A T matrix for scattering from a doubly infinite fluid–solid interface with doubly periodic surface roughness

Abstract

The T-matrix formalism is used to calculate scattering of a pressure wave from a doubly infinite fluid–solid interface with doubly periodic surface roughness. The Helmholtz–Kirchhoff integral equations are used to represent the scattered pressure field in the fluid and the displacement field in the solid. The boundary conditions are applied and a system of four coupled integral equations is obtained. The incident field, the surface fields, and scattered pressure field in the fluid and displacement field in the solid, are represented by infinite series of Floquet plane waves. This process discretizes the integral equations and transforms them into a system of four coupled doubly infinite linear equations. The extended boundary condition is applied and the T matrix that relates the spectral amplitudes of the incident field to the spectral amplitudes of the scattered fields is constructed. An exact analytic solution and numerical results are obtained for scattering from a doubly periodic rough surface constructed by superposing two sinusoids each of which depends on a single but different orthogonal coordinate.