2.3 ACOUSTIC PRESSURE AND PARTICLE VELOCITY

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The acoustic pressure (p) is defined as the instantaneous difference between

the local pressure (P) and the ambient pressure (Po) for a sound wave in the

Basics of Acoustics 15

FIGURE 2-2 Wavelength and period for a simple harmonic wave: (A) pressure vs.

time and (B) pressure vs. position.

Copyright © 2003 Marcel Dekker, Inc.

material. The acoustic pressure for a plane simple harmonic sound wave

moving in the positive x-direction may be represented by the following.

pрx; tЮ ј pmax sinр2_ft _ kxЮ р2-7)

The quantity pmax is the amplitude of the acoustic pressure wave.

Acoustic instruments, such as a sound level meter, generally do not

measure the amplitude of the acoustic pressure wave; instead, these instruments

measure the root-mean-square (rms) pressure, which is proportional

to the amplitude. The relation between the pressure wave amplitude and the

rms pressure is demonstrated in the following.

Suppose we define the variable _ ј 2_t=_, so d_ ј 2_ dt=_. The rms

pressure is defined as the square root of the average of the square of the

instantaneous acoustic pressure over one period of vibration _:

р prmsЮ2 ј

1

_

р_

0

p2рx; tЮ dt ј р pmaxЮ2

2_

р2_

0

sin2р_ _ kxЮ d_

Carrying out the integration, we find:

р prmsЮ2 ј р pmaxЮ2

2_

1

2 р_ _ kxЮ _ 1

4 sinр2_ _ 2kxЮ

_ _2_

0

р prmsЮ2 ј 1

2 р pmaxЮ2

The rms pressure is related to the pressure amplitude for a simple harmonic

wave by:

prms ј

pmaffiffiffix

p2 (2-8)

To avoid excessive numbers of subscripts, we will use the symbol p (without

the subscript rms) to denote the rms acoustic pressure in the following

material, except where stated otherwise.

The instantaneous acoustic particle velocity (u) is defined as the local

motion of particles of fluid as a sound wave passes through the material.

The rms acoustic particle velocity is the quantity used in engineering analysis,

because it is the quantity pertinent to energy and intensity measurements.

The rms acoustic pressure and the rms acoustic particle velocity are

related by the specific acoustic impedance рZsЮ:

p ј Zsu (2-9)

The specific acoustic impedance is often expressed in complex notation to

display both the magnitude of the pressure–velocity ratio and the phase

angle between the pressure and velocity waves. The SI units for specific

acoustic impedance are Pa-s/m. This combination of units has been given

16 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

the special name rayl, in honor of Lord Rayleigh, who wrote the famous

book on acoustics: i.e., 1 rayl _ 1Pa-s/m. In conventional units, the specific

acoustic impedance would be expressed in lbf -sec/ft3.

For plane acoustic waves, the specific acoustic impedance is a function

of the fluid properties only. The specific acoustic impedance for plane waves

is called the characteristic impedance (Zo) and is given by:

Zo ј _c=gc (2-10)

(Note that, since the quantity gc is a units conversion factor, it is often

omitted from equations, and it is assumed that consistent units will be

maintained when substituting values in the equations.) Values for the characteristic

impedance for several materials are given in Appendix B.

Example 2-2. A plane sound is transmitted through air (R ј 287 J/kg-K)

at 258C (298.2K or 778F) and 101.3 kPa (14.7 psia). The speed of sound in

the air is 346.1 m/s. The sound wave has an acoustic pressure (rms) of 0.20

Pa. Determine the rms acoustic particle velocity.

The density of the air may be determined from the ideal gas equation

of state:

_ ј

P0

RT ј р101:3Юр103Ю

р287Юр298:2Ю ј 1:184 kg=m3

The characteristic impedance for the air is:

Zo ј _c=gc ј р1:184Юр346:1Ю=р1Ю ј 409:8Pa-s=m ј 409:8rayl ј p=u

The acoustic particle velocity may be evaluated:

u ј

0:20

409:8 ј 0:488_10_3 m=s ј 0:488mm=s р0:0192 in=secЮ

We observe that the acoustic particle velocity (0.000448 m/s) is a rather

small quantity and is generally much smaller than the acoustic velocity

(346.1 m/s).

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