The Application of a Variational Method to the Calculation of Radio Wave Propagation Curves for an Arbitrary Refractive Index Profile in the Atmosphere

Abstract

The problem of determining the field strength produced by a microwave transmitter beyond the horizon has been intensively studied in recent years. Charts and tables, which are based on approximate solutions of the wave equation obtained by the phase-integral method, have been produced (Booker and Walkinshaw 1946), from which the field strength can be calculated when the modified refractive index of the air (the M-curve) varies with height according to a power law. In many cases, however, the M-curve cannot be adequately represented by a power law, and serious analytical difficulties are encountered if the problem is tackled by the phase-integral method. The basic problem is to find the eigenvalues Dm and eigenfunctions Um of the wave equation d2Um/ds2 + {s + f(s) + Dm}Um = 0, in which f(s), the M-anomaly, is a given function, which tends to zero as s tends to infinity. In this paper a method, previously described by the author (Macfarlane 1947), is applied to determine the first eigenvalue D1 for the class of M-anomalies that can be represented by a series of the form, Σn=1r An exp(-αns). Two numerical examples are given to illustrate the method. They were obtained by fitting one exponential and three exponentials in turn to an M-anomaly calculated from meteorological observations made at Kaikoura, New Zealand. In the first case the M-anomaly is represented by a surface duct and in the second case by an elevated duct. The exponential attenuation rate, which is determined by the imaginary part of D1, is calculated in both cases as a function of the wavelength. Finally, the theoretical height-gain function for a wavelength of 3 metres is calculated from the M-curve observed 47 miles off shore from Ashburton, South Island, New Zealand, in the late afternoon of 4th November 1946. It is found that there is close agreement between theoretical and observed height-gain curves.