Trapped modes in acoustic waveguides

Abstract

We consider the eigenvalue problem for the Laplacian on a cylinder perturbed by a compact obstacle (with Neumann boundary conditions) and look for eigenvalues of such a problem. In the case of a two-dimensional cylinder with symmetric obstacle we give sufficient conditions for both existence and non-existence of an eigenvalue. We also prove that in the case of a thin obstacle, parallel to the axis of the cylinder, there always exists an eigenvalue.