INTEGRAL-EQUATIONS FOR A CLASS OF PROBLEMS CONCERNING OBSTACLES IN WAVE-GUIDES

In this paper we consider the two-dimensional boundary-value problem that arises when the Helmholtz equation is solved in a parallel-plate waveguide on the centreline of which is placed an obstacle that is symmetric about the centreline but which has otherwise arbitrary shape. The normal derivative of the unknown potential phi is specified on the surface of the obstacle. Two problems are considered in detail. First the problem of determining any trapped-mode wavenumbers is considered and secondly the problem of the scattering of an incident wave by the obstacle is examined. The solutions to these problems are sought using integral equations. Both problems have relevance in acoustics and in water-wave theory.