Introduction to the Theory of VLF Propagation

This paper is a self-contained exposition of the conventional theory of propagation of VLF radio waves. The model is a spherical earth surrounded by a concentric ionosphere. Although the model is highly idealized, the theory is still quite involved. The complexities arise from the long wavelengths inherent in such problems. In Section I the elementary geometrical or hop theory for VLF is considered. This includes a discussion of ionospheric reflection coefficients and the influence of multiple reflections. The validity of the model is established by a comparison with experimental data. In Section II, the diffractive corrections near the caustics of the ray systems are obtained. These provide a means to extend the theory to points near and beyond the horizon point for the individual hops or rays. In Section III, the waveguide mode theory is expounded. A number of approximations and simplifications are introduced in order to illustrate principles. References to more detailed analyses which use higher order approximations are given. Finally, in Section IV, the influence of a stratified ionosphere is treated by using a two-layer model. The material in this paper was included in the Summer (1961) Lecture Course on Radio Propagation given at the Boulder Laboratories of the National Bureau of Standards, and in earlier graduate courses, at the University of Colorado.