wave equation in fluids

7/29/2019 Wave Equation in Fluids

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Fundamentals of Sound and Vibration

Indian Institute of Technology Roorkee

Wave Equation in Fluids

1. The wave equation in a source-free medium

Acoustic disturbances in fluids, i.e., gases and liquids, which cannotsupport shear stresses, propagate as longitudinal waves.

Plane wave propagation means that the acoustic variables, such as soundpressurep, have a constant instantaneous magnitude throughout any given

plane perpendicular to the direction of wave propagation

Direction of propagation

Pressurep/ p0 WavelengthRarefied Compressed

Direction of propagationp0

Figure 1 Symbolic depiction of plane longitudinal wave propagation for a sinusoidal

disturbance. The fluid particles are sketched in the compressed and in the rarefied regions.

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Simplifying assumptions are

(i) The medium is homogenous and isotropic, i.e., it has the same

properties at all points and in all directions.

(ii) The medium is linearly elastic, i.e., Hookes law applies.

(iii) Viscous losses are negligible.

(vi) Heat transfer in the medium can be ignored, i.e., changes of state canbe assumed to be adiabatic.

(v) Gravitational effects can be ignored, i.e., pressure and density are

assumed to be constant in the undisturbed medium.

(vi) The acoustic disturbances are small, which permits linearization of the

relations used.

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The following quantities are considered:

Pressure: ),(),( 0 trpptrptvv

+= (1)

where is the total pressure as a function of position ( ) and time (t),

p0 the pressure in the undisturbed medium, andthe pressure disturbance in the medium, the sound pressure.

),( trptv

rr

),( trp v

Particle Velocity zzyyxx eueueutrurrrrr

++=),( (2)

where is the particle velocity vector

ux, uy, uz are the corresponding velocity components,

is the unit vector

),( tru rr

zyx eeerrr

,,

Density:),(),(

0trtr

t

rr +=

(3)

where is the total density,

0 is the density in the undisturbed medium,

is the density disturbance in the medium.

),( trtr

),( tr

r

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Absolute Temperature:

A fundamental assumption is that we considersmall disturbances, i.e., small

variations in pressure. As a rule of thumb, for air at normal temperature and

pressure, the sound pressure level should not exceed 140 dB

Equations of continuity

one-dimensional case

x

yz

+x

tux

)(tuxtux x

y

z

x

Figure 2 Mass flow in thex-direction through a volume element fixed in space.

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According to figure 2, the mass in the volume element is , the mass

flow into the element is , and the mass flow out is . The

net flow in the element is therefore , and must

equal the mass change , so that a mass balance is received

zyxt

xxt zyu )( xxxt zyu + )(

xxxtxxt zyuzyu + )()( )( zyx

t

t

xxxtxxtt zyuzyuzyx +=

)()()( (4)

( )

+=

xzyux

zyuzyuzyxt x

xtxxtxxtt )()()( (5)

The second term on the right-hand side can be expanded into a series, for small

variations about the undisturbed equilibrium state, and if higher-order terms canbe neglected, then

This can be simplified to

( ) 0=

+

xt

t uxt

(6)

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Generalized to the three dimensional case,

( ) ( ) ( ) 0=

+

+

+

ztytxt

t uz

uy

uxt

(7)

Defining the del operator, as

+

+

=z

ey

ex

e zyxrrr

(8)

permits a simplified expression of the continuity equation, as

0=)( ut

tt r

+

(9)

Putting the total density (3) into the equation and taking advantage of the factthat the undisturbed density 0 is independent of time and position, and ignoring

second order terms, that consist of the product of two acoustic disturbances,

gives the linearized wave equation

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00 =+ ut

r

(10)

In one dimension

00 =

+

x

u

tx

(11)

Equation of motion

Consider a specific fluid particle, with a fixed mass Dm and a fixed volume

V = xyz, as in figure 6-3, that moves with the medium.

( )( )

x

y

z

+0p+p [ ] x0p+p( ) 0

p+p

y

z

x

y yz zx-

Figure 3 Force in thex-direction on a

particular fluid particle moving with the

medium.

