Integral equation methods for wave propagation over rough surfaces

Electromagnetic and acoustic waves propagating over irregular surfaces give rise to a high degree of multiple scattering, and treatment continues to present major theoretical and computational challenges. The difficulties are greatly increased where the surface is penetrable (such as fluid/solid or dielectric interfaces), and in waveguides (when repeated scattering takes place between adjacent boundaries). The solution can be expressed exactly in terms of integral equations, but numerical inversion of these equations is highly intensive computationally, and may become prohibitive when treating the coupled sets of equations arising for penetrable surfaces and waveguides. Work is in progress on the case of penetrable surfaces, for use in the study of imaging of submerged or buried objects. In order to illustrate the main ideas we give a brief description of the equations in the case of an irregular 2-dimensional horizontal waveguide