Exact spectral formalism for rough-surface scattering

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Abstract
The spectral amplitudes of the scattered and transmitted fields for plane-wave incidence on an arbitrary rough interface in one dimension are derived exactly and simply by using Green’s theorem. Results are stated in terms of integrals on values of the field and its normal derivative on the interface. The energy constraint is derived, and individual energy contributions in each region are also related to the surface-field values. The latter contributions can be calculated from coupled linear equations that are also derived using Green’s theorem. The interface separates media of different but constant densities and sound speeds (acoustics) or different dielectrics (electro-magnetics).