Variational principles in high-frequency scattering

Abstract

A pair of variational principles are formulated for two-dimensional scattering by obstacles. The first of these is in terms of the obstacle boundary values, and it is shown that a simple ‘optical’ trial function leads to an incorrect frequency dependence for the scattering cross-section. In the second, the obstacle is viewed as the analogue of an aperture coupling two half spaces. The geometric optics part of the cross-section can then be made explicit and is split off to leave a stationary form for the frequency correction. The zero-order calculation for the cross-section of a circle, using corresponding ‘optical’ trial functions, is found to have the correct (Ka)−2/3 frequency dependence.