## Abstract

A theoretical investigation is given of the phenomena arising when vertically polarized radio waves are propagated across a boundary between two homogeneous sections of the earth's surface which have different complex permittivities. The problem is treated in a two-dimensional form, but the results, when suitably interpreted, are valid for a dipole source. The earth's surface is assumed to be flat. In the first part of the paper one section of the earth is taken to have infinite conductivity and is represented by an infinitely thin, perfectly conducting half-plane lying in the surface of an otherwise homogeneous earth. The resulting boundary-value problem is initially solved for a plane wave incident at an arbitrary angle; the scattered field due to surface currents induced in the perfectly conducting sheet is expressed as an angular spectrum of plane waves, and this formulation leads to dual integral equations which are treated rigorously by the methods of contour integration. The solution for a line-source is then derived by integration of the plane-wave solutions over an appropriate range of angles of incidence, and is reduced to a form in which the new feature is an integral of the type G(a, b) = b e$^{ia^{2}}\int_{a}^{\infty}\frac{\text{e}^{-i\lambda ^{2}}}{\lambda ^{2}+b^{2}}$ d$\lambda $, where a and b are in general complex within a certain range of argument. The case when both the transmitter and receiver are at ground-level is considered in some detail. If the receiver is a large 'numerical distance' from the transmitter, further simplification is possible; the results then agree with some previously given by Feinberg, whose method, however, was quite different. The practical adequacy of Millington's graphical technique for deriving attenuation curves of the ground-to-ground field is demonstrated, and the possibility of an increase of field-strength with distance is confirmed. This 'recovery effect' is illustrated by a numerical example in which the phase curve is also shown to rise steeply just beyond the boundary, indicating a phase velocity in this region much greater than that in free space. A different approximate form of the general solution is obtained when the transmitter and receiver are sufficiently elevated; this is used to indicate the validity of the application of height-gain factors over an appreciable range of heights. In the second part of the paper the restriction that one of the earth media should be perfectly conducting is waived. A condition, usually met in practice, is assumed, namely, that the modulus of the complex permittivity of each section of the earth is large. Approximate boundary conditions are then likely to be valid, and their introduction makes possible an analytical treatment on the same lines as before. The solution is again reduced to a form only involving, apart from standard features, integrals of the type G(a, b). Various features of the expression for the ground-to-ground field are examined; in a numerical example the attenuation and phase curves are given, the former being compared with the results of an experiment previously reported by Millington and the agreement shown to be good. The different approximate form of the solution when the transmitter and receiver are sufficiently elevated is briefly considered. Finally, some ramifications of the theory are outlined.