Spectral domain solution and asymptotics for the diffraction by an impedance cone

The problem of diffraction of a plane wave by a cone with impedance condition on its surface is studied, for a scalar wave field satisfying the Helmholtz equation. The integral Kontorovich-Lebedev, Sommerfeld-Maliuzhinets and Fourier transformations are exploited to investigate the problem and to separate the radial variable. The problem in question is reduced to that for a spectral function satisfying a Laplace-Beltrami type equation on the unit sphere with a hole cut out by the conical surface. An impedance type boundary condition with a nonlocal impedance operator is determined on the boundary of the hole. Then the problem for the spectral function can be transformed to a second kind integral equation with a nonoscillatory kernel. In the particular case of a circular impedance cone the problem is simplified. As an application, a closed form expression for the scattering diagram (or pattern) is deduced in the narrow cone approximation. The leading and first correction terms are represented.