Electromagnetic Wave Propagation Over a Nonparallel Stratified Conducting Medium

Abstract

The objective is to calculate the electromagnetic field of a dipole located over a flat inhomogeneous ground whose upper layer has a variable thickness. Although a formally exact derivation for a lossy dielectric wedge model is available, its complexity prohibits immediate application. Instead, using a geometrical-optical technique, an approximate expression is developed for the effective surface impedance at the ground surface. This method takes full account of multiple reflections within the wedge region but neglects scattering from the wedge apex. Thus, the resulting surface impedance formula is valid everywhere except near the apex of the wedge. With this point kept in mind, the compensation theorem is employed to derive an integral equation for the electromagnetic field produced by a dipole located above this model of a nonparallel stratified ground. Using an appropriate reference field, the resulting integral equation of the Volterra type is solved by an iterative method. The excellent convergence of this procedure is demonstrated for a particular model. This solution yields quantitative results for the ground-wave attenuation function. In the case of a parallel stratified ground, it reduces to earlier known results.

Radiation From Layered Structures of Variable Thickness

“…The solutions are shown to be consistent with the reciprocity relationship. Some of the terms of the full-wave solution can be predicted directly through a geometrical optical approach [Schlak and Wait, 1967]. However, the geometrical-optical approach is limited in its application, since it does not satisfy the reciprocity relationship and the excitation of the surface waves is not accounted for.…”

“…This discrepancy, which decreases with increasing distance between the two dipoles over dissipative overburdens, is attributed to the approximations inherent in the geometrical-optical approach. The experimental results obtained by Schlak and Wait [1967] were not considered sufficiently accurate to resolve this problem. Recently King and Hustig [1971 ], who conducted controlled model experiments using low-loss dielectrics, showed that the surface impedance (which does depend upon the direction of propagation) cannot, in general, be evaluated by the geometrical-optical approach.…”

“…It is precisely these terms that make the full-wave solutions symmetric in W and c• • and therefore consistent with the reciprocity relationships. Schlak and Wait [1967], who had conducted a reciprocity test, show that the numerical values for the mutual impedances between two dipoles over a wedge-shaped overburden are dependent upon the direction of the transmitted wave. This discrepancy, which decreases with increasing distance between the two dipoles over dissipative overburdens, is attributed to the approximations inherent in the geometrical-optical approach.…”

A wave‐optical field solution for a line source over nonparallel stratified earth

“…This is justified when all significant dimensions are small compared with the wavelength. Now if we apply a geometrical-optical argument, such as used by Schlak an Wait [ 1967], the field at P is the sum total of all the ray contributions between the cable and the observer. The first four of these are depicted in Figure 1.…”

“…Notable work in this area is by Bahar [ 1970], who has treated such problems as coupled modes in a waveguide of variable cross section; by Geyer [1972], who has utilized Lebedev-Kontorovich transforms; and by Jones [1970] who solved the differential equations for a step-structure by a purely numerical method. Recently, Balling [ 1971 ] has suggested that the geometrical optical formulation of Schlak and Wait [ 1967] for wedge geometries be generalized to allow for the full spectrum of complex angles in the primary field. We wish to pursue this approach here from the standpoint of electromagnetic induction in a stratified conducting earth.…”