Sommerfeld and Zenneck wave propagation for a finitely conducting one-dimensional rough surface

Starting with Zenneck and Sommerfeld wave propagation over a flat finitely conducting surface has been extensively studied by Wait (see IEEE Antennas Propagat. Mag., vol.40, p.7-24, 1998) and many other authors. We examine propagation over a finitely conducting rough surface, also studied by many people including Feinberg (1944), Bass and Fuks (1979), and Barrick (see Radio Sci., vol.6, p.517-26, and vol.6., p.527-33). This paper extends the multiple scattering theories based on Dyson and Bethe-Salpeter equations and their smoothing approximations. The theory developed here applies to rough surfaces with small root-mean-square (RMS) heights (σ<0.1λ). We limit ourselves to the one-dimensional (1-D) rough surface with finite conductivity excited by a magnetic line source, which is equivalent to the Sommerfeld dipole problem in two dimensions (x-z plane). With the presence of finite roughness, the total field decomposes into the coherent field and the incoherent field. The coherent (average) field is obtained by using Dyson´s equation, a fundamental integral equation based on the modified perturbation method. Once the coherent field has been obtained, we determine the Sommerfeld pole, the effective surface impedance, and the Zenneck wave for rough surfaces of small RMS heights. The coherent field is written in terms of the Fourier transform, which is equivalent to the Sommerfeld integral. Numerical examples of the attenuation function are compared to Monte Carlo simulations and are shown to contrast the flat and rough surface cases. Next, we obtain the general expression for the incoherent mutual coherence functions and scattering cross section for rough conducting surfaces