# TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. Reducing the Problem to SLAE-II

**Theorem**

**1.**

**Theorem**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

## 4. Convergence of the Truncation Method

- Uniqueness of the solution to the exact equation;
- Uniqueness of the solution to the approximating equations;
- Error estimation of the approximate solution;
- Completeness of space X.

**Theorem**

**4.**

- 1.
- operator F has a bounded left inverse;
- 2.
- starting from a certain number (at least), the solutions $\overline{x}$ of the approximating equations are unique;
- 3.
- the sequence $\overline{y}$ S-converges to y;
- 4.
- sequence $\overline{F}$ strongly $TS$-converges to F;
- 5.
- sequence $ST$-converges uniformly to I;
- 6.
- operators S are uniformly bounded.

**Lemma**

**3.**

**Theorem**

**5.**

## 5. Resonance Frequencies

## 6. Results of Model Computations

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Cowley, J.M. Diffraction Physics, 3rd ed.; American Elsevier: New York, NY, USA, 1975; p. 410. [Google Scholar]
- Lewin, L. Theory of Waveguides; Newnes-Butterworths: London, UK, 1975; p. 346. [Google Scholar]
- Mittra, R.; Lee, S.W. Analytical Techniques in the Theory of Guided Waves, 1st ed.; Macmillan: New York, NY, USA, 1974; p. 327. [Google Scholar]
- Peterson, A.F.; Scott, L.; Mittra, R. Computational Methods for Electromagnetics; Wiley: Hoboken, NJ, USA; IEEE: Piscataway, NJ, USA, 1998; p. 592. [Google Scholar]
- Il’inskii, A.S.; Kuraev, A.A.; Slepyan, G.Y. Semi-reversal method in problems of wave diffraction on forks of plane irregular waveguides. Dokl. AN USSR
**1987**, 294, 1345–1349. [Google Scholar] - Poedinchuk, A.E.; Tuchkin, Y.A.; Shestopalo, V.P. The method of the Riemann-Hilbert problem in the theory of diffraction by shells of arbitrary cross section. J. Comput. Math. Math. Phys.
**1998**, 38, 1314–1328. [Google Scholar] - Poyedinchuk, A. Discrete spectrum of one class of open cylindrical structures. Dopovidi Ukr. Akad. Nauk
**1983**, 48–52. [Google Scholar] - Koshparenok, V.; Melezhik, P.; Poyedinchuk, A.; Shestopalov, V. Spectral theory of open two-dimensional resonators with dielectric inclusions. USSR Comp. Math. Math. Phys.
**1985**, 25, 151–161. [Google Scholar] [CrossRef] - Poyedinchuk, A.Y.; Tuchkin, Y.A.; Shestopalov, V.P. Diffraction on Curved Strips. IEEJ Trans. Fundam. Mater.
**1993**, 113, 139–146. [Google Scholar] [CrossRef] [PubMed] - Poyedinchuk, A.Y.; Tuchkin, Y.A.; Shestopalov, V.P. New numerical-analytical methods in diffraction theory. Math. Comput. Model.
**2000**, 32, 1029–1046. [Google Scholar] [CrossRef] - Stefanidou, E.; Vafeas, P.; Kariotou, F. An Analytical Method of Electromagnetic Wave Scattering by a Highly Conductive Sphere in a Lossless Medium with Low-Frequency Dipolar Excitation. Mathematics
**2021**, 9, 3290. [Google Scholar] [CrossRef] - Vinogradov, S.S.; Tuchkin, Y.A.; Shestopalov, V.P. Investigation of summation equations with kernel in the form of Jacobi polynomials. Dokl. AN USSR
**1980**, 253, 318–321. [Google Scholar] - Koshparenok, V.N.; Shestopalov, V.P. Rigorous solution of the problem of excitation of two circular cylinders with longitudinal slots. J. Comput. Math. Math. Phys.
**1978**, 18, 1196–1213. [Google Scholar] - Radin, A.M.; Shestopalov, V.P. Diffraction of a plane wave by a sphere with a circular hole. J. Comput. Math. Math. Phys.
**1974**, 14, 1232–1243. [Google Scholar] - Vinogradova, E.D.; Smith, P.D. Q Factor Enhancement of Open 2D Resonators by Optimal Placement of a Thin Metallic Rod in Front of the Longitudinal Slot. Mathematics
**2022**, 10, 2774. [Google Scholar] [CrossRef] - Shestopalov, Y. Cloaking: Analytical theory for benchmark structures. J. Electr. Waves Appl.
**2020**, 35, 485–510. [Google Scholar] [CrossRef] - Shestopalov, Y. Resonance Scattering by a Circular Dielectric Cylinder. Radio Sci.
**2021**, 56, 1–15. [Google Scholar] [CrossRef] - Afzal, M.; Akhtar, N.; Alkinidri, M.O.; Shutaywi, M. A Mode-Matching Tailored-Galerkin Approach for Higher Order Interface Conditions and Geometric Variations. Mathematics
**2023**, 11, 755. [Google Scholar] [CrossRef] - Il’inskii, A.S.; Shestopalov, Y. Applications of the Methods of Spectral Theory in the Problems of Wave Propagation; Moscow University Press: Moscow, Russia, 1989; p. 184. [Google Scholar]
- Veselov, G.I.; Temnov, V.M. On the application of the reduction method in solving algebraic systems in some diffraction problems. J. Comput. Math. Math. Phys.
