4.12 ABSORPTION OF SOUND

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In the previous discussion of sound transmission, we had assumed that the

energy dissipation within the medium was negligible. This assumption is

generally quite valid, except for high-frequency sound waves and for

sound transmitted over large distances. In this section, we will examine

the effect of energy attenuation as the sound wave moves through a

medium.

In the development of the wave equation in Sec. 4.1, we had considered

only pressure forces acting on a fluid element. The attenuation or

dissipation effects may be represented by a dissipation force, defined by:

Fd ј CDS

@c

@t ј _CDS

@2_

@x@t

(4-197)

where CD is the dissipation coefficient with units, Pa-s, and the quantity c is

the condensation, as defined by Eq. (4-5). With the dissipation effect

included, Eq. (4-2) for the net force becomes, as follows:

Fnet ј _

@p

@x

S dx ю CD

@3_

@x2 @t

S dx (4-198)

Transmission of Sound 139

Copyright © 2003 Marcel Dekker, Inc.

The force–balance equation, Eq. (4-3), has the following form if dissipative

forces are included:

@p

@x _ CD

@3_

@x2 @t ј __

@2_

@t2 (4-199)

The development presented in Sec. 4.1 may be carried out, using Eq.

(4-199), to obtain the one-dimensional wave equation with dissipative effects

included:

@2p

@x2 ю

CD

_c2

@3p

@x2 @t ј

1

c2

@2p

@t2 (4-200)

The coefficient on the second term in Eq. (4-200), corresponding to

dissipation effects, has time units, so we may define the relaxation time _ as

follows:

_ ј

CD

_c2 (4-201)

140 Chapter 4

TABLE 4-6 Selected Design Values of the STC for Partitions

According to HUD Criteria.

Partitions between dwellings

Sound transmission class, STC, dB

Grade I Grade II Grade III

Bedroom to bedroom 55 52 48

Corridor to bedroom 55 52 48

Kitchen to bedroom 58 55 52

Partitions within dwellings:

Bedroom to bedroom 48 44 40

Kitchen to bedroom 52 48 45

Office areas:

Normal office to adjacent office 37

Normal office to building exterior 37

Conference room to office 42

Schools, etc.:

Classroom to classroom 37

Classroom to corridor 37

Theater to similar area 52

Source: Berendt et al. (1967).

Copyright © 2003 Marcel Dekker, Inc.

If we substitute the expression from Eq. (4-201) for the relaxation time into

Eq. (4-200), we find an alternative form for the one-dimensional wave equation

including dissipation effects:

@2

@x2 p ю _

@p

@t

_ _

ј

1

c2

@2p

@t2 (4-202)

Let us consider a simple harmonic sound wave, given by:

pрx; tЮ ј рxЮ ej!t (4-203)

where рxЮ is the amplitude function, which is a complex quantity, in this

case. Making the substitution from Eq. (4-203) into the wave equation, Eq.

(4-202), we obtain the following differential equation:

р1 ю j!_Ю

d2

dx2 ю

!2

c2 ј 0 (4-204)

Let us define the complex quantity _ (complex wave number) as follows:

_2 ј

!2

c2р1 ю j!_Ю

(4-205)

The solution of Eq. (4-204) for waves traveling in the юx-direction is as

follows:

рxЮ ј Ae_j_x (4-206)

Let us write the complex wave number in terms of its real and imaginary

parts:

_ ј k _ j_ (4-207)

The term _ is called the attenuation coefficient. The amplitude function may

be written in terms of the attenuation coefficient, by combining Eqs (4-206)

and (4-207):

рxЮ ј Ae_jрk_j_Юx ј Ae__x e_jkx (4-208)

If we combine Eqs (4-205) and (4-207) and solve for the real and

imaginary parts, we obtain the following expressions:

_ ј

!

c

р1 ю !2_2Ю1=2 _ 1

2р1 ю !2_2Ю

" #1=2

(4-209)

k ј

!

c

р1 ю !2_2Ю1=2 ю 1

2р1 ю !2_2Ю

" #1=2

(4-210)

Transmission of Sound 141

Copyright © 2003 Marcel Dekker, Inc.

In many fluids, the relaxation time is quite small. For example, for

monatomic gases, _ is approximately 0.2 ns. In the acoustic range of frequencies

(20 Hz to 20 kHz), the term р!_Ю is much less than unity. In this

case, Eqs (4-209) and (4-210) reduce to the following relationships:

_ ј

!2_

2c ј

2_2 f 2_

c

(4-211)

k ј

!

c ј

2_ f

c

(4-212)

According to Eq. (4-211), if the relaxation time is independent of frequency,

then the attenuation coefficient is directly proportional to the frequency

squared. This behavior is observed for many gases over a wide range of

frequencies.

If we denote the amplitude of the acoustic pressure at x ј 0 by po, then

the magnitude of the acoustic pressure at any location is given by Eq.

(4-208):

pрxЮ ј po e__x (4-213)

For a plane acoustic wave with attenuation, the amplitude of the pressure

wave decays exponentially with distance from the source of the wave. The

intensity of the plane wave at any location may be written as follows:

IрxЮ ј

p2

_c ј

p2

o

_c

e_2_x ј Io e_2_x ј Io e_mx (4-214)

The quantity Io is the intensity at x ј 0 and m ј 2_. The quantity m is called

the energy attenuation coefficient. We note from Eq. (4-214) that, for a plane

wave, the intensity is not constant with position, because energy is being

dissipated in this case. The change in the intensity level with distance due to

energy attenuation is given by the following expression for a plane sound

wave:

_LI ј LIрx ј 0Ю _ LIрxЮ ј _10 log10рe_2_xЮ (4-215)

_LI ј р10Юр2_xЮ= lnр10Ю ј 8:6859_x (4-216)

For a spherical sound wave, the acoustic energy varies according to

the following expression, if dissipation effects are considered:

WрrЮ ј Wo e_2_r (4-217)

The intensity for a spherical sound wave with energy attenuation may be

expressed by the following:

I ј

W

4_r2 ј

Wo

4_r2 e_2_r (4-218)

142 Chapter 4

Copyright © 2003 Marcel Dekker, Inc.

The change in the intensity level for a spherical sound wave with dissipation

effects may be written as follows:

_LI ј LIрr ј roЮ _ LIрrЮ ј 20 log10рr=roЮ ю 8:6859_рr _ roЮ (4-219)

We observe from Eq. (4-219) that the change in intensity level for a spherical

wave involves two effects: (a) the spreading of the acoustic power over a

larger area and (b) the dissipation of acoustic energy in the material.

The attenuation term р_xЮ or р_rЮ has been given ‘‘units’’ of neper

(named after John Napier, who developed logarithms), with the abbreviation

Np (Pierce, 1981). Note that the quantity р_xЮ is actually dimensionless.

The ‘‘unit’’ radian for angular measure is also dimensionless. The attenuation

coefficient _ can be written with units neper per meter, Np/m. But the

intensity level change has units of decibel, dB, so the term (8.6859_) should

have units of dB/m in Eq. (4-219). From this observation, we see that the

conversion factor between the two dimensionless ‘‘units’’ is:

8:6859 dB=Np ј 20 log10рeЮ ј 20=lnр10Ю р4-220)

It is important to note that the attenuation coefficient may be reported in

the literature in either Np/m units or dB/m (or dB/km) units.

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