1. Higher order mode wave propagation in duct Yan Li机械系统与振动国家重点实验室 上海交通大学， 200240 ，上海

2. Research background Higher order mode In a rectangular duct In a circular duct In expansion chamber Conclustion Outline

3. Research 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

4. Research background Higher order mode wave propagation In a rectangular duct In a circular duct In expansion chamber Outline

5. Higher order mode in a rectangular duct Boundary condition at the wall of the duct requires: Transmission wave number for (m, n) mode: Cutoff frequencies: a x b z

6. Higher order mode in a rectangular duct 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

7. Higher order mode in a circular duct 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 R x y z

8. 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 duct

9. Cutoff frequency First asymmetric mode m=1,n=0 First circularly symmetric/radial mode m=0, n=1

10. Effects in expansion chamber Fig. 4. Schematic of illustrations of instantaneous axial pressure distribution for higher mode propagation, tunneling and leakage of one mode

11. Effects 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)

12. Research background Higher order mode In a rectangular duct In a circular duct In expansion chamber Conclustion Outline

13. Thanks ！