On scattering and reflection of electromagnetic waves by rough surfaces

Simple approximations for the reflection coefficients and differential scattering cross sections per unit area of a random distribution of arbitrary protuberances on a ground plane are given in terms of the scattering amplitude of an isolated protuberance, their average number in unit area, and the given incident wave. These functions take into account multiple coherent scattering, and are mutually consistent in fulfilling the energy principle. It is shown, in general, that if the horizon angle approaches zero, then the reflection coefficients approach unity linearly, and the horizontal/vertical back scattering vanishes like the fourth/second power of the angle. General results are then specialized to arbitrary hemispheres and circular semicylinders, and applied to limiting cases of perfect conductors with radii very small or very large compared to wavelength.