Electromagnetic Wave Propagation Over a Constant Impedance Plane

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Abstract
The classical Sommerfeld problem of electromagnetic wave propagation over a flat earth is reformulated by using the electromagnetic compensation theorem. The solution is carried through without recourse to the usual spectral representation of the potential function, and therefore complicated contour integrations are avoided in the subsequent evaluation. Using reciprocity and duality, the complete electromagnetic fields are calculated for elemental vertical or horizontal dipoles of both electric and magnetic types, which are elevated above an arbitrary but passive surface impedance such as that corresponding to homogeneous, horizontally stratified, or uniformly rough surfaces. The results are identical to the little known but more accurate work of Norton, and under certain approximations they reduce to the more well‐known results of Norton, Hufford, Bremmer, and others. The restrictions on the impedance properties of the surface are given, and certain direction finding errors are discussed. It is shown that the index of refraction contrast between the upper and lower medium need not be large.