## On the Reduction of Maxwell’s Equations in Waveguidesto the System of Coupled Helmholtz Equations

**Authors:**Malykh M.D., Sevastianov A.L., Sevastianov L.A., Tyutyunnik A.A.**Issue:**Vol 26, No 1 (2018)**Pages:**39-48**URL:**http://journals.rudn.ru/miph/article/view/17892**DOI:**http://dx.doi.org/10.22363/2312-9735-2018-26-1-39-48

#### Abstract

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 wedon’t 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 Maxwell’s 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 Maxwell’sequations 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.

### M D Malykh

**Principal contact for editorial correspondence.**

malykhmd@yandex.ru

Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

### A L Sevastianov

sevastianov_al@rudn.university

Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

### L A Sevastianov

sevastianov_la@rudn.university

Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

### A A Tyutyunnik

tyutyunnik_aa@rudn.university

Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation

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