Electromagnetic scattering by a smooth convex impedance cone

摘要

The problem of the diffraction of an electromagnetic plane wave by a convex cone of arbitrary smooth cross-section with impedance (Leontovich) boundary conditions is studied. The vector problem is reduced to that for the Debye potentials. By means of Kontorovich-Lebedev integrals, two spectral functions are introduced and the corresponding boundary value problem is formulated. The spectral functions for the potentials are found to satisfy the Helmholtz equations on the unit sphere and to be coupled through non-traditional boundary conditions of the impedance type with shifts on the spectral variable. The use of the Green theorem permits us to establish an integral formulation of the boundary value problem for the spectral functions. The formal asymptotic solution of the problem is then given for the case of a narrow cone. For this, two different methods are given: a method of perturbation applied to the spectral integral equations and an adaptation of the method of matching the asymptotic series in spectral domain. Both methods lead to the same closed-form result for the leading term of the scattering diagram asymptotics.