Higher order mode wave propagation in duct

Yan Li

机械系统与振动国家重点实验室 机械系统与振动国家重点实验室 上海交通大学，上海交通大学， 200240200240 ，上海，上海

Research background

Higher order mode • In a rectangular duct

• In a circular duct

• In expansion chamber

Conclustion

OutlineOutline

Research BackgroundResearch Background

Sound transmission characteristics of duct system

• Building, automotive and aeronautic noise control

• Bends, constrictions, expansions & side branches

• Many study at low frequency

• Inaccurate estimates at first cut-on frequency

• Higher order mode wave propagation

Research background

Higher order mode wave propagation • In a rectangular duct

• In a circular duct

• In expansion chamber

OutlineOutline

Higher order mode in a rectangular Higher order mode in a rectangular ductduct

)cos()cos(~ ykxkAp nm

0)cos()sin(~ ykakBv nmx

Boundary condition at the wall of the duct requires:

0)sin()cos(~ bkxkCv nmy

Transmission wave number for (m, n) mode:

0])/()/([ 2/12220,, bnamkk nmz

,2,1,0, mmakm

,2,1,0, nnbkn

2/122 ])/()/)[(2/( bnamcfc

Cutoff frequencies:

a x

b z

Higher order mode in a rectangular Higher order mode in a rectangular ductduct

Fig. 1. Nodal lines for transverse pressure distribution in a rectangular duct up to m=3, n=3 higher order mode

the number of pressure nodal lines in the x direction

the number of pressure nodal lines in the y direction

Higher order mode in a circular ductHigher order mode in a circular duct

)(~ krAJp

0)(~ ' kRBJv m

Fig. 2. Bessel function of the first kind for order m=0,1,2. Boundary condition met when slopes goes to zero at wall of

circular duct

Rx

y

z

Fig. 3. Nodal lines for transverse pressure distribution in circular duct up to m=3, n=3 higher node with preferred

notation for m, n

Higher order mode in a circular ductHigher order mode in a circular duct

kR

Higher order mode in a circular ductHigher order mode in a circular duct

Cutoff frequency• First asymmetric mode m=1,n=0

• First circularly symmetric/radial mode m=0, n=1

DcDcfc /586.0/84.1

DcDcfc /22.1/83.3

Effects in expansion chamberEffects in expansion chamber

Fig. 4. Schematic of illustrations of instantaneous axial pressure distribution for higher mode propagation,

tunneling and leakage of one mode

Effects in expansion chamberEffects in expansion chamber

Fig. 5. Insertion loss measurements on expansion chamber using broad band noise excitation on impedance tube with anechoic

termination (c=345m/s)

Research background

Higher order mode • In a rectangular duct

• In a circular duct

• In expansion chamber

Conclustion

OutlineOutline

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