Matrix operators of convolution type in the problems of diffraction by wedges of finite size (Q2754772)

An ideally conducting wedge \(0\leq r\leq c\), \(\varphi=0\), \(\varphi= 2\gamma\), \(-\infty<z<\infty\) is irradiated by a TM-wave independent of \(z\). Time dependence is \(\exp(-i\omega t)\). The electrodynamics problem is reduced to the Dirichlet problem for the Helmholtz equation for the electric component of the scattered field, which satisfies boundary conditions, the radiation condition and the Meixner condition at the edge. This component is represented by a Lebedev integral. Assuming that the field is excited by a filament of in-phase current parallel to its edge and passing through a point with coordinates \(r_0, \varphi_0\), the integral equation of the problem, containing Bessel functions of imaginary argument and McDonald functions, is given. For the solution of this equation, a semi-inversion technique is proposed. It is based on exact analytical inversion of convolution operators extracted from the original equation and results in infinite systems of algebraic equations. Possible generalization of this approach is demonstrated.