Effective Earth's Radius for Radiowave Propagation Beyond the Horizon

Abstract

The geometrical optical arguments underlying the equivalent earth's radius approximation are extended to non-horizontal rays. The appropriate wave equation for a non-uniform, but spherically symmetric, region is derived in a natural way. The notion of a ``primary field'' with the form eikR/R at the source is dropped in favor of a solution to the wave equation in terms of a Green's Function. The formal solution thus obtained is seen to contain the known solutions for a uniform medium. For the non-uniform case, the solutions to the radial equation are found by a technique due to Langer. These functions account for the index variation near the earth without making unwarranted assumptions about the behavior at greater heights. The formal series obtained is summed by the Watson technique. The first term in this series (lowest mode) alone determines the field at large distances and indicates that one may account for standard atmospheric refraction by using an effective earth's radius.