42.2 Fundamentals of Sound Absorption

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42.2.1 Attenuation of Sound

When an acoustic wave propagates in a medium, the sound energy attenuates due to such reasons as

viscosity, heat conduction, and the effects of molecular absorption. In a medium of small volume

surrounded by a boundary surface, the attenuation is particularly considerable, for example, when the

medium is a thin tube. This is because there is the dissipation of the energy controlled by the viscosity of the

medium and heat conduction between the material and the medium of tube wall. A sound-absorbing

material may be utilized to adjust such dissipation of acoustic energy.

42.2.1.1 Absorption Coefficient and Normal Acoustic Impedance

Some amount of energy is lost when an acoustic

wave hits the surface of a sound-absorbing

material. Figure 42.1 illustrates an infinite medium

of absorbing material separated by air and the

reflected wave (sound pressure pr) from the

boundary surface with the air where a plane

wave of sound pressure pi is emitted in the

direction indicated by an arrow, at an angle u:

When u ¼ 0; sound pressure p in air is given by

p ¼ pi þ pr ¼ ðAe2jkx þ Bejkx Þejvt ð42:1Þ

where

A; B ¼ the amplitude of sound pressure of incident

and reflected waves (in Pa),

j ¼

ffiffiffiffi

21 p ;

k ¼ 2pf =c; wave number (1/m),

v ¼ angular frequency (rad/sec).

The sound pressure, pm; in the absorbing material

may be expressed using a complex propagation

constant, by the equation:

pm ¼ px¼0e2gx e jvt ð42:2Þ

where

g ¼ the propagation constant in the absorbing material (m21). Note: g ¼ d þ jb: g is a property of the

material itself and is not dependent on the mounting conditions when large areas of material are

considered.

d ¼ attenuation constant. Note: d tells us how much of the sound wave will be reduced as it travels

through the material.

b ¼ phase constant. Note: b is a measure of the velocity of propagation of the sound wave through the

material.

The relation for determining the velocity of sound in the material is given by

cm ¼ v=b ð42:3Þ

FIGURE 42.1 Plane wave incidence on an infinite

absorbing material.

42-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Boundary conditions must be satisfied on the boundary surface. The acoustic impedance of a unit area of

air and of absorbing material are, respectively, denoted by z and za: The pressure and the particle velocity

on both sides of the boundary are equal. We have

pi þ pr ¼ px¼0

pi 2 pr

za ¼

px20

z

9>=

>;

ð42:4Þ

The amplitude of reflectance of sound pressure, r; is obtained from Equation 42.4, and is given by

r ¼

pr

pi ¼

za 2 z

za þ z ð42:5Þ

The reflectivity is the energy reflection rate. The absorption coefficient, a; of an absorbing material is

defined as

a ¼ 1 2 lr2l ð42:6Þ

The impedance, zn; through a surface is the quantity that represents the dissipation of energy of sound as

well as the absorption coefficient. It is given as a ratio between sound pressure and particle velocity on

boundary surface in the reflecting acoustic wave:

zn ¼

p

u

􀀏 􀀐

x¼0¼

rc

cos u

􀀏 􀀐

pi þ pr

pi 2 pr ð42:7Þ

Note that zn is a complex quantity and involves both amplitude and phase, both of which depend on the

sound pressure at the boundary surface in the reflecting acoustic wave.

In the case of oblique incidence, the surface impedance can be expressed by following equation:

zn ¼ Zgz=q ð42:8Þ

where z ¼ the acoustic impedance (Pa sec/m3). Here,

Z ¼

zl coshðqlÞ þ ðgz=qÞsinhðqlÞ

zl sinhðqlÞ þ ðgz=qÞcoshðqlÞ

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2 þ k2 sin2u

q

The absorption coefficient, aðuÞ; for an oblique incidence with angle u may be expressed by

aðuÞ ¼ 1 2

zn cos u 2 rc

zn cos u þ rc

􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈 􀀈

2

ð42:9Þ

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