Direct methods for the problem of an electromagnetic resonator

Abstract

IN boundary value problems of electrodynamics the representation of fields is given in the form of a series of eigenvector functions, arising from some electromagnetic problem which has an analytical solution. On this basis various direct methods are constructed. In particular we can study in such a way free and forced vibrations of fields of electromagnetic resonators in cases where the corresponding boundary value problem is, as we usually say, “irregular”, i.e. does not permit of a separation of the variables, the reason for which can be either the complicated form of the region or its non-uniformity, and also anisotropy of the internal medium (or all these factors combined). In connection with the use of high speed computers, direct methods for problems of electromagnetic oscillations take on a significance which they could not have earlier, because of the great complexity of the subjects which offered real interest, and now they attract the particular attention of specialists. Up to recent times, however, questions on the basis of direct methods in electrodynamics have been set on one side. Therefore, the work of Sveshnikov [1], who investigated the convergence of the process of reducing the problem of the perturbation of an irregular waveguide to a system of ordinary differential equations is of special interest. As regards this paper, its aim is to reduce to a system and develop methods of the Galerkin — Ritz type for irregular resonators, previously put forward by the author [2–5]. Here the basis of the methods is tested, partly by using the methodological features contained in [1].