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The force in thex-direction is

zyxppx

zyxppx

ppzyppFx +

=

+

+++= )())(()( 0000

(12)wherep0 is constant, so that

zyxx

pFx

= (13)

In three dimensions, the force vector becomes

zyxz

pe

y

pe

x

peF zyx

+

+

=

rrrr(14)

Putting in the del operator, as

+

+

=z

ey

ex

e zyxrrr

(15)

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and using the relation xyz= V, then equation (14) reduces to

VpF = r

(16)

For a given fluid particle, the velocity is a function of position and time.At time t and position (x, y, z), the velocity is (x, y, z, t). At a later instant t+t,

the position is (x+ x, y+ y, z+ z) and the velocity is .

The differential change in position (x, y, z) can be written x=uxt, y= uytand z= uzt, so that the acceleration can be written

),( tru

rr

( tzzyyxxu ++++ ,,,r

t

tzyxutttuztuytuxua

zyx

t

++++=

),,,(),,,(lim

0

rrr

(17)

The first term is reformulated with the help of a Taylor series, so that equation

(17) can be rewritten

t

tzyxutt

utu

z

utu

y

utu

x

utzyxu

azyx

t

+

+

+

+

+=

),,,(),,,(

lim0

rK

rrrrr

r(18)

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The acceleration of the fluid particle becomes

t

u

z

uu

y

uu

x

uua zyx

+

+

+

=rrrr

r(19)

With simplifying notation this is

uut

ua

rrr

r)( +

= (20)

In one dimension,

x

uu

t

ua xx

xx

+

= (21)

For acoustic fields with small disturbances, the second term in equation(20)can be neglected, so that

t

ua

=r

r

(22)

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In one dimension,

t

ua xx

=

(23)

Making use of equations (16) and (22), as well as m = (0 + ) V, theequation of motion can be formulated as

t

uVVp

+=r

)( 0 (24)

If second order terms can be ignored, then the linear, inviscid equation of

motion is

00 =+

pt

ur

(25)

In one dimension,

00 =+

xp

tux (26)

The equation of motion, gives a relation between pressure and particle velocity

in a sound field.

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The thermodynamic equation of state

For an ideal gas, the ideal gas lawapplies,

MRTpp )()( 00 +=+ (27)

where (p0 +p) [Pa] is the total pressure,

(0 + ) [kg/m3] is the total density,

R= 8.315 [J/(mol K)] is the ideal gas constant,M[kg] is the mass of a mole of gas, and

T[K] is the absolute temperature.

Two idealizations can be considered: an isothermal process, implying such

good heat conduction in the medium that the temperature is constant

throughout; or, an adiabatic process, in which no heat conduction occurs.

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where = cp/cvis the ratio of the specific heatof the gas at constant

pressure, to that at constant volume. For the sake of simplicity,pt= (p0 +p)denotes total pressure, and t= (0 + ) denotes total density, in the series

expansion. The total pressure can be expanded as

...2

1

00

2

22

00 +

+

+=+===

tt

t

t

t

tt

pp

pppp

where the partial derivatives are constants that remain to be determined for

adiabatic disturbances about 0, i.e., the density in the undisturbed medium.

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(29)

( ) ( )

+=+

0

0

0

0

p

pp(28)

For small disturbances below a certain frequency limit, it can be shown that

the process can, to a good approximation, be regarded as adiabatic. For an

adiabatic change of state,


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where

0

0

=

=

tt

tp(32)

is called the adiabatic bulk modulus.

