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Scattering cross section from a rough surface for small grazing angles

The particular problem of wave scattering at low grazing angles is of great interest because of it importance for radiowave long distance propagation along the Earth surface and radar observation of near surface objects. One of the main questions is-how the scattering amplitude and specific cross section behave for extremely small grazing angles? In V.I. Tatarskii et al. (1998) a general answer to this question was obtained for the scattering cross section by arbitrary rough surfaces of two types: with Dirichlet and Neumann boundary condition. For the latter case (the Neumann boundary condition) the main result of such general consideration, obtained by V.I. Tatarskii et al. is the following: the scattering amplitude tends to a constant without any assumptions on the relationship between wave length and the geometrical scales of surface roughness. This result was obtained for an infinite plane interface, the central part of which contains the bounded domain with roughness. In this case, on large distances from the rough domain, according to the Neumann boundary condition, the normal derivative of the field (equal to z-derivative) is equal to zero. This means that the field (and the scattering amplitude, which is proportional to the field in the far zone) does not depend on the grazing angle of the scattered wave for small grazing angles. The results by D.E. Barrick (1998), including both numerical calculations and a general proof, contain the statement: the scattering amplitude for both surface types mentioned above tends to zero as the second power of grazing angle. This result was obtained for the periodical, i.e., infinite, rough surface. Because of differences in the formulations of the problem disagreement between the results of theses authors does not mean that one of them is wrong. The present author considers the process of wave scattering by statistically rough surfaces with a Neumann boundary condition. This model corresponds to sound scattering from a perfectly "hard" surface. In the case of EM waves and one-dimensional surface, this model describes the scattering of "vertically" polarized waves.