43.3 Effects of Reactive Mufflers

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The acoustic behavior of a reactive muffler may be expressed in term of the insertion loss, the difference

in the noise levels measured at some external point with and without the muffler in the system. The

transmission loss is defined as the insertion loss for a nonreflecting source and the end of exhaust duct.

43.3.1 Insertion Loss

A single expansion-type muffler installation is shown schematically in Figure 43.2. At the open end of a

pipe, as in Figure 43.2, the traveled and reflected waves of the source become A0 e2jkl; B0 e jkl; over a

length l; where the amplitudes are denoted by A0; B0: The reflective coefficient for length l0 is given by

R0 ¼

B0 e jkl0

A0 e2jkl0 ð43:11Þ

This is obtained from Equation 43.6 with x ¼ l0:

The energy, W0; of the acoustic wave escaping from the open end of the pipe is given by

W0 /

S0A20

ð1 2 R20

Þ

r0c0 ð43:12Þ

in which r0 ¼ density of air, and c0 ¼ speed of sound in air. The equation of the sound-pressure level

measured at an open point at some distance is given by

p0 ¼ 10 log10

Qd

4pr2

0 þ

4

Rr

􀁻 !

þ PWLr0 ð43:13Þ

FIGURE 43.1 (a) Magnitude of the reflection coefficient at open end of an unflanged circular pipe; (b) End

correction for an unflanged circular pipe. (Source: Levine, H. and Schwinger, J., On the radiation of sound from an

unflanged circular pipe, J. Phys. Rev., 73, 383, 1948. With permission.)

1

0.8

0.6

0.4

0.2

0

0.0 1.0 2.0 3.0 4.0

ka

|R|

(a)

0.6

0.4

0.2

0

0.0 1.0 2.0 3.0 4.0

ka

α/a

(b)

Design of Reactive Mufflers 43-3

© 2005 by Taylor & Francis Group, LLC

where

r0; ¼ distance

Rr ¼ A=ð1 2 aÞ; room constant

A ¼ the indoor sound absorbing power (indoor surface area times indoor average absorption coefficient)

a ¼ the indoor average absorption coefficient

Qd ¼ the directivity factor from the open end

Therefore, the measured value of insertion loss can be obtained from Equation 43.14, when Qd values

are equal. Power level is defined as

PWL ¼ 10 log10

W

10212

􀀏 􀀐

Now,

IL ¼ PWLr0

2 PWLr ð43:14Þ

Using Equation 43.14, it can be shown that IL can be expressed by

IL ¼ 10 log10

S0

S3

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

A0

A3

2􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

1 2 R20

1 2 R23

􀀈 􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈 􀀈

ð43:15Þ

Sound source

A0

A1

A2

A3

B1

B2 B3

B2e jkl2

l1 l2 l3

B1e jkl1

A1e −jkl1

A2e −jkl2

B0

B0e jkl0

l0

A0e −jkl0

S

S

L

p0(dB)

p0(dB)

Expansion

chamber

Inlet pipe Outlet (or tail) pipe

L

I II

FIGURE 43.2 Measurement of insertion loss.

43-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

3.2 Transmission Loss

It is desirable to eliminate the source and radiation characteristics from the system in Figure 43.3, and to

look only at some property of the muffler itself. This may be accomplished by defining a quantity called

“transmission loss” (TL) as follows:

TL ¼ 10 log10

A1

A2

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

2 S1

S2 ð43:16Þ

In the measurement of TL, it is difficult to separate the reflected wave. The theoretical calculation is easy

and useful.

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