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The Quarterly Journal of Mechanics and Applied Mathematics 2002 55(2):313-326; doi:10.1093/qjmam/55.2.313
© 2002 by Oxford University Press
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The Branch Structure of Embedded Trapped Modes in Two-Dimensional Waveguides

M. McIver1, C. M. Linton1 and J. Zhang1

( 1 Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU )

In this paper we investigate the existence of branches of embedded trapped modes in the vicinity of a rectangular block which is placed on the centreline of a two-dimensional acoustic waveguide. Modes are sought which are antisymmetric about the centreline of the channel and which have frequencies that are above the first cut-off for antisymmetric wave propagation down the guide. In previous work (McIver et al., Q. Jl Mech. Appl. Math. 54 (2000)) an integral equation method for finding such modes was developed and it was shown numerically that a branch of trapped modes exists for an ellipse which starts from a flat plate on the centreline of the guide and terminates with a flat plate perpendicular to the guide walls. In this work we use a matched eigenfunction method to show that further branches of such modes exist for rectangular blocks, each of which starts with a plate of different length on the centreline of the guide. Approximations to the trapped mode wave numbers are derived from a two-term matched eigenfunction expansion and good agreement with the full numerical results is obtained. The transition from trapped mode to standing wave which occurs at one end of each of the branches is investigated in detail.

Received 10 October 2000. Revised 12 July 2001.


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