2.4 ACOUSTIC INTENSITY ANDACOUSTIC ENERGY DENSITY

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The acoustic intensity рIЮ is defined as the average energy transmitted

through a unit area per unit time, or the acoustic power (W) transmitted

per unit area. The SI units for acoustic intensity are W/m2. The conventional

units ft-lbf /sec-ft2 are not used in acoustic work at the present time.

For plane sound waves, as shown in Fig. 2-3, the acoustic intensity is

related to the acoustic power and the area (S) by:

Basics of Acoustics 17

Copyright © 2003 Marcel Dekker, Inc.

I ј

W

S р2-11)

For a spherical sound wave (a sound wave that moves out uniformly in all

directions from the source), the area through which the acoustic energy is

transmitted is 4_r2, where r is the distance from the sound source, so the

intensity is given by:

I ј

W

4_r2 (2-12)

For the general case in which the sound is not radiated uniformly from

the source, but the acoustic intensity may vary with direction, the intensity is

given by:

I ј

QW

4_r2 (2-13)

18 Chapter 2

FIGURE 2-3 Intensity for (A) plane waves and (B) spherical waves.

Copyright © 2003 Marcel Dekker, Inc.

The quantity Q is called the directivity factor, which is a dimensionless

quantity that generally depends on the direction and the frequency of the

sound wave.

The acoustic intensity may be related to the rms acoustic pressure. The

average acoustic power per unit area, averaged over one period for the

acoustic wave, is given by:

I ј

1

_

р_

0

pрx; tЮuрx; tЮ dt ј

1

2_

р2_

0

pрx; tЮuрx; tЮ d_ (2-14)

where _ ј 2_ft ј р2_=_Юt. Let us use the following expressions for the acoustic

pressure and acoustic particle velocity for a plane wave:

pрx; tЮ ј

ffiffiffi

2 p prms sinр2_t _ kxЮ

uрx; tЮ ј р

ffiffiffi2p prms=_cЮ sinр2_t _ kxЮ

р2-15)

Making these substitutions into Eq. (2-14), we find:

I ј

1

2_

р2_

0

2р prmsЮ2

_c

sin2р_ _ kxЮ d_

I ј

2р prmsЮ2

2__c

1

2 р_ _ kxЮ _ 1

4 sinр2_ _ 2kxЮ

_ _2_

0

The final expression for the acoustic intensity becomes:

I ј

p2

_c р2-16)

where p ј prms. We will show that this same expression also applies for a

spherical sound wave and for a non-spherical sound wave.

When making sound measurements in a room or other enclosure, one

parameter of interest is the acoustic energy density (D), which is the total

acoustic energy per unit volume. The SI unit for the acoustic energy density

is J/m3. The total acoustic energy is composed of two parts: the kinetic

energy, associated with the motion of the vibrating fluid; and the potential

energy, associated with energy stored through compression of the fluid.

The kinetic energy per unit volume, averaged over one wavelength,

may be expressed in terms of the acoustic particle velocity:

KE ј

1

_

р_

0

1

2

_u2рx; tЮ dx ј

1

2_

р2_

0

1

2

_u2р_; _Ю d_

where _ ј kx. If we use the acoustic particle velocity expression from Eq.

(2-15) for a plane wave, we find:

Basics of Acoustics 19

Copyright © 2003 Marcel Dekker, Inc.

KE ј

p2

2_c2 (2-17)

For a spherical sound wave, the acoustic pressure and acoustic particle

velocity are not in-phase. We will show that the kinetic energy per unit

volume for a spherical wave is dependent on the frequency (or the wave

number, k) for the sound wave, and the distance from the sound source, r, as

follows.

KE ј

p2

2_c2 1 ю

1

k2r2

_ _

(2-18)

The potential energy may also be related to the acoustic pressure. For

a plane sound wave, the potential energy per unit volume, averaged over one

wavelength, is given by:

PE ј

1

_

р_

0

p2рx; tЮ

2_c2 dx ј

1

2_

р2_

0

p2р_; _Ю

2_c2 d_

Using the expression for the acoustic pressure from Eq. (2-15), we obtain the

following equation for the potential energy per unit volume:

PE ј

p2

2_c2 (2-19)

By comparison of Eqs (2-17) and (2-19), we see that, for a plane sound wave,

the kinetic and potential contributions to the total energy are equal. The

total acoustic energy is half kinetic and half potential, for a plane sound

wave: this is not the case for a spherical wave.

For a plane sound wave, the acoustic energy density is found by adding

the kinetic energy, Eq. (2-17), and the potential energy, Eq. (2-19):

D ј

p2

_c2 (2-20)

If we compare Eq. (2-20) with Eq. (2-16), we see that (for a plane sound

wave) the acoustic intensity and acoustic energy density are related:

D ј

I

c р2-21)

For a spherical sound wave, the acoustic energy density is given by:

D ј

p2

_c2 1 ю

1

2k2r2

_ _

(2-22)

20 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

Example 2-3. A plane sound wave is transmitted through air (speed of

sound, 346.1 m/s; characteristic impedance, 409.8 rayl) at 258C (298.2K or

778F) and 101.3 kPa (14.7 psia). The sound wave has an acoustic pressure

(rms) of 0.20 Pa. Determine the acoustic intensity and acoustic energy density

for the sound wave.

The acoustic intensity is given by Eq. (2-16):

I ј

p2

_c ј р0:20Ю2

р409:8Ю ј 97:6_10_6W=m2 ј 97:6mW=m2

The SI prefixes are listed in Appendix A.

The acoustic energy density is given by Eq. (2-20):

D ј

p2

_c2 ј

p2

Zoc ј р0:20Ю2

р409:8Юр346:1Ю ј 0:282 _ 10_6 J=m3 ј 0:282 mJ=m3

This is actually an extremely small quantity of energy. The specific heat of

air at 258C is cp ј 1005:7 J/kg-8C. The thermal capacity per unit volume is:

_cp ј р1:184Юр1005:7Ю ј 1190:7 J=m3-8C

If all of the acoustic energy in this problem were dissipated into the air, the

temperature of the air would rise by:

_T ј

D

_cp ј р0:282Юр10_6Ю

р1190:7Ю ј 0:24 _ 10_98C р0:43 _ 10_98FЮ

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