2.5 SPHERICAL WAVES

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In many situations, the size of the source of sound is relatively small, and the

sound is radiated from the source uniformly in all directions. In this case,

the sound waves would not be planar; instead, the sound waves are called

spherical waves. By combining Eq. (2-12) and Eq. (2-16) for spherical waves,

we see that the acoustic pressure varies inversely with the distance from the

sound sosurce, r, because the acoustic power W is constant for the case of

zero energy dissipation:

I ј

p2

_c ј

W

4_r2 (2-23)

The acoustic power is spread over a larger area as the sound wave moves

away from the source, so the acoustic intensity decreases inversely proportional

to r2.

Basics of Acoustics 21

Copyright © 2003 Marcel Dekker, Inc.

From the solution of the acoustic wave equation in Chapter 4, we find

that the magnitude of the specific acoustic impedance for a spherical sound

wave is given by:

Zs ј

_ckr

р1юk2r2Ю1=2 ј

Zokr

р1юk2r2Ю1=2 р2-24)

where k ј р2_=_Ю ј р2_f =cЮ ј wave number. The phase angle (      ) between

the acoustic pressure and the acoustic particle velocity is found from:

tan       ј

1

kr р2-25)

We may note two limiting cases for the acoustic impedance of spherical

waves. For long wavelengths or low frequencies (kr_1), the acoustic

impedance approaches рZokrЮ ј 2__fr, and the phase angle approaches

1

2_rad ј 908. This regime, kr < 0:1 approximately, is called the near-field

regime. The acoustic pressure and acoustic particle velocity are almost 908

out of phase, and the acoustic pressure produced by a spherical source is

very small near the source, for a given acoustic particle velocity.

For short wavelengths (high frequencies) or for distances far from the

source рkr_1Ю, the specific acoustic impedance approaches the characteristic

impedance рZs _ ZoЮ, and the phase angle is approximately zero. This

region, kr > 5 approximately, is called the far-field regime. In this regime,

the spherical wave appears to behave almost as a plane sound wave.

Because the acoustic pressure and acoustic particle velocity are not inphase

for a spherical wave, the potential energy and kinetic energy of the

acoustic wave are not equal, as is the case for a plane wave. The acoustic

energy density for a spherical wave is given by:

D ј

p2

_c2 1 ю

1

2k2r2

_ _

(2-26)

The kinetic energy contribution is given by Eq. (2-18), and the potential

energy contribution is given by Eq. (2-19). For the near-field regime

р1=2k2r2 _1Ю, the kinetic energy contribution predominates; whereas, in

the far-field regime р1=2k2r2 _1Ю, the kinetic energy and potential energy

contributions are equal.

Example 2-4. Aspherical source of sound produces an acoustic pressure of

2Pa at a distance of 1.20m(3.937 ft or 47.2 in) fromthe source in air at 258C

(778F) and 101.3 kPa (14.7 psia). The frequency of the sound wave is 125 Hz.

Determine the rms acoustic particle velocity, the acoustic energy density,

and acoustic intensity for the sound wave at 1.20mfromthe source and at a

distance of 2.50m (8.202ft) from the source.

22 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

The characteristic acoustic impedance is Zo ј 409:8 rayl from

Appendix B. The wave number is:

k ј

2_f

c ј р2_Юр125Ю

р346:1Ю ј 2:269m_1

The parameter kr ј р2:269Юр1:20Ю ј 2:723. This value is neither in the nearfield

nor the far-field regime. The specific acoustic impedance may be evaluated

from Eq. (2-24):

Zs ј

Zokr

р1 ю k2r2Ю1=2 ј р409:8Юр2:723Ю

р1 ю 2:7232Ю1=2 ј р409:8Юр0:9387Ю ј 384:7 rayl

The acoustic particle velocity at a distance of 1.20m from the source is:

u ј

p

Zs ј р2:00Ю

р384:7Ю ј 0:00520m=s ј 5:20mm=s р0:205 in=secЮ

The phase angle between the acoustic pressure and acoustic particle

velocity is given by Eq. (2-25):

             ј tan_1р1=krЮ ј tan_1р1=2:723Ю ј 20:28

The acoustic intensity is found from Eq. (2-23):

I ј

p2

Zo ј р2:00Ю2

р409:8Ю ј 0:00976W=m2 ј 9:76mW=m2

The acoustic power radiated from the source is:

W ј 4_r2I ј р4_Юр1:20Ю2р9:76Юр10_3Ю ј 0:1766W

For a distance of 1.20m from the source, the acoustic energy density is given

by Eq. (2.-26):

D ј р2:00Ю2

р1:184Юр346:1Ю2 1 ю

1

р2Юр2:723Ю2

_ _

ј р28:2Юр10_6Юр1:0674Ю

D ј 30:1 _ 10_6 J=m3 ј 30:1 J=m3

The acoustic energy dissipation is negligible for sound transmitted

through a few meters in air; therefore, the acoustic power at a distance of

2.50m from the source is also 0.1766 W. The acoustic intensity at a distance

of 2.50m from the source is:

I ј

W

4_r2 ј р0:1766Ю

р4_Юр2:50Ю2 ј 0:00225W=m2 ј 2:25mW=m2

Basics of Acoustics 23

Copyright © 2003 Marcel Dekker, Inc.

The acoustic pressure at a distance of 2.50m from the source is:

p ј рZoIЮ1=2 ј Ѕр409:8Юр0:00225Ю_1=2 ј 0:960 Pa

We note that both the intensity and acoustic pressure decrease for a spherical

wave as we move away from the source of sound, because the area

through which the energy is distributed is increased.

The wave number is not affected by the position, so:

kr ј р2:269Юр2:50Ю ј 5:673

The specific acoustic impedance becomes:

Zs ј р409:8Юр5:673Ю

р1 ю 5:6732Ю1=2 ј р409:8Юр0:9848Ю ј 403:6 rayl

The acoustic particle velocity at 2.50m from the source is:

u ј

p

Zs ј р0:960Ю

р403:6Ю ј 0:00238m=s ј 2:38mm=s р0:937 in=secЮ

The acoustic energy density is:

D ј р0:960Ю2

р1:184Юр346:1Ю2 1 ю

1

р2Юр5:673Ю2

_ _

ј р6:50Юр10_6Юр1:0155Ю

D ј 6:60 _ 10_6 J=m3 ј 6:60 mJ=m3

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