43.2 Fundamental Equations

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43.2.1 Analytical Model

The physical behavior of a reactive muffler may be adequately modeled by linear differential equations.

The law of conservation of mass must hold, while three simultaneous equations, Newton’s, Boyle –

Charles, and that of conservation, must be satisfied. When these equations are combined, we obtain the

wave equation for the plane, one-dimensional sound-pressure wave:

›2j

›t2 ¼ c2 ›2j

›x2 ð43:1Þ

p ¼ rc2 ›j

›x ð43:2Þ

where

j ¼ displacement of particle motion (m)

c ¼ velocity of sound (m/sec)

p ¼ sound pressure (Pa)

r ¼ density of air (kg/m3)

t ¼ time (sec)

x ¼ coordinate system along which wave travels (m)

The stationary solutions for angular frequency v of Equation 43.1 and Equation 43.2 are given by

j ¼ ðA e2jkx 2 B e jkx Þe jvt ð43:3Þ

p ¼ 2rc2kðA e2jkx þ B e jkx Þe jvt ð43:4Þ

where A; B ¼ amplitudes of sound pressure or particle motion for traveling and reflecting waves,

k ¼ 2pf =c; wave number, and j ¼

ffiffiffiffi

21 p .

43.2.2 Boundary Conditions

The boundary conditions are given below.

(1) Sound source. The sound source is assumed to be independent of the existence of the muffler, and

the volume rate of the particles is assumed constant, as given by

S_j ¼ const ð43:5Þ

in which · denotes the time derivative.

(2) Open end of duct. The reflection coefficient, R; at the open end of an unflanged circular pipe is

available, and is given by

R ¼

B e jkx

A e2jkx ð43:6Þ

The magnitude of the reflection coefficient, lRl; is shown in Figure 43.1a as a function of ka; where a is the

pipe radius. The phase shift can be determined from Figure 43.1b, which is a plot of a=a as a function of

ka: Also, the reflection coefficient is [1]:

R ¼ 2lRle22jka ð43:7Þ

For the small values of ka that are most often encountered in reactive muffler design, lRl < 1 and

a=a ¼ 0:613:

(3) Closed end. The displacement of particle motion is zero at a rigid wall. Hence, we have

j ¼ 0 ð43:8Þ

43-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

(4) Junction conditions. The following equations correspond to the continuity of volume flow rate of

the particles and the continuity of pressure, even if the cross section changes suddenly:

Si

_j

i ¼ Siþ1

_j

iþ1 ð43:9Þ

pi ¼ piþ1 ð43:10Þ

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