AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS
Subject and Purpose. Considered in the paper is diffraction of a plane wave by a structure involving a half-plane and two disks. The disks and the half-plane, lying within parallel planes, are assumed to be infinitely thin and perfectly conducting. The problem is to be analyzed for two cases, namely...
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Kaliberda, M. E. Lytvynenko, L. M. Pogarsky, S. A. AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
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Article |
| author |
Kaliberda, M. E. Lytvynenko, L. M. Pogarsky, S. A. |
| author_facet |
Kaliberda, M. E. Lytvynenko, L. M. Pogarsky, S. A. |
| author_sort |
Kaliberda, M. E. |
| title |
AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
| title_short |
AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
| title_full |
AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
| title_fullStr |
AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
| title_full_unstemmed |
AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS |
| title_sort |
operator method for the problem of plane wave diffraction by infinitely thin, perfectly conducting half-plane and two disks |
| title_alt |
ОПЕРАТОРНИЙ МЕТОД У ЗАДАЧІ ПРО ДИФРАКЦІЮ ПЛОСКОЇ ХВИЛІ НА НЕСКІНЧЕННО ТОНКИХ ІДЕАЛЬНО ПРОВІДНИХ НАПІВПЛОЩИНІ ТА ДВОХ ДИСКАХ |
| description |
Subject and Purpose. Considered in the paper is diffraction of a plane wave by a structure involving a half-plane and two disks. The disks and the half-plane, lying within parallel planes, are assumed to be infinitely thin and perfectly conducting. The problem is to be analyzed for two cases, namely for that of both disks located on the same side with respect to the half-plane, and for the other where they are placed on opposite sides against the half-plane. The purpose of the paper is to develop a suitable operator method for performing the analysis of the structure described.Methods and Methodology. The solution to the problem has been sought for within the operator method suggested. The electric field components tangential to the half-plane and the disks are expressed, with the aid of Fourier integrals, via some unknown functions having the sense of amplitudes. The unknown amplitudes shall obey the operator equations formulated in terms of wave scattering operators for individual disks and the sole half-plane.Results. When subjected to certain transformations, the operator equations allow obtaining integral equations relative amplitudes of the spherical waves involved. The integral equations permit investigating scattered wave fields for the cases where the disks stay in the shadow region behind the half-plane or in the penumbra, or else in the region which is illuminated by the incident wave. As has been shown, in the case of plane wave scattering at the edge of the half-plane the resulting cylindrical waves possess non-zero amplitudes even with the disks placed totally in the shadow region, hence not illuminated by the incident plane wave.Conclusions. Making use of an operator method, an original solution has been obtained for the problem of plane wave diffraction by a structure consisting of a perfectly conducting, infinitely thin half-plane and two disks. The operator equations of the problem have been shown to be reducible to integral equations, further solvable numerically with the use of discretization based on quadrature rules. The behavior of far and near fields relative to the disks has been studied for a variety of values of the disk radii and their positions relative to the half-plane.Keywords: half-plane, disk, operator method, diffraction, integral equationsManuscript submitted 25.04.2022Radio phys. radio astron. 