Soft Half Plane Research Articles

The factorization process in the modified theory of physical optics

The factorization process is integrated to the procedure of the modified theory of physical optics. The scattering integral for the diffraction problem of waves by a soft half-plane is taken into account. The values of the kernel is obtained for two special values of the scattering angle β. By considering the behavior of the integrand at these special values, a technique for the factorization process is determined. The new approach is applied to the problems of hard, impedance and conductive half-screens.

Diffraction of a spherical acoustic wave due to the coupling of pressure release and absorbing half planes in a moving fluid

We investigate the diffraction of a spherical acoustic wave (emanating due to a point source) from the coupling of a pressure release half plane characterized by zero impedance (soft half plane) and an absorbing (locally reacting) half plane satisfying Myers’ impedance condition in a moving fluid. Myers’ condition (a generalization of Ingard’s impedance condition) is now the accepted form of the boundary condition for impedance barriers with flow and yields a correct form of the diffracted field. The method of solution is based on the Wiener–Hopf technique and asymptotic approximation methods.

Three-dimensional scalar beam diffraction by a half plane

The complex source point technique greatly simplifies the inclusion of source directivity in scattering and diffraction solutions. By replacing its coordinates by appropriate complex coordinates, an omnidirectional source becomes a beam source. Any diffraction solution for the source becomes the corresponding solution for beam diffraction. Geometrical diffraction theory solutions for local beam sources can be most easily obtained in this way. Here the complex source point technique is applied to a canonical solution in geometrical diffraction theory: three-dimensional scalar beam diffraction by a half plane. First, a general three-dimensional solution for a scalar point source and its evolution to a beam source is reviewed. An exact uniform asymptotic solution for point and beam source diffraction by a rigid or soft half plane is then given for arbitrary incidence in three dimensions. A method for numerical evaluation of the diffraction integrals is shown. In this method the Fresnel integrals with complex arguments are converted to complementary error functions with complex arguments, for which efficient computer subroutines are available. The procedure is illustrated by a numerical example and three-dimensional numerical plots are presented. Electromagnetic (vector) field three-dimensional beam diffraction by half planes will yield to the same approach. So also will three-dimensional beam diffraction by wedges using the uniform geometrical theory of diffraction but the resulting expressions are more complicated.

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

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.