4.4 SOLUTION FOR SPHERICAL WAVES

Back

If we develop the wave equation for spherical coordinates, in which the

quanitities are functions of the radial coordinate r and time t only, the

following expression is obtained:

1

r2

@

@r

r2 @p

@r

_ _

ј

1

c2

@2p

@t2 (4-50)

The wave equation for one-dimensional spherical waves may be written in

the following alternative form:

@2рrpЮ

@r2 ј

1

c2

@2рrpЮ

@t2 (4-51)

Let us try a solution for Eq. (4-51) in the form:

pрr; tЮ ј рrЮ ej!t (4-52)

where рrЮ is the amplitude function. If we substitute the expression from

Eq. (4-52) into the wave equation, Eq. (4-51), we obtain the following

ordinary differential equation:

d2рr Ю

dr2 ј

j2!2рr Ю

c2 ј _k2рr Ю (4-53)

The general solution for the amplitude function is:

r рrЮ ј Ae_jkr ю B ejkr р4-54)

The general solution for the instantaneous acoustic pressure may be found

by substituting the amplitude function expression from Eq. (4-54) into Eq.

(4-52).

pрr; tЮ ј

1

r рAe_jkr ю B ejkrЮ ej!t (4-55)

We observe that the amplitude of the acoustic pressure is not constant for a

spherical wave; instead, the amplitude varies inversely with distance from

the source, r.

If we consider only waves moving radially outward from the source or

the case for no waves reflected back toward the origin, we must have B ј 0.

pрr; tЮ ј рA=rЮ ejр!t_krЮ (4-56)

88 Chapter 4

Copyright © 2003 Marcel Dekker, Inc.

The real component from Eq. (4-56) is:

pрr; tЮ ј рA=rЮ cosр!t _ krЮ (4-57)

The rms acoustic pressure for a spherical wave is given by:

prms ј A=

ffiffiffi2p r or A ј

ffiffiffi2p prms r (4-58)

Next, let us determine the expression for the instantaneous particle

velocity for a spherical wave. Integrating the Newton’s law expression, as

given by Eq. (4-17), we may evaluate the particle velocity:

uрr; tЮ ј _

1

_o

р

@pрr; tЮ

@r

dt ј

1

j_o!

A

r

1

r ю kj

_ _

ejр!t_krЮ (4-59)

If we introduce ! ј kc, and the acoustic pressure expression, Eq. (4-56), we

may write Eq. (4-59) as follows:

uрr; tЮ ј _

jpрr; tЮ

_ockr р1 ю jkrЮ (4-60)

The specific acoustic impedance, which is a complex quantity for a

spherical wave, may be found from Eq. (4-60):

Zs ј

p

u ј

j_ockr

1 ю jkr ј

_ockrрkr ю jЮ

1 ю рkrЮ2 ј jZsj ej     (4-61)

According to Eq. (4-23), the magnitude of the specific acoustic impedance

is the square root of the sum of the squares of the real and imaginary

parts of the complex expression in Eq. (4-61):

jZsj ј j pj

juj ј

_ockrрk2r2 ю 1Ю1=2

1 ю k2r2 ј

Zokr

р1 ю k2r2Ю1=2 (4-62)

The tangent of the phase angle between the acoustic pressure and particle

velocity is the ratio of the imaginary to the real parts of the complex quantity,

as given by Eq. (4-24):

tan        ј 1=kr (4-63)

We note that the magnitude of the specific acoustic impedance

approaches рZokrЮ and the phase angle approaches 908 for kr very small

(less than about 0.15). This condition occurs for positions very near the

spherical source or for very low frequencies. On the other hand, we note

that the specific acoustic impedance approaches the characteristic impedance

and the phase angle approaches 08 for kr large (greater than about 7).

This condition occurs for positions far away from the spherical source or for

high frequencies.

Transmission of Sound 89

Copyright © 2003 Marcel Dekker, Inc.

The expressions for the acoustic intensity and energy density may be

found for a spherical wave. Using the complex notation, the intensity may

be evaluated from the following:

I ј Reрprmsu_rmsЮ (4-64)

The quantity u_rms is the complex conjugate of the rms particle velocity. The

complex conjugate of a complex quantity z ј x ю jy ј r ej_ is:

z_ ј x _ jy ј r e_j_ (4-65)

Let us write the rms quantities (with time integrated out) as follows:

prms ј jpj e_jkr (4-66)

urms ј j pj

jZsj

e_jрkrю           Ю (4-67)

u_rms ј j pj

jZsj

eюjрkrю          Ю (4-68)

Making these substitutions into Eq. (4-64), we obtain the expressions for the

intensity of a spherical wave:

I ј j pj2

jZsj

Reрej   Ю (4-69)

I ј j pj2 cos     

jZsj

(4-70)

The tangent of the phase angle  is given by Eq. (4-63). The cosine of the

phase angle is given by the following:

cos        ј

kr

р1 ю k2r2Ю1=2 (4-71)

Using the expression for the magnitude of the specific acoustic impedance

from Eq. (4-62), we obtain the final expression for the intensity of a

spherical wave:

I ј j pj2

Zo ј

p2

_oc

(4-72)

The kinetic energy per unit volume for a sound wave is given by:

KE ј Reр1

2 _ourmsu_rmsЮ (4-73)

90 Chapter 4

Copyright © 2003 Marcel Dekker, Inc.

We may evaluate this expression, using Eqs (4-67) and (4-68):

KE ј ReЅ1

2 _oрj pj2=jZsj2Ю e_jрkrю   Ю eюjрkrю     Ю_ (4-74)

KE ј j pj2р1юk2r2Ю

2_oc2k2r2 ј j pj2

2_oc2 1 ю

1

k2r2

_ _

(4-75)

The potential energy per unit volume for a sound wave is given by:

PE ј Reр1

2 _oprmsp_rms=KsЮ (4-76)

The quantity Ks is the adiabatic compressibility given by the following

expression for an ideal gas:

Ks ј _Po ј __oRT ј _oc2 (4-77)

The quantity _ ј cp=cv is the specific heat ratio. The potential energy may

be evaluated as follows:

PE ј ReЅрj pj2=2_oc2Ю e_jkr eюjkr_ ј j pj2

2_oc2 (4-78)

The acoustic energy density or the total energy per unit volume is the

sum of the kinetic and potential energies:

D ј KEюPE ј

p2

_oc2 1 ю

1

2k2r2

_ _

(4-79)

Навигация