Embedded trapped modes for obstacles in two-dimensional waveguides

In this paper we investigate the existence of embedded trapped modes for sy mmetric obstacles which are placed on the centreline of a two-dimensional a coustic waveguide. Modes are sought which are antisymmetric about the centr eline of the channel but which have frequencies that are above the first cu t-off for antisymmetric wave propagation down the guide. In the terminology of spectral theory this means that the eigenvalue associated with the trap ped mode is embedded in the continuous spectrum of the relevant operator. A numerical procedure based on a boundary integral technique is developed t o search for embedded trapped modes for bodies of general shape. In additio n two approximate solutions For trapped modes are found; the first is for l ong plates on the centreline of the channel and the second is for slender b odies which are perturbations of plates perpendicular to the guide walls. I t is found that embedded trapped modes do not exist for arbitrary symmetric bodies but if an obstacle is defined by two geometrical parameters then br anches of trapped modes may be obtained by varying both of these parameters simultaneously. One such branch is found for a family of ellipses of varyi ng aspect ratio and size. The thin plates which are parallel and perpendicu lar to the guide walls are found to correspond to the end points of this br anch.