Comparison of the Rayleigh–Fourier and extinction theorem methods applied to scattering and transmission at a rough solid–solid interface

Abstract

The Rayleigh–Fourier and extinction theorem mathematical formalisms are established and compared in the case of scattering from and transmission through a rough solid–solid interface. By using the reciprocity properties of the solution resulting from the extinction theorem formalism, it is shown that, if they have unique solutions, the two formalisms are theoretically equivalent when used to obtain the displacement fields beyond the maximum excursions of the rough interface. As a result, the perturbation approaches of the two formalisms yield identical results. From a numerical point of view, it is shown that the two methods can give close results regardless of the verification of the Rayleigh hypothesis. The closeness of the two approaches, which has already been established and numerically confirmed in the case of a fluid–solid interface, does not seem to be known or even conjectured in the literature in the case of a solid–solid interface. It is indicated that, with the mathematics included in this paper, analogous results can be obtained for electromagnetic scattering and transmission at a rough interface.