The Mode Theory of VLF Radio Propagation for a Spherical Earth and a Concentric Anisotropic Ionosphere

The theory of the propagation of electromagnetic waves in the space between a spherical conducting earth and a concentric ionospheric reflecting layer is treated. The problem is idealized to the extent that, at the boundaries of this spherical wave guide, the tangential field components are related in a prescribed way. In the case of the ground, the relation involves a scalar surface impedance while, for the ionosphere, it is necessary to employ a reflection coefficient or surface impedance in the form of a matrix. On this basis a solution is sought for the total field produced by a vertical electric dipole. Explicit results are obtained under the assumption of azimuthal symmetry. By applying the Watson transformation a representation in terms of guided waves is obtained. This form of the solution was used previously by the author to obtain specific results.