Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia1789210.22363/2312-9735-2018-26-1-39-48Research ArticleOn the Reduction of Maxwell’s Equations in Waveguidesto the System of Coupled Helmholtz EquationsMalykhM Dmalykhmd@yandex.ruSevastianovA Lsevastianov_al@rudn.universitySevastianovL Asevastianov_la@rudn.universityTyutyunnikA Atyutyunnik_aa@rudn.universityPeoples’ Friendship University of Russia (RUDN University)15122018261394828022018Copyright © 2018, Malykh M.D., Sevastianov A.L., Sevastianov L.A., Tyutyunnik A.A.2018<p>The investigation of the electromagnetic ﬁeld in a regular homogeneous waveguide reducesto the investigation of two independent boundary value problems for the Helmholtz equation,corresponding to TE- and TM-modes. In the case of an inhomogeneous waveguide TE- andTM-modes are connected to each other, which in numerical experiments can not always be fullytaken into account. In this paper we show how to rewrite the Helmholtz equations in vectorform to express this relationship explicitly.In the article the cylindrical waveguide with perfectly conducting walls is considered, but wedont make any assumptions about ﬁlling of waveguide. The introduced approach is based ontwo-dimensional analogue of the theorem known in the theory of elastic bodies as the Helmholtzdecomposition. On its basis, we introduce four potentials, instead of two potentials, usuallyused in the theory of hollow waveguides. It is proved that any solution of Maxwells equationsin a waveguide that satisﬁes the boundary conditions of ideal conductivity on the boundariesof a waveguide can be represented with the help of these potentials. The system of Maxwellsequations is written with respect to these potentials and it is shown that this system has theform of two independent Helmholtz equations in the case of a hollow waveguide.</p>SagemathwaveguideMaxwell’s equationsHelmholtz EquationnormalmodesSagemathволноводуравнения Максвеллауравнение Гельмгольцанор-мальные моды[A. A. Samarskiy, A. N. Tikhonov, On the Representation of a Field in a Waveguide in the Form of a Sum of Fields TE and TM, Zhurnal tekhnicheskoy ﬁziki 18 (7) (1948) 959–970, in Russian.][A. G. Sveshnikov, The Basis for a Method of Calculating Irregular Waveguides, USSR Computational Mathematics and Mathematical Physics 3 (1) (1963) 219–232.][A. G. Sveshnikov, A Substantiation of a Method for Computing the Propagation of Electromagnetic Oscillations in Irregular Waveguides, USSR Computational Mathematics and Mathematical Physics 3 (2) (1963) 413–429.][A. G. Sveshnikov, Incomplete Galerkin Method, DAN USSR 236 (5) (1977) 1076–1079, in Russian.][A. S. 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