Chapter 9 - On The Propagation of Long Wavelength Terrestrial Radio Waves—Two Theoretical Techniques

Abstract

A multiplicity of waves or modes of electrical vibration propagate in the space between a terrestrial sphere and a concentric isotropic plasma reflector. Such a theoretical model is used to represent the earth and the ionosphere, and the propagation for such a model can be treated theoretically at frequencies less than approximately 50 kc/s by a direct summation of the classical series of zonal harmonics. Indeed, this is the rigorous solution to such a propagation problem. An alternate method is found by summing the residues at the poles in the complex plane of integration of a contour integral—a procedure known as the Watson transformation. This alternate solution to the problem is exactly equivalent to the series of zonal harmonics by the theory of functions of the complex variable. However, the latter method is complicated by the search for the roots of a transcendental determinant equation, the elements of which comprise Hankel functions of complex order and argument. The procedure resulting from such a transformation can be applied to long wavelengths, especially l.f. (>50 kc/s or 30–300 kc/s). The zonal harmonics procedure, although simple, is more appropriate at frequencies below l.f., such as v.l.f. and e.l.f. The areas of overlap of the two procedures provide an independent computation check on the entire analysis.