Higher-order Modal Reflectionand Transmission in Acoustic

Waveguide Step Discontinuities

Ralph T. Muehleisen

The Applied Research Lab andThe Graduate Program in AcousticsThe Pennsylvania State University

PO Box 30, State College, PA 16804

rtm@sabine.acs.psu.eduhttp://www.acs.psu.edu/users/rtm

October 8, 1997

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics

Introduction

• Little work on the general scattering higher ordermodes.

• Important for HVAC and condition monitoring athigher frequencies

Goal: Develop expressions for the scattering of modes ata size discontinuity of a duct

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 1

Geometry of Step Discontinuity

S1

Region 1

S2

Region 2

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 2

Modal Decomposition at Discontinuity

The pressure p can be written

p1 =∑N

(A1N +B1N)Ψ1N(x, y)

p2 =∑N

(B2N +A2N)Ψ2N(x, y).

The axial velocity uz can be written

uz1 =∑N

Y1N(A1N −B1N)Ψ1N(x, y)

uz2 =∑N

Y2N(B2N −A2N)Ψ2N(x, y)

where Y1N and Y2N are the characteristic impedancesof mode N in region 1 and 2.

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 3

Boundary Conditions

• p and uz are continuous at discontinuity

Orthogonality leads to (in Matrix Form):

H(A1 + B1) = A2 + B2

Y1(A1 − B1) = HT Y2(B2 − A2)

where

HMN =∫∫S2

Ψ2MΨ1NdS2.

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 4

Scattering Parameters

Sµν are defined by matrix equation[B1

B2

]=[S11 S22

S21 S22

] [A1

A2

].

• Reflection Coefficients: S11 and S22

• Transmission Coefficients: S12 and S21

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 5

Solution

S11 = [Y1 + Y ′2]−1[Y1 − Y ′2]

S21 = 2H[Y1 + Y ′2]−1Y1

S12 = 2G[Y2 + Y ′1]−1Y2

S22 = [Y2 + Y ′1]−1[Y2 − Y ′1]

whereY ′2 = HT (Y2)H

Y ′1 = GT (Y1)G

and G is the generalized inverse of H.

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Example: Asymmetric Step

a

a/2Region 2

Region 1

• Duct width a → a/2

• Plots are computational results

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Region 1 Reflection Coefficients

S1100

S1111

S1101

S1110

0 2 4 6 80

0.5

1

1.5

2

ka

|S11µν | for Asymmetric Step

|S11 µν

|

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 8

Region 2 Reflection Coefficients

S2200

S2211

S2201

S2210

0 2 4 6 80

0.2

0.4

0.6

0.8

1

ka

|S22µν | for Asymmetric Step

|S22 µν

|

Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 9

Region 1→ 2 Transmission Coefficients

S1200

S1211

S1201

S1210

0 2 4 6 80

0.5

1

1.5

2

2.5

3|S12

mm| for Asymmetric Step

ka

|S12 m

m|

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Conclusions

• In matrix form, equations for Sµν are same formas a simple plane wave discontinuity

• Sµν form is independent of duct shape

• Modal Coupling is not negligible in general

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