Optical approximation for residue series of terminal gain in radiowave propagation over inhomogeneous earth

Abstract
The method of geometric-optical approximation is applied to the residue series of the obstacle gain, or ‘terminal gain’, which has been previously introduced. This approximation leads to a simple formula for the terminal gain which consists of two terms representing, respectively, the direct ray and the reflected ray from the ground, in the same way as does the corresponding formula of the field strength in the case of radiowave propagation over homogeneous Earth. The optical values obtained from this formula, and the exact values given by the residue series, are numerically compared with the corresponding flat-earth values (obtained by the formula previously published) for various obstacle heights and distances in the case of a ridge and a bluff on homogeneous earth. A very good agreement is found in the region slightly before the horizon, when the obstacle is high enough from the ground, particularly in the ridge case. When the obstacle height is not large enough, on the other hand, the flat-earth values show a very good agreement with the residue values. Finally, an empirical formula is introduced to estimate the gain caused by a ridge or a bluff on homogeneous earth, in terms of the values found from the diagrams of the terminal gains, without important restrictions for its location in the wavepath.

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