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 and many other auth ors. In this paper, we examine propagation over a finitely conducting rough surface, also studied by many people including Feinberg, Bass, Fuks, and B arrick. 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 (sigma < 0.1<lambda>). We limit ourselves to the one-dimensional ( 1-D) rough surface with finite conductivity excited by a magnetic line sour ce, which is equivalent to the Sommerfeld dipole problem in two dimensions (x-z plane). With the presence of finite roughness, the total field decompo ses into the coherent field and the incoherent field. The coherent (average ) field is obtained by using Dyson's equation, a fundamental integral equat ion based on the modified perturbation method. Once the coherent field has been obtained, we determine the Sommerfeld pole, the effective surface impe dance, and the Zenneck wave for rough surfaces of small rms heights. The co herent field is written in terms of the Fourier transform, which is equival ent to the Sommerfeld integral. Numerical examples of the attenuation funct ion are compared to Monte Carlo simulations and are shown to contrast the f lat and rough surface cases. Next, we obtain the general expression for the incoherent mutual coherence functions and scattering cross section for rou gh conducting surfaces.