THE DIFFRACTION OF A DIPOLE FIELD BY A PERFECTLY CONDUCTING HALF-PLANE

Abstract

The correct solution for the particular case of an electric dipole with axis normal to the half plane was obtained in 1953. The procedure was to represent the source field as an angular spec· trum of plane waves, and thereby synthesize the solution from the known (Sommerfeld) solution for the diffraction of a plane wave. It was found that the expressions for the field components are compose<! of terms which are derivatives of the scalar solutions for the diffraction of a point­ source field by an acoustically hard or soft half plane, plus terms corresponding to a source-free solution of Maxwell's equations. Many of the later investigations result~d in this same type of representation of the solution, and the additive or source-free contribution essential for the cor­ rect edge behavior is now known for all orientations of dipoles, both electric and magnetic (Van­ dakurov, 1954; Woods, 1957; Williams, 1957; Jones, 1964). In contrast with this mode of solution, most of the Russian literature has been directed at a representation of the Hertz vector in the form of a Sommerfeld contour integral. S~veral quite elegant results for the exact solutions appropriate to electric and magnetiC dipoles have been obtained, as well as various uniform asymptotic developments of the solutions (Malyuzhinets and Tuzhilin, 1963; Tuzhilin, 1964). Nevertheless, some of the elementary expressions that are deducible from these do not seem to have appeared in the literature, and, in addition, the rela­ tionships between the results of the two contrasting types of representation have not been fully explored. It is the purpose of this paper to unify thes·e forms of representation of the solution, and, at the same time, to give some additional results which are either new or do not appear in the open literature. An attempt has been made throughout to correct the numerous, and sometimes elusive, errors that have crept into the various treatments of the problem. For brevity, much of the analysis will be omitted.