4.1 THE WAVE EQUATION

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To understand fully the principles governing various noise control procedures,

one should be familiar with the governing equation for acoustic

wave transmission, or the wave equation, which may be written in cartesian

coordinates as follows:

@2p

@x2 ю

@2p

@y2 ю

@2p

@z2 ј

1

c2

@2p

@t2 (4-1)

where p is the instantaneous acoustic pressure, c is the speed of sound, and t

is the time coordinate (Norton, 1989).

The wave equation has two important restrictions: (a) energy dissipation

effects are neglected, and (b) the pressure wave amplitude must be

Copyright © 2003 Marcel Dekker, Inc.

relatively small in comparison with atmospheric pressure. One may employ

elaborate solution techniques to the wave equation, but if these two conditions

are not met in the physical situation analyzed, the results of the analysis

will be worthless.

As we will show in this chapter, energy dissipation effects are most

pronounced for high-frequency sound and for sound transmission through

large distances. For example, if the frequency of the sound being transmitted

through atmospheric air is 500Hz or less, the attenuation of acoustic energy

is less than 0.1 dB for transmission distances smaller than about 25m(82 ft).

We see that there are many practical situations in which the effect of

attenuation is negligible. On the other hand, if the frequency of the sound

being transmitted through atmospheric air is 8kHz and the distance transmitted

is 125m, the attenuation of acoustic energy can be as large as 40dB.

For most noise control work, the amplitude of the sound wave is

small. For example, for a sound pressure level of 150 dB, the rms acoustic

pressure is 632 Pa (0.092 psi) or the peak pressure amplitude is 894 Pa (0.13

psi). These values are less than 1% of atmospheric pressure (101.3kPa or

14.7 psia). Unless one is dealing with low-level sonic booms or sound from

nearby blasts, for example, the sound pressure levels encountered in industrial

or environmental conditions are usually less than 150 dB, and the

restriction of ‘‘small’’ pressure amplitude is met.

Let us develop the one-dimensional wave equation for plane sound

waves transmitted in the x-direction. An elemental layer of fluid initially

having thickness dx and surface area S is shown in Fig. 4-1. After a small

increment of time dt, one face moves from position x to a new position

рx ю _Ю, where _ is the instantaneous particle displacement. The other face

moves from рx ю dxЮ to a new position:

x ю dx ю _ ю

@_

@x

dx

_ _

The fluid moves from one position to another as a result of forces

applied to the element. Now, we introduce the first restriction:

Restriction 1: frictional forces are negligible, so that the only forces

acting on the element of fluid are the pressure forces. The net force acting on

the element becomes:

Fnet ј pS _ p ю

@p

@x

dx

_ _

S ј _

@p

@x

dxS (4-2)

The particle velocity for the layer of material is the change in displacement

per unit time:

Transmission of Sound 79

Copyright © 2003 Marcel Dekker, Inc.

u ј

@_

@t

The acceleration of the layer of material is the change in velocity per unit

time:

a ј

@u

@t ј

@2_

@t2

The mass of the small layer is its mass per unit volume (or density, _) times

the element volume рS dxЮ, or dm ј _S dx.

If we make these substitutions into Newton’s second law of motion,

Fnet ј ma, we obtain the following:

@p

@x ј __

@2_

@t2 (4-3)

The final form of the wave equation may be written in terms of several

different variables. In the following, let us develop the wave equation in

terms of the acoustic pressure.

80 Chapter 4

FIGURE 4-1 Initial and displaced positions of a fluid element as a sound wave

passes through the element.

Copyright © 2003 Marcel Dekker, Inc.

The speed of sound in any fluid may be determined from the following

derivative, taken at constant entropy (van Wylen et al., 1994):

c2 ј

@p

@_ ј

@p=@t

@_=@t

or

@p

@t ј c2 @_

@t

(4-4)

Let us define the property condensation c as the fractional change in

the fluid density:

c ј

_ _ _o

_o ј

_

_o _ 1 (4-5)

The quantity _o is the density of the undisturbed fluid (usually at atmospheric

pressure). For a fixed mass of the fluid element, we may write the

condensation in the following form, in terms of the particle displacement:

c ј

Vo _ V

V ј

S dx _ S dx ю

@_

@x

dx

_ _

dx ю

@_

@x

dx

_ _

S

ј _

@_

@x

1 ю

@_

@x

(4-6)

The quantity Vo is the initial volume of the element.

Now, we will introduce the second important restriction:

Restriction 2: the displacement of the fluid particles is very small, such

that:

@_

@x _ 1

With this condition, the condensation may be written in the following form:

c ј _

@_

@x

(4-7)

The fluid density may be written in terms of the condensation, using Eq.

(4-5):

_ ј _oр1 ю cЮ

And,

@_

@t ј _o

@c

@t

(4-8)

If we use this result in Eq. (4-4), we obtain the following:

@p

@t ј c2 @_

@t ј _oc2 @c

@t

(4-9)

Taking the second partial derivative both sides of Eq. (4-9),

Transmission of Sound 81

Copyright © 2003 Marcel Dekker, Inc.

@2p

@t2 ј _oc2 @2c

@t2 (4-10)

Similarly, taking the second partial derivative of Eq. (4-3), the force balance

equation, assuming that the displacement is very small, such that the density

is practically constant,

_ ј _oр1 ю cЮ _ _o ј constant

we obtain:

@2p

@x2 ј __o

@3_

@t2@x ј __o

@2

@t2

@_

@x

_ _

(4-11)

Using Eq. (4-7),

@2p

@x2 ј ю_o

@2c

@t2 (4-12)

By comparing Eqs (4-10) and (4-12), we obtain the wave equation in

terms of the instantaneous acoustic pressure p:

@2p

@x2 ј

1

c2

@2p

@t2 (4-13)

This equation, when solved subject to the pertinent initial and boundary

conditions, yields expressions for all of the acoustic quantities in a particular

situation. The wave equation may also be written in terms of the instantaneous

particle displacement (Randall, 1951):

@2_

@x2 ј

1

c2

@2_

@t2 (4-14)

The displacement formulation may be more convenient to use in the solution

of problems in which displacements are known at the boundaries.

After the wave equation has been solved for either the acoustic pressure

or the particle displacement, the other quantities may be found by

operating mathematically on the solution of the wave equation. For

example, suppose we have solved the wave equation for the instantaneous

acoustic pressure, pрx; tЮ. By integrating both sides of Eq. (4-9), we obtain

the condensation expression:

p ј _oc2c (4-15)

By integrating Eq. (4-7) and using Eq. (4-15), we obtain the expression for

the instantaneous particle displacement:

_ ј _

р

cdx ј _

1

_oc2

р

pрx; tЮ dx (4-16)

82 Chapter 4

Copyright © 2003 Marcel Dekker, Inc.

Finally, if we integrate the force balance, Eq. (4-3), with respect to time and

use the velocity-displacement relation, we may evaluate the instantaneous

particle velocity:

uрx; tЮ ј _

1

_o

р

@pрx; tЮ

@x

dx (4-17)

On the other hand, if we have obtained the solution in terms of the

instantaneous particle displacement, _рx; tЮ, we may evaluate the other

acoustic quantities from the following expressions:

uрx; tЮ ј

@_

@t

(4-18)

cрx; tЮ ј _

@_

@x

(4-19)

pрx; tЮ ј _oc2c ј __oc2 @_

@x

(4-20)

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