scP

n

a

(b)(a)Figure 4.20 Scattering by (a) arbitrary scatterer and by (b) a sphere. (Pi, Pscrepresent the complex amplitude of incident andscattered waves;0 and n represent the surface of the scatterer and a unit vector normal to the surface, and a is thecharacteristic length of the scatterer)

,i scP P0S n a



• The total pressure has to satisfy the following boundary condition, that is,

44

0.t n P (4.59)

The boundary is acoustically rigid and is a unit normal vector on the surface.

• Let P propagate in the direction of the wave number vector k. The incidentwave at the position r can then be written as

iP k

r

,jk ri e

P B (4.60)

where B is the complex amplitude of the incident wave (Figure 4.20).

.jk rsc n j k n e

P B

• Equations 4.58, 4,59 and 4.60 lead us to a relation between the scatteredwave and the incident wave, that is,

(4.61)

n



45

cos ,jk rsc jk en

P B

0 ,sc scj

P U

,j tsc sce

u U

• If we rewrite Equation 4.61 in terms of the velocity of the scatterer by usingthe linearized Euler equation, then we obtain

(4.62)

(4.63)

where we assume that the scattering velocity ( ) is harmonic in time, thatis,

scu

where is the magnitude of the scattering velocity in vector form.sc

U

• By starting with the simplest case, we consider the scattering of a rigidsphere to explore what is meant by Equations 4.61 and 4.62. RewritingEquation 4.61 using the coordinate of Figure 4.20(b), we obtain

(4.64)

where represents the angle between the propagation vector and thenormal vector .

k

n



46

00

1cos ,jk rsc

on S

n jk ejk c

U B

cos

0

cos .jkasc n e

c

BU

0

cos 1 cos .sc n jkac

BU

• The velocity in the normal direction, which is , that is,sc n U

where is the surface of the sphere (Figure 4.20(b)). Since the radius ofthe sphere is a, we can rewrite Equation 4.65 as

0Sa

(4.66)

(4.65)

(4.67)

• If the size of the scatterer is substantially smaller than the wavelength ofinterest (i.e., if ), then Equation 4.66 can be approximately rewrittenas

1ka

• This means that the scattering field is essentially induced by the radiationof the sphere, which vibrates with a velocity as described by Equation 4.67.



47

Figure 4.21 Directional component ( ) of the normal velocity ( ) on the rigid sphere with respect to kacoscos jkae scU n ka



48

r

0r

b O

R

r

z

y

0y

r

0r

b O

R

r

z

y

0y

x

z

rr

r

0r0r

O00

00 , yx 00 , yx

R

0r0r

x

y

z

rr

r

0r0r

O0

R

ψ

Figure 4.22 (a) Two-dimensional and (b) three-dimensional rectangular slit with corresponding nomenclature

(b)(a)

• Figure 4.22 exhibits two fundamental scatterers which can demonstratethe scatterers’ geometrical effect on the radiation. The first is a two-dimensional case and the second is its extension to three dimensions.



• The radiated sound pressure for the two-dimensional slit can be found as

49

0 0

sin / 2 sin, 2 1

/ 2 sinjkr kbkr j c e

r kb

P U (4.70)

• The scattered field of the rectangular slit can be obtained as

(4.71)

00

sin / 2 sin cos sin / 2 sin cos, , .

2 / 2 sin cos / 2 sin cos

jkr ka kbk c er j abr ka kb

P U

• We can also consider that Equations 4.70 and 4.71 describing a diffractionfield due to the slits or scatterers. It can also be argued that the diffractionsin these cases strongly depend on the non-dimensional scale factors (kaand kb) and observation angle .kb

ka

(see Section 4.7.4.5 for details).

(See Section 4.7.4.5 for the detailed derivation.)



• To understand the diffraction phenomenon, we begin by studying a typicalexample: diffraction phenomenon by a sound barrier as shown in Figure4.23, for example.

