Rough surface propagation and scatter 1. General formulation and solution for periodic surfaces

A general formulation or the electromagnetic propagation and scattering problem for rough surfaces is presented. The method is based on a decomposition of both the physical properties and the various electromagnetic field quantities associated with two media in terms of characteristic (or step) functions determined by the geometry. Maxwell equations are invoked and are interpreted in a generalized function (or distributional) sense. The results of the analysis are two coupled integral equations for the fields above and below the surface separating the media, together with a boundary condition The boundary condition is generated as a by-product of the analysis. The integral equations are simplified by the use of a spatial Fourier transform It is shown that when certain assumptions are made on the refractive index and spatial spectra of the surface profile and fields, an approximate solution of one integral equation may be obtained This last result is of the same nature as the “surface impedance” boundary condition. Using the formulation, a series solution is proposed for the surface electric field in the spatial and temporal Fourier transform domain for a good conducting two dimensional periodic surface. The choice of a finite source is kept arbitrary. As an application of the series solution in ground wave propagation and scattering we have then considered the source to be an elementary vertical pulsed electric dipole. It is assumed that the surface slopes are small compared to unity. For this source, the zero-, first- and second-order approximations of the vertical component of the surface field are asymptotically transformed back into coordinate space. These solutions are in the form of ground waves with modified surface impedances