The homogenous linearized wave equation

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For small disturbances, second and higher order terms can be neglected,

and a linear relation is obtained as

0

=

=

t

t

tpp

0=por

(30)

(31)


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,

One dimension Three dimensions

Time derivative of continuity eq. (11), Time derivative of continuity equation (10),

Using

Spatial derivative of eq. of motion (26) Divergence of eq. of motion (25),

0

2

02

2

=

+

tx

u

t

x

(33)

02

22

0 =

+

x

p

tx

ux (35)

( ) 00

=+

p

t

ur

002

2

=

+

t

u

t

r

(34)

( ) 00 =+

pt

ur

(36)

In abbreviated notation,

020 =+

pt

ur

(37)

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One dimension Three dimensions

Subtraction of (33) from (35) gives

The wave equation in one dimension

(38)

Subtraction of (34) from (37) gives

02

22

=

tp

(39)

Equation of state (31) eliminates

(40)02

2

02

2

=

t

p

x

p

02

2

2

2

=

tx

p

Equation of state (31) eliminates

02

2

02 =

t

pp

(41)

The wave equation in three dimensions

01

2

2

22

2

=

p

cx

p(42) 0

12

2

22 =

t

p

cp (43)

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The constant cis defined as

0/=c (44)

and is the propagation speed of a disturbance in the medium, the speed ofsound. According to equation (32)

0

0/

=

==

tt

tpc (45)

For an ideal gas, equation (28) implies

)ln(lnlnln 0000

=

= tttt ppp

p(46)

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so that

t

t

t

t pp

=

(47)

Put into the expression for the speed of sound (45), this yields

0

=

=t

ttpc

i.e.,

00 pc =

(48)

(49)

The temperature dependence of the speed of sound is obtained by putting the

ideal gas law (27) into (48)

MRTc = (50)

If the speed of sound at 0 C (273 K) is denoted c0, then for other temperatures

2730 Tcc = (51)

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The speed of sound increases with temperature, a relationship with great

significance for sound propagation outside, where the temperature often varies

with distance to the ground.

isothermal bulk modulus T . the relation between these is b = T , after

which the sound speed in liquids is found from

0 Tc=(52)

Solutions to the wave equation

General solution for free plane one-dimensional wave propagation

Forplane wave propagation in thex-direction, the wave equation (42) applies.

Assume a general solution form of

( ) ( )cxtgcxtftxp ++=),( (53)

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where fand gare arbitrary functions and (tx / c) and (t+x / c) are theirrespective arguments. That assumed solution is known as dAlembert s

solution. Derivation of equation (53) with respect toxgives

( ) ( )cxtgc

cxtfcx

p++=

11

(54)

( ) ( )cxtgc

cxtfcx

p++=

222

2 11 (55)

and similarly with respect to tgives

( ) ( )cxtgcxtft

p++=

(56)

( ) ( )cxtgcxtft

p ++=

2

2(57)

Putting (55) and (57) into (42) shows that the solution fulfills the wave equation.

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To interpret (53), consider a special point (x1,t1) in the wave, in figure 4. It

represents a certain sound pressurep1(x1,t1). To have the same sound pressur

at another point (x1 + x), at a later instant in time (t1 + t), then the arguments

have to be the same, i.e.,

( ) ( )cxxttcxt )( 1111 ++= (58)

which gives the condition

tcx = (59)

Thus, the solution f(tx / c) implies wave propagation in the positive

direction along thex-axis, with speed c.

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Figure 4 Instantaneous picture of the wave propagation in the positivex-direction at time

instants t1 and (t1 + t). The propagation speed of the disturbance is c.

Similarly, g(t+x / c) implies propagation in the negativex- direction, so that

tcx = (60)The propagation speed cof a disturbance is called the speed of sound(or

sound speed), wave speedorphase velocity.

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Harmonic solution for free, plane, one-dimensional wave propagation

From Fourier analysis, it is known that everyperiodic process can be built up

of the summation of harmonic, sinusoidal processes with different

frequencies, the set of which is called a Fourier series.

The harmonic solution we seek for the angular frequency = 2f must, for

a certain x-value, say x1 + x, give the same sound pressure at time tas it

does one period later at time t + T, where T is called the period. For a

harmonic function, that implies that the argument, i.e., the angle, increases

by 2. The solution must also, for a certain time value, say t1

= t, give the

same sound pressure at points separated along thex-axis a distance equal

to the wavelength . Even in that case, the argument must increase by 2,

see figures 5a and 5b. Thus, we attempt a solution of the form

)(cos),( cxtptxp = + (61)

where is the amplitude, i.e., the highest value of the sound pressure. The

argument (t - x/c) = t kxis called thephase and+p

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k= /c (62)

is called the wave number. The first condition above gives

))((cos)(cos 11 kxTtpkxtp += ++ (63)

i.e.,

T = 2p , = 2 /T (64)

