Edge Excited Rays on Convex and Concave Structures – 2d and 3d Problems

In this paper, we present a review on edge excited rays on convex and concave perfectly conducting structures with sharp edges or curvature discontinuities. The incident wave is either a space wave, a creeping wave or a whispering gallery wave. The observation point may be at large distance from the structure or located on its surface. In addition to more classical formulas, the paper contains also some new solutions established by the author. 1. EDGE TYPE SINGULARITY IN A CONVEX SURFACE Formulas for edge excited creeping waves by an incident space wave or a creeping wave with emission of a space wave at large distance, which are known as hybrid diffraction coefficients, have been derived by Pathak and Koujoumjian [1], Albertsen [2], Molinet [3], Albertsen and Christiansen [4]. They have been extended to curved edges in a 3D surface by Hill and Pathak [5]. A uniform solution valid at grazing incidence has been derived by Liang [6]. Michaeli [7] obtained an asymptotic expansion of the solution established by Idemen and Felsen [8] for the canonical diffraction problem of a plane wave by a truncated cylindrical shell which he extended to the wedge problem at grazing incidence. He showed that the field in the transition region is described by new Fock functions involving incomplete Airy functions. A threedimensional version of Michaeli’s uniform solution, valid at grazing incidence has been established by Molinet [9]. When the observation point is located on the surface, or reciprocally when the source is located on the surface, the Fock functions which are involved in the former solutions are replaced by Nicholson’s functions. At grazing incidence new transition functions have been established by Michaeli [10]. The field on the surface or the corresponding asymptotic currents have been compared on 2D geometries with the Method of Moments results and an excellent agreement has been obtained [11]. 2. CURVATURE DISCONTINUITY IN A CONVEX SURFACE The asymptotic theory of diffraction by a discontinuity in curvature on a smooth convex perfectly conducting surface was initially restricted to situations where both the incident and the diffracted rays were bound to make sufficiently wide angles with the tangent plane at the point of diffraction on the discontinuity [12] [13] [14]. More recently, the theory has been extended by Michaeli [15] to cover the case where one of the two rays is nearly tangent to the surface or represents a creeping ray. A further augmentation of the theory has been established by Michaeli and Molinet [16] who extended the solution to the case where both rays are now permitted to be nearly tangent to the surface or represent a creeping ray, provided both of them are situated on the same side of the discontinuity in question. This situation excludes forward scattering, but includes back scattering. Recently, Michaeli’s procedure for grazing incidence on a curved convex wedge has been extended by Molinet [17] to a curvature discontinuity and gives the creeping wave excited at grazing incidence in forward scattering. The same approach gives also the field on the surface for gazing incidence. 3. EDGE TYPE SINGULARITY IN A CONCAVE SURFACE The formulas for the diffraction of a Whispering Gallery Mode by the edge of a thin cylindrically curved surface have been established by Idemen and Felsen [8] using the solution of the canonical problem of the curved shell obtained by the same authors and the Green’s function of a line source located on a concave cylindrical surface established earlier by Ishihara, Felsen and Green [18]. Later on Idemen [19] established modal edge diffraction coefficients and Idemen and Büyükaksoy [20] obtained formulas for surface currents induced by a line source located near either the convex or the concave side of a cylindrical reflector. An interesting expression has been obtained in [8] and [20] for the sum of the currents excited on both faces in a thin cylindrical shell : the variation of the sum with respect to the position is given by the incomplete Gamma function. Starting with the integral form of the Green’s function established in [18] and [22] and performing a spectral decomposition of the field radiated by a line source in front of a concave surface, in the vicinity of the edge, new formulas for the diffraction coefficients of edge excited whispering gallery waves have been obtained by Molinet [23].