Scattering of a scalar wave from a slightly random surface

Scalar wave scattering from a slightly random surface is analyzed by a probabilistic method. We make use of the homogeneity of an infinite random surface, that is, the shift invariance property of the strictly homogeneous random field. By the group-theoretic consideration of such a shift invariance property, the wave solution proves to be a homogeneous random field multiplied by an exponential function. Then such a homogeneous random field is approximately solved for a slightly random surface to yield a wave solution involving multiple scattering. Several statistical properties of the scattering are calculated and shown in the figures. The accuracy of the approximate solution is examined in terms of the error of the boundary-value equation.