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
[email protected]://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
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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|
Ralph Muehleisen, Applied Research Lab and Graduate Program in Acoustics 10
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