On the Representation of Electromagnetic Fields in Discontinuously Filled Closed Waveguides by Means of Continuous Potentials

Abstract

A closed waveguide of a constant cross section \(S\) with perfectly conducting walls is considered. It is assumed that its filling is described by function \(\varepsilon \) and \(\mu \) invariable along the waveguide axis and piecewise continuous over the waveguide cross section. The aim of the paper is to show that, in such a system, it is possible to make a change of variables that makes it possible to work only with continuous functions. Instead of discontinuous transverse components of the electromagnetic field \({\mathbf{E}}\), it is proposed to use potentials \({{u}_{e}}\) and \({{v}_{e}}\) related to the field as \({{{\mathbf{E}}}_{ \bot }} = \nabla {{u}_{e}} + \tfrac{1}{\varepsilon }\nabla {\kern 1pt} '{{v}_{e}}\) and, instead of discontinuous transverse components of the magnetic field \({\mathbf{H}}\), to use the potentials \({{u}_{h}}\) and \({{v}_{h}}\) related to the field as \({{{\mathbf{H}}}_{ \bot }} = \nabla {{v}_{h}} + \tfrac{1}{\mu }\nabla {\kern 1pt} '{{u}_{h}}\). It is proven that any field in the waveguide admits the representation in this form if the potentials \({{u}_{e}},{{u}_{h}}\) are elements of the Sobolev space \(\mathop {W_{2}^{1}}\limits^0 (S)\) and \({{v}_{e}},{{v}_{h}}\) are elements of the space \(W_{2}^{1}(S)\).