Spherical-multipole analysis of scattering by finite and semi-infinite elliptic cones

Abstract

The interest in scattering by elliptic cones is mainly motivated by the role which the diffraction coefficients play in asymptotic high frequency theories like the GTD (geometrical theory of diffraction) or the uniform theory of diffraction. Since the elliptic cone possesses a two-parametric tip, the pertinent field must contain the information of a very general tip diffraction coefficient (TDC). This TDC is obtained by analysis of the field scattered by a semi-infinite elliptic cone and is then validated by comparing the exact field scattered by a finite elliptic cone with the corresponding complete GTD-result (including the TDC). By applying the spherical-multipole technique the exact solutions for the scattering of EM waves by a finite as well as by a semi-infinite perfectly conducting elliptic cone are deduced. The vector problems are reduced to scalar problems. Products of spherical Bessel functions and so-called Lame products, the vector spherical-multipole functions can be derived, which form a complete base to construct any EM field outside the sources. The boundary-value problem for the finite elliptic cone is formulated as a standard two-domain problem. In each domain the EM field is described by an appropriate spherical multipole expansion, while the corresponding multipole amplitudes are found by enforcing the boundary- and continuity conditions of the field and by employing the orthogonality relations of the vector spherical-multipole functions. The problem of plane wave scattering by a semi-infinite elliptic cone is solved via the pertinent dyadic Green's function. Suitable sequence transformations are applied which enforce the convergence and yield the limiting value for these series.