Green´s function for rough surface with Dirichlet, Neumann, and impedance boundary conditions

This paper presents an analytical theory of rough surface Green´s functions based on the extension of the diagram method of Bass and Fuks (1979), and Ito (1985) with the smoothing approximation used by Watson and Keller (1983, 1984). The method is a modification of the perturbation method and is applicable to rough surfaces with small RMS height. But the range of validity is considerably greater than the conventional perturbation solutions. We consider one-dimensional rough surfaces with Dirichlet, Neumann, and impedance boundary conditions. The coherent Green´s function is obtained from the smoothed Dyson´s equation by using a spatial Fourier transform. The mutual coherence function for the Green´s function is obtained by the first-order iteration of the smoothing approximation applied to the Bethe-Salpeter equation in terms of a quadruple Fourier transform. These integrals are evaluated by the saddle-point technique. The equivalent bi-static cross section per unit length of the surface is compared to the conventional perturbation method and Watson-Keller´s result. With respect to Watson-Keller´s result, it should be noted that our result is reciprocal while the Watson-Keller result is nonreciprocal. Included in this paper is a discussion of the specific intensity at a given observation point. The theory developed will be useful for the RCS signature related problems and LGA (low grazing angle) scattering when both the transmitter and object are close to the surface.