**1984**, 24, 63–69. [Google Scholar] [CrossRef] - Il’inskii, A.S.; Fomenko, E.Y. Investigation of infinite-dimensional systems of linear algebraic equations of the second kind in wave guide diffraction problems. Comput. Math. Math. Phys.
**1991**, 31, 1–11. [Google Scholar] - Abgaryan, G.V. Finite Element Method and Partial Area Method in One Diffraction Problem. Lobachevskii J. Math.
**2022**, 43, 1228–1235. [Google Scholar] [CrossRef] - Pleshchinskii, N.; Abgaryan, G.V.; Vildanov, B. On Resonant Effects in the Semi-Infinite Waveguides with Barriers. Lect. Notes Comput. Sci. Eng.
**2021**, 141, 391–401. [Google Scholar] - Abgaryan, G.V. Electromagnetic Wave Diffraction on a Metal Diaphragm of Finite Thickness. Lobachevskii J. Math.
**2021**, 42, 1327–1333. [Google Scholar] [CrossRef] - Abgaryan, G.V. On the Resonant Passage of Electromagnetic Wave through Waveguide with Diaphragms. Lobachevskii J. Math.
**2020**, 41, 1315–1319. [Google Scholar] [CrossRef] - Abgaryan, G.V.; Pleshchinskii, N.B. On Resonant Frequencies in the Diffraction Problems of Electromagnetic Waves by the Diaphragm in a Semi-Infinite Waveguide. Lobachevskii J. Math.
**2020**, 41, 1325–1336. [Google Scholar] [CrossRef] - Abgaryan, G.V.; Pleshchinskii, N.B. On the Eigen Frequencies of Rectangular Resonator with a Hole in the Wall. Lobachevskii J. Math.
**2019**, 40, 1631–1639. [Google Scholar] [CrossRef] - Abgaryan, G.V.; Khaybullin, A.N.; Shipilo, A.E. A method for partial estimation of electromagnetic wave diffraction by a longitudinal baffle in an endless waveguide. Univ. Proc. Volga Reg. Phys. Math. Sci.
**2022**, 4, 3–16. [Google Scholar] [CrossRef] - Zhao, M.; He, J.; Zhu, N. Fast High-Order Algorithms for Electromagnetic Scattering Problem from Finite Array of Cavities in TE Case with High Wave Numbers. Mathematics
**2022**, 10, 2937. [Google Scholar] [CrossRef] - Tognolatti, L.; Ponti, C.; Santarsiero, M.; Schettini, G. An Efficient Computational Technique for the Electromagnetic Scattering by Prolate Spheroids. Mathematics
**2022**, 10, 1761. [Google Scholar] [CrossRef] - Antosik, P.; Swartz, C. Matrix Methods in Analysis; Springer: Berlin, Germany, 1985; p. 114. [Google Scholar]
- Baggett, L.W. Functional Analysis; Marsel Dekker: New York, NY, USA, 1992; p. 267. [Google Scholar]
- Lavrent’ev, M.M.; Savel’ev, L.Y.; Balakin, S.V. Matrix operator equations. Sib. J. Ind. Math.
**2006**, 9, 105–124. [Google Scholar] [CrossRef] - Kerimov, M.K.; Skorokhodov, S.L. Some asymptotic formulas for cylindrical Bessel functions. USSR Comput. Math. Math. Phys.
**1990**, 30, 126–133. [Google Scholar] [CrossRef] - Vorobyov, N.N. The Theory of Series; Izd. Nauka: Moscow, Russia, 1979; p. 408. [Google Scholar]
- Pleshchinskii, N.B. Toward an abstract theory of approximate methods for solving linear operator equations. J. Comput. Math. Mat. Phys.
**1990**, 30, 1775–1784. [Google Scholar] - Pleshchinskii, N.B. On the abstract theory of approximate methods for solving linear operator equations. Izv. Vyss. Uchebnykh Zaved. Mat.
**2000**, 3, 39–47. [Google Scholar] - Shestopalov, V.; Shestopalov, Y. Spectral Theory and Excitation of Open Structures; IEE Publisher: London, UK, 1995; p. 412. [Google Scholar]
- Shestopalov, Y. Trigonometric and Cylindrical Polynomials and Their Applications in Electromagnetics. Appl. Anal.
**2020**, 99, 2807–2822. [Google Scholar] [CrossRef] - Shestopalov, Y. Resonance frequencies of arbitrarily shaped dielectric cylinders. Appl. Anal.
**2021**, 62, 1–15. [Google Scholar] [CrossRef] - Vinogradov, S.S.; Tuchkin, Y.A.; Shestopalov, V.P. Effective solution of pairwise summation equations with kernel in the form of associated Legendre functions. Dokl. AN USSR
**1978**, 242, 80–83. [Google Scholar] - Shestopalov, Y.; Smirnov, Y.; Chernokozhin, E. Logarithmic Integral Equations in Electromagnetics; VSP: Utrecht, The Netherlands, 2000. [Google Scholar]

