Reducing the problem of waveguide excitation by currents in cross-section to a system of integral volterra equations

Abstract

The problem of excitation of a cylindrical metal waveguide by a source located in the cross section is considered. We assume that the source is surface currents on a flat, infinitely thin metal plate with a smooth boundary. The plate is connected to the generator of non-harmonic oscillations. The boundary of the cross section of a waveguide filled with a homogeneous dielectric is a closed piecewise-smooth contour. The initial physical problem is formulated as a mixed boundary problem for the system of the Maxwell equations. Components of the desired solution for the problem is presented in the form of a series in two sets of two-dimensional eigenfunctions of the Laplace operator. The first set of the eigenfunctions corresponds to the operator with Dirichlet boundary conditions, the second set to the operator with Neumann boundary conditions. We show that the expansion coefficients of the longitudinal components (components directed along the waveguide axis) of the electric and magnetic intensity vectors must be solutions to the jump problem for a system of telegraph equations. The problem of finding the unknown coefficients of the expansion of the longitudinal component of the vector of electric intensity is reduced to solving a system of the Volterra integral equations of the first kind with respect to the derivatives of the desired coefficients. The unknown coefficients of the expansion of the longitudinal component of the vector of magnetic intensity are found by solving a system of the Volterra integral equations of the second kind.