The expansion of the solution of the rough-surface scattering problem in powers of quasi-slopes

Abstract

The theory of rough-surface scattering is formulated for the Dirichlet boundary conditions on the basis of the exact integral equations for the surface sources density. The theory of the small-slope approximation, proposed by Voronovich, is revised on the basis of this theory. Some of the general properties of the solution, suggested by Voronovich, are proved on the basis of this equation. The equation for the function describing modified (invariant) surface source density is obtained. Only the small-scale spectral components of the surface elevations are essential for this modified sources. We call the corresponding expansion a quasi-slope expansion. The two first term of expansion in powers of quasi-slopes are presented: The expansion for the modified wave sources induces the corresponding expansion for the scattering amplitude, which is consistent both with perturbation theory and the Kirchhoff approximation. In the high-frequency limit the solution obtained coincides with Voronovich's expression. The reciprocity-theorem consequences for the solution obtained are analysed.