The second condition gives

( )( ) += ++ xktpkxtp 11 cos)(cos (65)

i.e.,

2,2 == kk (66)From (62), (64) and (66), one obtains the relation

fc= (67)

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f S


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which applies to all types of wave propagation, and in which the frequency is

f= 1 / T (68)

p(x1,t) $p+

Period T

t

Figure 5a The variation of sound pressure with time at a fixed positionx1.

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F d t l f S d d Vib ti


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p(x,t1)

Wavelength

$p+x

Figure 5b The variation of sound pressure with position at a fixed time t1.

complex notation

)()( ),( kxtikxti epeptx +

+ +=p (69)

in which bold print means that the variable concerned is complex. The

first term on the right-hand side refers to propagation in the positive x-

direction, and the second term to propagation in the negativex-direction.

For the sake of physical interpretation, the real part of (69) is needed, i.e.,

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)cos()cos()Re(),()()(

kxtpkxtpepeptxpkxtikxti ++=+= +

++

(70)The equation of motion (26)

00 =+ xp

tux (71)

relates the particle velocity to the sound pressure; rearranged, it gives

dtx

pux

1

0

=

(72)

Next, putting in (69) gives the particle velocity

( )

+

= ++ )()(

0

1, kxtikxtix epi

ikepi

iktx u (73)

Since k/ = 1/c, the particle can be written

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)(

0

)(

0

),( kxtikxtix e

c

pe

c

ptx ++ =

u (74)

The two terms refer to wave propagation in the positive and negative x-

directions, respectively. The ratio of pressure to particle velocity is called thespecific impedance Z,

xupZ = (75)

and, for the free plane wave case, is therefore

cZ 00 =+

(76)

and

cZ 00 = (77)

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for propagation in the positive and negative directions, respectively. The

quantity 0cis called the wave impedance.



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Sound intensity for free, plane, one-dimensional wave propagation

The sound intensity is defined as the sound energy per unit time that

passes through a unit area perpendicular to the propagation direction. the

instantaneous power can be written

)()()W( tutFtrr

= (78)

A general expression for sound intensity is therefore that

),(),(),( trutrptrIrrrrr = (79)

),(),(),(x txutxptxI x=

For propagation in thex-direction,

(80)

The time-averaged sound intensity is

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=T

xx dttxutxpT

xI

0

),(),(1

)( (81)

Making use of the expression for pressure, i.e., the real part of (69), and the

particle velocity, i.e., the real part of (74), in (81), gives the intensity in theform

cppI x 022

2)( + = (82)

The first term refers to a wave moving in the positive x-direction, and thesecond term to a wave moving in the negative x-direction. For harmonic

waves, the relation between the rms amplitude and the peak value

is , so that (82) can be written2pp=

c

p

c

pI x

0

2

0

2 ~

~

+ = (83)

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F ndamentals of So nd and Vibration


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Energy and energy density in free, plane, one-dimensional

wave propagation

The energy Eassociated with a sound wave consists of two parts, the kinetic

and the potential energy. The kinetic energy Ek

can be related to the velocity of

a fluid particle. The potenti-al energy Ep is due to the compression, i.e., the

elasticity. Considera part icular mass of gas that has density 0 and volume V0in the undisturbed medium, so that its mass can be written

VV t =00 (85)For wave propagation in thex-direction, its kinetic energy is, from elementary

mechanics,

2

),(),(

200 txuVtxE xk

= (86)

The potential energy of the fluid mass comes from the work that expended to

compress it. Consider, now, the fluid volume shown in figure 7. The forceapplied to the piston is equal to the product of the pistons area S and the

pressurep against its inside surface, i.e., Fx=p(x,t)S. When the piston moves a

distance dx, the differential amount of work dEp = Fx dx = p(x,t) S dx is

performed.

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Figure 7 Work performed to compress a fluid volume.

