Propagation and scattering of low grazing skimming waves over conducting rough surfaces

If both the transmitter and the observation points are located close to the rough conducting surface, the wave incident upon a point on the surface is a mixture of the coherent and incoherent waves and is no longer the incident plane or spherical wave in free space. If the surface is flat, this is the Sommerfeld problem which has been studied extensively. This paper considers the Sommerfeld problem for rough surfaces. First, the authors consider the coherent field over the one-dimensional rough surface which satisfies the Dyson equation. Using the flat surface Green's function, the coherent field is expressed in a spatial Fourier transform which is equivalent to the Sommerfeld integral. From the complex reflection coefficient in the Fourier domain, the authors obtain the Sommerfeld pole and the final expressions are given for the attenuation function. Numerical examples are given for rough ocean and land surfaces showing the additional attenuation due to the scattering. The results are then compared with Monte Carlo simulations showing good agreement. Next, the incoherent field is formulated based on the Bethe-Salpeter equation. The first-order solution indicated that the coherent wave propagates to a point on the surface where the incoherent wave is excited and is propagated to the observation point. The total incoherent field is a sum of contributions from all scattering points on the surface.