Application of conformal mapping to earth-flattening procedures in radio propagation problems

The propagation of electromagnetic waves around a cylinder is rigorously treated by the application of conformal mapping. The cylinder is surrounded by a nonmagnetic isotropic medium whose dielectric constant is a function of the radius. It is transformed into a Cartesian system filled with a fictitious medium whose dielectric constant and magnetic permeability are functions of height. The wave equations for a TM and a TE wave are given. Wave propagation around a cylinder represents an approximation to wave propagation around a sphere (e.g., the earth) when the influence of the curvature of the sphere perpendicular to the direction of propagation can be neglected. It is shown that the conformal mapping applied to the cylinder yields an earth-flattening technique which agrees with first-order approximations obtained in spherical coordinates. Pryce treats wave propagation around the earth in spherical coordinates using a range transformation suggested by Pekeris and a height transformation suggested by Copson. It will be shown in this paper that both independently proposed transformations follow directly from the application of conformal mapping.