The branch structure of embedded trapped modes in two-dimensional waveguides

In this paper we investigate the existence of branches of embedded trapped modes in the vicinity of symmetric obstacles which are placed on the centreline of a twodimensional 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 [1], a procedure 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 show that further branches of such modes exist for both ellipses and 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 for rectangular blocks are derived from a two-term matched eigenfunction expansion and these are compared to the results from the numerical scheme described in [1]. The transition from trapped mode to standing wave which occurs at one end of each of the branches is investigated in detail.