On the scheme for seeking the solution to a system of Maxwell's equations in a spherically symmetric model of the Earth-ionosphere waveguide

A classical scheme for solving the boundary-value problem for a system of Maxwell's equations in a spherically stratified model of the Earth-ionosphere waveguide has been known since the beginning of the 20th century, and implies the following. The elementary electric or magnetic dipoles are usually assumed as emitters. In each of these cases the Hertz potentials are introduced, which satisfy partial differential equations with separable variables. Solutions to this equations are sought either in the form of a series in terms of eigenfunctions of the angular operator or in the form of a series in terms of eigenfunctions of the radial operator. The transition from one series to the other is accomplished through the Watson transformation. Attempts to generalize this scheme to the case of more sophisticated waveguide and emitter models led the author to the conclusion that the scheme should be modified. To elucidate the essence of the problem, a very simple model is considered. Unlike the classical scheme, we abandoned the idea of introducing any potentials, and formulated boundary-value problems for the components of fields. The objective of this paper is to devise a reasonably rigorous mathematical scheme for seeking the formal expressions for solving a system of Maxwell's equations.