Embedded trapped modes for obstacles in two-dimensional waveguides

In this paper we investigate the existence of embedded trapped modes for symmetric obstacles which are placed on the centreline of a two-dimensional acoustic waveguide. Modes are sought which are antisymmetric about the centreline of the channel but which have frequencies that are above the first cut-off for antisymmetric wave propagation down the guide. In the terminology of spectral theory this means that the eigenvalue associated with the trapped mode is embedded in the continuous spectrum of the relevant operator. A numerical procedure based on a boundary integral technique is developed to search for embedded trapped modes for bodies of general shape. In addition two approximate solutions for trapped modes are found;the first is for long plates on the centreline of the channel and the second is for slender bodies which are perturbations of plates perpendicular to the guide walls. It is found that embedded trapped modes do not exist for arbitrary symmetric bodies but if an obstacle is defined by two geometrical parameters then branches of trapped modes may be obtained by varying both of these parameters simultaneously. One such branch is found for a family of ellipses of varying aspect ratio and size. The thin plates which are parallel and perpendicular to the guide walls are found to correspond to the end points of this branch.