Since Sdx= dV, dEp can, with complete generality, be written

dVtxpdEp ),(= (87)

where the minus sign implies that a positive sound pressure p(x,t), which

gives a negative volume change (i.e., a volume reduction or compression),

corresponds to a positive potential energy. We choose to express dVas a

function of the sound pressure. Differentiating (85),

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tt

t

tt

dV

dV

dVVV

== 2

0000

1(88)

Since equation (46) states that

= 00

tt

p

p

tt

tt

t

t

t

tdppdp

dpd

==1

then

(89)

equations (88) and (89) give

tt

dpp

VdV

= (90)

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For small pressure and volume disturbances, the undisturbed pressure andthe undisturbed volume can be used

dpp

VdV

0

0

= (91)

The work (87) can now be written as

dppp

VdEp

0

0

= (92)

The potential energy is, finally, obtained by integrating from 0 to the soundpressurep; thus,

2

0

0

2p

p

VEp

=(93)

According to (49), ; thus, for plane wave propagation in thex-

direction, a general expression for the energy is002 pc =

( ) ),(2

, 22

0

0 txpc

VtxEp

= (94)

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The concept ofenergy density [J/m3] refers to the energy per unit volume. Ofcourse this quantity, just as the total energy does, consists of two parts: kinetic

and potential energy densities, i.e.,

),(),(),( trtrtr pkrrr

+= (95)

Using expressions for the kinetic energy (86) and the potential energy (94), the

energy density for propagation in thex-direction is then

2

0

220

2

),(

2

),(),(

c

txptxutx x

+= (96)

According to (76) and (77), applies to plane waves.

Using this in (96) yields the instantaneous value of the energy density

)(),(),( 0ctxptxux =

20

2

20

2

20

2 ),(

2

),(

2

),(),(

c

txp

c

txp

c

txptx

=+=

(97)The time average is obtained by integrating with respect to time

2

0

2

0

)(~),(

1)(

c

xpdttx

Tx

T

== (98)

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For a plane wave however, since it doesnt decay with distance, the rmspressure is independent of positionx; that naturally applies to the time-averaged

energy density as well, i.e.,

20

2~

c

p

= (99)

A comparison to (83) gives the relation between the sound intensity and the

energy density of a plane wave, as

cI x = (100)

General solution for free spherical wave propagation

The spherical wave is a basic cornerstone in the study of acoustic fields.

More complex sound fields can be built up of combinations of spherical waves

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For spherical wave propagation, it is of course more convenient to make use ofthe latter; see figure 9.

x

y

z

r sinsin

r sincos

r cos

r

(r,, )

Figure 9 Relation between spherical and Cartesian coordinate systems.

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In spherical coordinates, the wave equation (43) takes the form

2

2

22

2

2222

2

1

sin

1sin

sin

11

t

p

cp

rrrr

rr

=

+

+

(101)

For spherical symmetry, the sound pressure has no angular dependence, and

equation (101) reduces to

2

2

22

2

11

t

p

c

p

r

r

rr

=

(102)

or

2

2

22

2 12

t

p

cr

p

rr

p

=

+

(103)

Unlike plane wave propagation, the sound pressure amplitude decays with

increasing radius, since the sound power in the wave is divided over an

ever-expanding spherical surface of area 4 r2. In a plane wave, the sound

intensity is, according to (82),

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i.e., proportional to the squared sound pressure amplitude. Assuming thatapplies to spherical waves as well, their amplitude would therefore have to

decay at a rate of 1/raccording to the energy principle; by analogy to the plane

wave case, an assumed solution might therefore take the form

( ) ( )crtgrcrtfrtrp ++=11

),( (105)

Here, the first term represents an outgoing diverging wave and the second term

an incident converging wave. The incoming wave seldom exists in connection

with acoustic radiation from machines. The solution is verified by inserting it intothe wave equation (103). On a term-by-term basis, we have

( ) ( ) ( )

( ) ( ) ( )crtgr

crtg

cr

crtg

rc

crtfr

crtfcr

crtfrcr

p

+++++

+++=

322

3222

2

221

221

(106)

