Study of the Kernels of Integral Equations in Problems of Wave Diffraction in Waveguides and by Periodic Structures
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GRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
Study of the Kernels of Integral Equations in Problems of Wave Diffraction in Waveguides and by Periodic Structures A. S. Il’inskii1∗ and T. N. Galishnikova1∗∗ 1
Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
Received March 16, 2020; revised March 16, 2020; accepted May 14, 2020
Abstract—We consider the problem of diffraction of a waveguide wave by an impedance rod in a rectangular waveguide with perfectly conducting walls and the problem of diffraction of a plane two-dimensional electromagnetic wave and the field of a point source by an evenly spaced array formed by infinite cylinders of arbitrary cross-section with perfectly and well conducting walls. Both problems are reduced to solving contour Fredholm integral equations. Such reduction is based on using the Green’s function of an empty planar waveguide and a quasiperiodic Green’s function, which are infinite series in the eigenfunctions of the cross-section of the planar waveguide and in the eigenfunctions satisfying the Floquet conditions. To calculate the kernels of the resulting integral equations, depending on both the Green’s functions themselves and their derivatives, we have developed special algorithms to improve the convergence of the series and explicitly isolate the logarithmic singularity occurring in the series. DOI: 10.1134/S0012266120090074
INTRODUCTION The issues of mathematical modeling of problems on propagation of steady-state oscillations in irregular waveguides are dealt with in a large number of publications. The irregularity of a waveguide can be due to various causes, for example, variations in its cross-section area, anisotropic properties of the filling medium, and the presence of a metallic, dielectric, ferrite, or another inclusion inside the waveguide. Dedicated methods are being developed [1–8] to study specific types of waveguide irregularities. Numerous papers also deal with open arrays formed by cylindrical structures placed in various media. Depending on the geometry of the physical model and its electrodynamic parameters, there exist various means to investigate these arrays [8–11]. In the present paper, we consider the problems of diffraction of electromagnetic waves by an inductive rod in a rectangular waveguide and by an evenly spaced array formed by infinite cylinders of arbitrary cross-section. These problems are explored by a unified method, namely, the method of integral equations derived using the Green’s function of a planar waveguide and a quasiperiodic Green’s function, both represented in the form of infinite conditionally convergent series. To construct computational algorithms for solving these integral equations by approximation and collocation methods, special series convergence acceleration algorithms have been developed that allow isolating the logarithmic singularity contained in these equations. The algorithms constructed permit one to calculate not only the integral characteristics of the diffraction field
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