2022, 27(3):167-180REFERENCES1. Jones, D.S., 1950. Note on diffraction by an edge. Q. J. Mech. Appl. Math., 3(4), pp. 420—434. DOI:https://doi.org/10.1093/qjmam/3.4.4202. Jones, D.S., 1952. A simplifying technique in the solution of a class of diffraction problems. Q. J. Math., 3(1), pp. 189—196. DOI: https://doi.org/10.1093/qmath/3.1.1893. Copson, E.T., 1946. On an integral equation arising in the theory of diffraction. Q. J. Math., os-17(1), pp. 19—34. DOI: https://doi.org/10.1093/qmath/os-17.1.194. Copson, E.T., 1950. Diffraction by a plane screen. Proc. R. Soc. Lond. A, 202(1069), pp. 277—284. DOI: https://doi.org/10.1098/rspa.1950.01005. Noble, D., 1958. Methods based on the Wiener-Hopf technique for the solution of partial differential equations. London: Pergamon Press.6. Khestanov, R.K., 1970. Diffraction of an arbitrary field on a half-plane. Radiotekhnika i elektronika, 15(2), pp. 289—297 (in Russian).7. Bertoni, H.L., Green, A., Felsen, L.B., 1978. Shadowing an inhomogeneous plane wave by an edge. J. Opt. Soc. Am., 68(7), pp. 983—989. DOI: https://doi.org/10.1364/JOSA.68.0009838. Bouwkamp, C.J., 1950. On the diffraction of electromagnetic waves by small circular disks and holes. Philips Research Reports, 5, pp. 401—422.9. Maixner, J., Andrejewski, W., 1950. Strenge theorie der Beugung ebener elektromagnetischer Wellen an der vollkommen leitenden Kreisscheibe und an der kreisförmigen Öffnung im vollkommen leitenden ebenen Schirm. Ann. Phys., 442(3—4), pp. 157—168. DOI:https://doi.org/10.1002/andp.1950442030510. Nomura, Y., Katsura, S., 1955. Diff raction of electromagnetic waves by circular plate and circular hole. J. Phys. Soc. Jpn., 10(4), pp. 285—304. DOI: https://doi.org/10.1143/JPSJ.10.28511. Lytvynenko, L.M., Prosvirnin, S.L., Khizhnyak, A.N., 1988. Semiinversion of the operator with the using of method of moments in the scattering problems by the structures consisting of the thin disks. Preprint, 19. Institute of Radio Astronomy Academy of Sciences Ukr SSR, pp. 1—8 (in Russian).12. Hongo, K., Naqvi, Q.A., 2007. Diffraction of electromagnetic wave by disk and circular hole in a perfectly conducting plane. PIER, 68, pp. 113—150. DOI: https://doi.org/10.2528/PIER0607310213. Losada, V., Boix, R. R., Horno, M., 1999. Resonant modes of circular microstrip patches in multilayered substrates. IEEE Trans. Microw. Theory Tech., 47(4), pp. 488—498. DOI: 10.1109/22.75488314. Losada, V., Boix, R.R., Horno, M., 2000. Full-wave analysis of circular microstrip resonators in multilayered media containing uniaxial anisotropic dielectrics, magnetized ferrites, and chiral materials. IEEE Trans. Microw. Theory Tech., 48(6), pp. 1057—1064. DOI:https://doi.org/10.1109/22.90474515. Losada, V., Boix, R.R., Medina, F., 2003. Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space. IEEE Trans. Geosci. Remote Sens., 41(5), pp. 988—997. DOI: https://doi.org/10.1109/TGRS.2003.81067816. Di Murro, F., Lucido, M., Panariello, G., Schettino, F., 2015. Guaranteed-convergence method of analysis of the scattering by an arbitrarily oriented zero-thickness PEC disk buried in a lossy half-space. IEEE Trans. Antennas Propag., 63(8), pp. 3610—3620. DOI:https://doi.org/10.1109/TAP.2015.243833617. Lucido, M., Panariello, G., Schettino, F., 2017. Scattering by a zero-thickness PEC disk: A new analytically regularizing procedure based on Helmholtz decomposition and Galerkin method. Radio Sci., 52(1), pp. 2—14. DOI: https://doi.org/10.1002/2016RS00614018. Balaban, M.V., Sauleau, R., Benson, T.M., Nosich, A.I., 