50

Figure 4.23 Two-dimensional diffraction problem for a plane wave source (it is assumed that the wavelength is much larger than the thickness of the wall, and the wall is acoustically rigid): (a) semi-infinite and (b) finite barrier case

(b) (a)



• The closed form solution that describes diffraction due to a semi-infinite barrier (Figure 4.23(a)) can be obtained (see Section 4.7.6.3):

51

/4

sin cos

0 ,, , ,

8 ,

j kr

j k y j k z

ey z Dkr e

P

where

1 1, .cos / 2 / 2 sin / 2 / 2

D

(4.72)

(4.73)

• We can say that the diffraction is the result of the edge scattering. In otherwords, we hear sound coming from the sound sources. This means thatthe edge condition strongly affects the diffraction in the shadow zone.

When , we cannot see the sound source. In other words, we are inthe shadow zone and the diffraction is dominated by the scattered fieldinduced at the edge of the wall.When , we can see the sound source on the positive z axis. Thediffraction field is composed of two parts: the scattered field from the edgeof the wall and the direct sound field.



52

Figure 4.24 Diffraction of barriers when we have a monopole source, obtained using FDTD (Finite Difference Time Domain): (a) by a straight barrier and (b) by a curved barrier with respect to time (S denotes a monopole source on the ground). (Photographs courtesy of H. Tachiban, University of Tokyo.)



• Figures 4.25 and 4.26 effectively provide the practical parameters that areassociated with the diffraction of a barrier.

• As the wavelength becomes increasingly larger (much larger than thedistance A+B), the barrier’s height would lose its presence in terms ofdiffraction.

• On the other hand, if the wavelength is much smaller than A+B, then thelistener would perceive that the sound comes from the edge of the barrier.

53

A B

A B

Figure 4.25 The sound barrier and associated nomenclature, where NF is the Fresnel number, S represents the source positionand R is the receiver’s location. We also assume that the wall thickness is small relative to the wavelength, and acousticallyhard

FN SR



• The Fresnel number is defined

54

./ 2F

A B dN

(4.74)

• The transmission loss of a barrier ( ) is generally expressed as

10

220log 5 0 .

tanh 2F

br FF

NTL dB N

N

brTL

(4.75)

Figure 4.26 The sound attenuation due to a sound barrier (NF is the Fresnel number)FN



• Refraction is generally due to a change in the media’s characteristicimpedance. For example, as illustrated in Figure 4.27, we can observe therefraction of sound due to the inhomogeneous characteristics of the mediabecause of varying temperature.

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Source

H

T

Source

H

T

yx

o

yx

o

(a) Sound propagation during the day

(b) Sound propagation during the night

Source

H

T

Source

H

T

yx

o

yx

o

yx

o

yx

o

(a) Sound propagation during the day

(b) Sound propagation during the night

Figure 4.27 Refraction due to the media’s impedance change (T is temperature and H is height) (a) during the day and (b) during the night



• To understand this in more depth, consider the case of multi-layer mediaas illustrated in Figure 4.28.

• As the propagation vectors depict, the propagation becomes stiffer in the xdirection as we have greater propagation speed in the y direction.

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1z 2z 3z 4z1c 2c 4c3c

xk1

yk1

1k

x

yo

Propagation trajectory

Change in wave number vector

1z 2z 3z 4z1c 2c 4c3c

xk1

yk1

1k

x

yo

Propagation trajectory

Change in wave number vector

Figure 4.28 Sound propagation in a medium where the characteristic impedance changes smoothly


4.6 Summary

• The breathing sphere and the trembling sphere are basic units that cancreate any radiation field by their linear combinations.

• The radiations are mainly dominated by the relative size (ka) of theradiator compared to the wavelength, and the radiated sound field ismainly governed by the relative distance (kr ) from the radiator comparedto the wavelength of interest.

• The scattered sound field depends on- the angle of the incident wave- the boundary condition of a scatterer- the ratio of the wavelength relative to the size of a scatterer

• Diffraction also depends on- the angle of the incident wave- the angle from the edge

• The Fresnel number is widely accepted as a practical means to design abarrier.

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ka

kr

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Chapter 4