Mode decomposition of fields into solenoidal and potential components in a closed waveguide
Abstract
Background. At the end of the last century, it became necessary to solve the problem of excitation of unsteady fields in closed and open electrodynamic structures. In the 1980s, the Department of Theoretical Radiophysics proposed an alternative to the Fourier method, the mode basis method for closed hollow resonators. In this paper, we consider an electrodynamic problem on the excitation of an unsteady field in an open cylindrical waveguide. At the very beginning, the original system of Maxwell's equations is divided into two systems of equations for solenoidal and potential vectors.
Objectives. To show the possibility of applying the mode basis method for superposition of solenoidal and potential fields in a waveguide with dielectric filling.
Materials and methods. When solving some spectral electrodynamic problems for solenoidal and potential components of the electromagnetic field vectors, the orthogonality of these eigenvectors in the waveguide volume is proved, taking into account the condition of field boundedness at the waveguide ends when the longitudinal component of a cylindrical waveguide tends to plus/minus infinity.
Results. Under the assumption of completeness of the found eigenfunctions, the desired field is represented as the sum of the expansion by the elements of the mode basis for solenoidal and potential fields with time-dependent coefficients. The first-order differential equations are obtained for the time coefficients.
Conclusions. In this paper, using the example of an infinite cylindrical waveguide with dielectric and diamagnetic filling, a type of mode basis is proposed, when at the initial stage of solving Maxwell's equations, the vector quantities of the initial problem are represented as the sum of solenoidal and potential components. After proving the orthogonality of the eigenvector functions for the potential and solenoidal fields and assuming their completeness, the paper presents the solutions for waveguide excitation by nonstationary sources in the form of an expansion by eigenfunctions with unknown time coefficients.
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References
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