2009. Dual integral equations technique in electromagnetic wave scattering by a thin disk. PIER B, 16, pp. 107—126. DOI:https://doi.org/10.2528/PIERB0905070119. Kaliberda, M.E., Lytvynenko, L.M., Pogarsky, S.A., 2022. Electromagnetic wave scattering by half-plane and disk placed in the same plane or circular hole in half-plane. J. Electromagn. Waves Appl., 36(10), pp. 1463—1483. DOI:https://doi.org/10.1080/09205071.2022.203237920. Kaliberda, M.E., Lytvynenko, L.M., Pogarsky, S.A., 2022. Scattering by PEC half-plane and disk placed in parallel planes. Int. J. Microw. Wirel. Technol., pp. 1—11. DOI:https://doi.org/10.1017/S175907872200047221. Muskhelishvili, N.I., 1972. Singular integral equations. Boundary problems of functions theory and their applications to mathematical physics. Wolters-Noordhoff. The Netherlands, Groningen. Revised transl. from Russian.22. Lifanov, I.K., 1996. Singular integral equations and discrete vortices. Utrecht: VSP. DOI:https://doi.org/10.1515/978311092604023. Hills, N.L., Karp, S.N., 1965. Semi-infinite diff raction gratings–I. Commun. Pure App. Math., 18(1—2). pp. 203—233. DOI: https://doi.org/10.1002/cpa.3160180119 |
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Видавничий дім «Академперіодика» |
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2023 |
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http://rpra-journal.org.ua/index.php/ra/article/view/1391 |
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oai:ri.kharkov.ua:article-13912023-06-20T14:17:41Z AN OPERATOR METHOD FOR THE PROBLEM OF PLANE WAVE DIFFRACTION BY INFINITELY THIN, PERFECTLY CONDUCTING HALF-PLANE AND TWO DISKS ОПЕРАТОРНИЙ МЕТОД У ЗАДАЧІ ПРО ДИФРАКЦІЮ ПЛОСКОЇ ХВИЛІ НА НЕСКІНЧЕННО ТОНКИХ ІДЕАЛЬНО ПРОВІДНИХ НАПІВПЛОЩИНІ ТА ДВОХ ДИСКАХ Kaliberda, M. E. Lytvynenko, L. M. Pogarsky, S. A. Subject and Purpose. Considered in the paper is diffraction of a plane wave by a structure involving a half-plane and two disks. The disks and the half-plane, lying within parallel planes, are assumed to be infinitely thin and perfectly conducting. The problem is to be analyzed for two cases, namely for that of both disks located on the same side with respect to the half-plane, and for the other where they are placed on opposite sides against the half-plane. The purpose of the paper is to develop a suitable operator method for performing the analysis of the structure described.Methods and Methodology. The solution to the problem has been sought for within the operator method suggested. The electric field components tangential to the half-plane and the disks are expressed, with the aid of Fourier integrals, via some unknown functions having the sense of amplitudes. The unknown amplitudes shall obey the operator equations formulated in terms of wave scattering operators for individual disks and the sole half-plane.Results. When subjected to certain transformations, the operator equations allow obtaining integral equations relative amplitudes of the spherical waves involved. The integral equations permit investigating scattered wave fields for the cases where the disks stay in the shadow region behind the half-plane or in the penumbra, or else in the region which is illuminated by the incident wave. As has been shown, in the case of plane wave scattering at the edge of the half-plane the resulting cylindrical waves possess non-zero amplitudes even with the disks placed totally in the shadow region, hence not illuminated by the incident plane wave.Conclusions. Making use of an operator method, an original solution has been obtained for the problem of plane wave diffraction by a structure consisting