**Figure 2.**Dependence of the modulus of the coefficients (

**a**) ${z}_{0}$ and (

**b**) ${z}_{1}$ on the spectral parameter $\kappa $, which varies over (

**a**) $\left(\right)$ and (

**b**) $\left(\right)$ with a step $0.001$.

**Figure 3.**Dependence of the modulus of the coefficients (

**a**) ${z}_{2}$ and (

**b**) ${z}_{3}$ on the spectral parameter $\kappa $, which varies over (

**a**) $\left(\right)$ and (

**b**) $\left(\right)$ with a step $0.001$.

**Figure 4.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /30]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{01},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{02}$, which correspond to approximate values of the first two zeros ${\mu}_{01}\approx 2.404,{\mu}_{02}\approx 5.52$ of the Bessel function ${J}_{0}\left(z\right)$.

**Figure 5.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /30]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{11},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{12}$, which correspond to approximate values of the first two zeros ${\mu}_{11}\approx 3.831,{\mu}_{12}\approx 7.015$ of the Bessel function ${J}_{1}\left(z\right)$.

**Figure 6.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /30]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{21},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{22}$, which correspond to approximate values of the first two zeros ${\mu}_{21}\approx 5.135,{\mu}_{22}\approx 8.417$ of the Bessel function ${J}_{2}\left(z\right)$.

**Figure 7.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /2]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{01},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{02}$, which correspond to approximate values of the first two zeros ${\mu}_{01}\approx 2.404,{\mu}_{02}\approx 5.52$ of the Bessel function ${J}_{0}\left(z\right)$.

**Figure 8.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /2]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{11},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{12}$, which correspond to approximate values of the first two zeros ${\mu}_{11}\approx 3.831,{\mu}_{12}\approx 7.015$ of the Bessel function ${J}_{1}\left(z\right)$.

**Figure 9.**Distribution of the amplitude of the electromagnetic field component ${E}_{z}$ in the cross-section of the circular slotted scatterer for the $\mathcal{N}=[0,\pi /2]$ and lowest modes $\left(\mathbf{a}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{21},\left(\mathbf{b}\right)\phantom{\rule{0.166667em}{0ex}}{E}_{22}$, which correspond to approximate values of the first two zeros ${\mu}_{21}\approx 5.135,{\mu}_{22}\approx 8.417$ of the Bessel function ${J}_{2}\left(z\right)$.

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**MDPI and ACS Style**

Abgaryan, G.V.; Shestopalov, Y.V.
TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder. *Mathematics* **2023**, *11*, 1991.
https://doi.org/10.3390/math11091991

**AMA Style**

Abgaryan GV, Shestopalov YV.
TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder. *Mathematics*. 2023; 11(9):1991.
https://doi.org/10.3390/math11091991

**Chicago/Turabian Style**

Abgaryan, Garnik V., and Yury V. Shestopalov.
2023. "TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder" *Mathematics* 11, no. 9: 1991.
https://doi.org/10.3390/math11091991