43.4 Calculation Procedure

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For the reactive muffler shown in Figure 43.2, the following equations are obtained from Equation 43.5

to Equation 43.7 and Equation 43.9. The inlet pipe, cavity, and tail pipe are denoted by suffix in the

figure [2].

(1) The sound source:

S0ðA0 2 B0Þ ¼ S1ðA1 2 B1Þ ð43:17Þ

(2) The open end of pipes:

B0 e jkl0 ¼ A0 e2jkl0 R0

B3 e jkl3 ¼ A3 e2jkl3 R3

9=

; ð43:18Þ

(3) The sudden expansion (junction I):

A1 e2jkl1 þ B1 e jkl1 ¼ A2 þ B2

S1ðA1 e2jkl1 2 B1 e jkl1 Þ ¼ S2ðA2 2 B2Þ

9=

; ð43:19Þ

(4) The sudden contraction (junction II):

A2 e2jkl2 þ B2 e jkl2 ¼ A3 þ B3

S2ðA2 e2jkl2 2 B2 e jkl2 Þ ¼ S3ðA3 2 B3Þ

9=

; ð43:20Þ

From these linear equations, the following equation can be obtained:

A0

B0 ¼

m10

1 2 R0 e22jkl0

j 1 þ R3 e22jkl3

􀀍 􀀎

sin kl1 cos kl2 þ m21kl1 sin kl2

n 􀀄 􀀅

þm32 1 2 R3 e22jkl3

􀀍 􀀎

m21 cos kl1 cos kl2 2 sin kl1 sin kl2

􀀄 􀀅o

ð43:21Þ

Muffler

S1 A1

B1 B2

A2 S2

TL = 10 log10

A1

2 S1

A2 S2

FIGURE 43.3 Definition of transmission loss for a muffler.

Design of Reactive Mufflers 43-5

© 2005 by Taylor & Francis Group, LLC

When the reflection coefficients are absent, R0 ¼ R3 ¼ 0; we have

IL ¼ 10 log10m03{m2

10ðsin kl1 cos kl2 þ m21 cos kl1 sin kl2Þ2 þ m2

10m2

32

􀀐 ðm21 cos kl1 cos kl2 2 sin kl1 sin kl2Þ2} ð43:22Þ

where m10 ¼ S0=S1; m21 ¼ S2=S1; m32 ¼ S3=S2; m03 ¼ S0=S3; which are ratios of cross-sectional areas of

the pipes.

For the reflection factors, R0 ¼ R3 ¼ 21; S0 ¼ S1 ¼ S3; and m21 ¼ m; the equation of insertion loss

becomes

IL ¼ 20 log10

cos kl1

cos kl0

cos kl21 cos kl3 2 m sin kl2 sin kl3 2 tan kl1 cos kl2 sin kl3 þ

1

m

sin kl2 cos kl3

􀀏 􀀐 􀀘 􀀙 􀀒 􀀓 􀀈 􀀈 􀀈 􀀈

�� 􀀈 􀀈 􀀈

ð43:23Þ

The ratio lA1=A3l may also be obtained from Equation 43.17 to Equation 43.20. When the magnitude of

reflection coefficients R0 ¼ R3 ¼ 0; S0 ¼ S1 ¼ S3; and expansion ratio of the cross-sectional open area of

the pipes m21 ¼ m; the transmission loss is given by

TL ¼ 10 log10 1 þ

1

4

m 2

1

m

􀀏 􀀐2

sin2 kl2

􀀘 􀀙

ð43:24Þ

When R0 ¼ R3 ¼ 21; S0 ¼ S1 ¼ S3; and m21 ¼ m; TL is obtained from the following equation:

TL ¼ 10 log10 1 þ

1

m2 ðm2 2 1Þ{ðm2 þ1Þsin2 kl3 2 1}sin2 kl2 þ

1

2

2m þ

1

m

􀀏 􀀐

sin 2kl2 sin 2kl3

􀀈 􀀈 􀀈 􀀈

􀀈 􀀈 􀀈 􀀈

ð43:25Þ

Computation formulas of insertion loss and transmission loss for the case of an expansion chamber

with insertion pipe and resonator are shown in Table 43.1.

The principal structures of several reactive mufflers are shown in Figure 43.4.

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