( ) ( ) ( ) ( )crtgr

crtgcr

crtfr

crtfcrr

p

r+++=

3232

22222

(107)

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( ) ( )crtgrc

crtfrct

p

c++=

222

2

2

111(108)

Putting all of these into (103) shows that the assumption does indeed fulfill

the wave equation. The equation of motion (25) relates the particle velocityto the sound pressure. In spherical coordinates, it takes the form

0sin

110 =

+

+

+

per

er

et

ur

rrrr

(109)

With spherical symmetry and , the equations of motion can be

reduced to

00 =

+

r

p

t

ur (110)

and the particle velocity expressed as

(111)dtr

pur

1

0

=

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rr etruurr

),(=



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The first term on the right-hand side refers to an outgoing, diverging, wave,

and the second to an incident, converging, wave. The sound pressure

amplitude in the outgoing wave isA+/ r, whereA+ is a constant. The amplitude

is therefore a function ofr. From (111), the particle velocity becomes

( ) ( ) dter

ik

r

Adt

r

r,ttr krtir

1

1, )(

200

+

+

=

=

pu

Harmonic solution for free, spherical wave propagation

A complex harmonic solution is obtained, as

)()(

),(

krtikrti

er

A

er

A

tr

++

+=

p

(112)

(113)

Integrating, and applying the relation (62), k= /c, gives the particle velocity

)(

0

11),( krtir e

rkicr

Atr +

+=

u (114)

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By analogy to the definition (75) of the specific impedance of a free plane wave,we define here the complex quantity Z as the ratio of the complex sound

pressure to the complex radial particle velocity at a point in a sound field

ru

pZ = (115)

For an outgoing spherical wave, the specific impedance, using equations (112)

and (114), becomes

ikr

ikrc

ikr

c

r+

=

+

==11

1

100

u

pZ (116)

Multiplying the numerator and denominator by the complex conjugate of the

latter, gives

++

+=

+

+=2222

22

022

22

0111 rk

kri

rk

rkc

rk

ikrrkc Z (117)

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and the specific impedance Z can be divided into a resistive part Rand areactive partX, i.e.,

iXR+=Z (118)Some observations that follow from the preceding are:

(i) Nearfield

In the acoustic near field (kr = 2r/ 1), i.e., when the radius is small in

comparison to the wavelength, both the resistance and the reactance approach

zero, but the resistance does so more quickly. The reactance therefore

dominates, and the impedance approaches

krcir )( 0Z (119)

That means that for a given sound pressure, the particle velocity becomes largeand its phase shift approaches 90 with respect to the sound pressure. Forkr=1, both the resistance and the reactance are equally large, 0c/ 2,

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(ii) Far field

In the acoustic far-field(kr= 2r/ 1), i.e., where the radius is large with

respect to the wavelength, the resistance approaches 0cand the reactance

approaches zero as rgoes to infinity.

The resistance dominates and the impedance approaches the same

expression as for plane waves

cr 0)( Z (120)

That means that the phase difference between the sound pressure and the

particle velocity approaches zero, as is the case for plane waves.

The curvature of the spherical waves in the farfield, with increasing distance

to the source, becomes all the less significant and the situationasymptotically approaches that of plane waves.

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kr 201510500

0,8

0,6

0,4

0,2

1,0

R/0c

X/0c

Figure 10 Normalized resistance R/0c, and normalized reactanceX/0c, for outgoing

spherical wave propagation.

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Sound intensity for spherical wave propagation

The time-averaged sound intensity of outgoing spherical waves can be

determined by the same methods as for plane waves. Write the intensity as

2)*Re(

2)*)(Re()( rrr rI upup == (121)

Putting equations (112) and (114) into equation (121) gives

20

2

20

2

211Re

21)(

crA

ikrcrArI r

++ =

+= (122)

The time-averaged energy flow through a closed, spherical control surface

of radius ris

cArrrrIW0

2

2 24)(

+== (123)

For a loss-free medium, the sound power is therefore independent of the

radius, which is in agreement with the energy principle.

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Thank you

wave equation in fluids

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