of a perfectly conducting, infinitely thin half-plane and two disks. The operator equations of the problem have been shown to be reducible to integral equations, further solvable numerically with the use of discretization based on quadrature rules. The behavior of far and near fields relative to the disks has been studied for a variety of values of the disk radii and their positions relative to the half-plane.Keywords: half-plane, disk, operator method, diffraction, integral equationsManuscript submitted 25.04.2022Radio phys. radio astron. 2022, 27(3):167-180REFERENCES1. Jones, D.S., 1950. Note on diffraction by an edge. Q. J. Mech. Appl. Math., 3(4), pp. 420—434. DOI:https://doi.org/10.1093/qjmam/3.4.4202. Jones, D.S., 1952. A simplifying technique in the solution of a class of diffraction problems. Q. J. Math., 3(1), pp. 189—196. DOI: https://doi.org/10.1093/qmath/3.1.1893. Copson, E.T., 1946. On an integral equation arising in the theory of diffraction. Q. J. Math., os-17(1), pp. 19—34. DOI: https://doi.org/10.1093/qmath/os-17.1.194. Copson, E.T., 1950. Diffraction by a plane screen. Proc. R. Soc. Lond. A, 202(1069), pp. 277—284. DOI: https://doi.org/10.1098/rspa.1950.01005. Noble, D., 1958. Methods based on the Wiener-Hopf technique for the solution of partial differential equations. London: Pergamon Press.6. Khestanov, R.K., 1970. Diffraction of an arbitrary field on a half-plane. Radiotekhnika i elektronika, 15(2), pp. 289—297 (in Russian).7. Bertoni, H.L., Green, A., Felsen, L.B., 1978. Shadowing an inhomogeneous plane wave by an edge. J. Opt. Soc. Am., 68(7), pp. 983—989. DOI: https://doi.org/10.1364/JOSA.68.0009838. Bouwkamp, C.J., 1950. On the diffraction of electromagnetic waves by small circular disks and holes. Philips Research Reports, 5, pp. 401—422.9. Maixner, J., Andrejewski, W., 1950. Strenge theorie der Beugung ebener elektromagnetischer Wellen an der vollkommen leitenden Kreisscheibe und an der kreisförmigen Öffnung im vollkommen leitenden ebenen Schirm. Ann. Phys., 442(3—4), pp. 157—168. DOI:https://doi.org/10.1002/andp.1950442030510. Nomura, Y., Katsura, S., 1955. Diff raction of electromagnetic waves by circular plate and circular hole. J. Phys. Soc. Jpn., 10(4), pp. 285—304. DOI: https://doi.org/10.1143/JPSJ.10.28511. Lytvynenko, L.M., Prosvirnin, S.L., Khizhnyak, A.N., 1988. Semiinversion of the operator with the using of method of moments in the scattering problems by the structures consisting of the thin disks. Preprint, 19. Institute of Radio Astronomy Academy of Sciences Ukr SSR, pp. 1—8 (in Russian).12. Hongo, K., Naqvi, Q.A., 2007. Diffraction of electromagnetic wave by disk and circular hole in a perfectly conducting plane. PIER, 68, pp. 113—150. DOI: https://doi.org/10.2528/PIER0607310213. Losada, V., Boix, R. R., Horno, M., 1999. Resonant modes of circular microstrip patches in multilayered substrates. IEEE Trans. Microw. Theory Tech., 47(4), pp. 488—498. DOI: 10.1109/22.75488314. Losada, V., Boix, R.R., Horno, M., 2000. Full-wave analysis of circular microstrip resonators in multilayered media containing uniaxial anisotropic dielectrics, magnetized ferrites, and chiral materials. IEEE Trans. Microw. Theory Tech., 48(6), pp. 1057—1064. DOI:https://doi.org/10.1109/22.90474515. Losada, V., Boix, R.R., Medina, F., 2003. Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space. IEEE Trans. Geosci. Remote Sens., 41(5), pp. 988—997. DOI: https://doi.org/10.1109/TGRS.2003.81067816. Di Murro, F., Lucido, M., Panariello, G., Schettino, F., 2015. Guaranteed-convergence method of analysis of the scattering by an arbitrarily oriented zero-thickness PEC disk buried in a lossy half-space. IEEE Trans. Antennas Propag., 63(8), pp. 3610—3620. DOI:https://doi.org/10.1109/TAP.2015.243833617. Lucido, M., Panariello, G., Schettino, F., 2017. Scattering by a zero-thickness PEC disk: A new analytically regularizing procedure based on Helmholtz decomposition and Galerkin method. Radio Sci., 52(1), pp. 2—14. DOI: https://doi.org/10.1002/2016RS00614018. Balaban, M.V., Sauleau, R., Benson, T.M., Nosich, A.I., 2009. Dual integral equations technique in electromagnetic wave scattering by a thin disk. PIER B, 16, pp. 107—126. DOI:https://doi.org/10.2528/PIERB0905070119. Kaliberda, M.E., Lytvynenko, L.M., Pogarsky, S.A., 2022. Electromagnetic wave scattering by half-plane and disk placed in the same plane or circular hole in half-plane. J. Electromagn. Waves Appl., 36(10), pp. 1463—1483. DOI:https://doi.org/10.1080/09205071.2022.203237920. Kaliberda, M.E., Lytvynenko, L.M., Pogarsky, S.A., 2022. Scattering by PEC half-plane and disk placed in parallel planes. Int. J. Microw. Wirel. Technol., pp. 1—11. DOI:https://doi.org/10.1017/S175907872200047221. Muskhelishvili, N.I., 1972. Singular integral equations. Boundary problems of functions theory and their applications to mathematical physics. Wolters-Noordhoff. The Netherlands, Groningen. Revised transl. from Russian.22. Lifanov, I.K., 1996. Singular integral equations and discrete vortices. Utrecht: VSP. DOI:https://doi.org/10.1515/978311092604023. Hills, N.L., Karp, S.N., 1965. Semi-infinite diff raction gratings–I. Commun. Pure App. Math., 18(1—2). pp. 203—233. DOI: https://doi.org/10.1002/cpa.3160180119 Предмет і мета роботи. Розглянуто дифракцію плоскої хвилі на системі, яка складається з напівплощини та двох дисків. Диски та напівплощина, що вважаються ідеально провідними та нескінченно тонкими, розташовано в паралельних площинах. Досліджуються два випадки, а саме коли обидва диски лежать по один бік від напівплощини, та коли вони є розміщеними по різні боки від напівплощини. Метою роботи є розвинення операторного методу та дослідження з його використанням дифракції на вказаній структурі.Методи і методологія. Розв’язок задачі отримано операторним методом. Дотичні до напівплощини та дисків компоненти електричного поля за допомогою інтегралів Фур’є виражаються через невідомі функції, які мають сенс амплітуд. Для невідомих амплітуд записано операторні рівняння, які використовують оператори розсіяння на поодиноких дисках і напівплощині.Результати. Після перетворень операторних рівнянь отримано інтегральні рівняння відносно амплітуд сферичних хвиль. Досліджено розсіяні поля при розміщенні дисків у області тіні від напівплощини, в напівтіні та в області, що повністю освітлюється хвилею, яка падає. Показано, що за рахунок розсіяння плоскої хвилі на краю напівплощини поле циліндричних хвиль має відмінну від нуля амплітуду навіть у випадку, якщо диски знаходяться цілком у області тіні та не освітлюються падаючою плоскою хвилею.Висновки. З використанням операторного методу вперше отримано розв’язок задачі про дифракцію плоскої хвилі на системі, яка складається з ідеально провідних і нескінченно тонких напівплощини та двох дисків. Показано, що операторні рівняння цієї задачі можливо звести до інтегральних, які можуть буди чисельно розв’язані за допомогою дискретизації на базі квадратурних формул. Досліджено поведінку поля в ближній і далекій зонах відносно дисків при різних значеннях радіусів дисків і при зміні їхнього положення відносно напівплощини.Ключові слова: напівплощина, диск, операторний метод, дифракція, інтегральні рівнянняСтаття надійшла до редакції 25.04.2022Radio phys. radio astron. 2022, 27(3):167-180БІБЛІОГРАФІЧНИЙ СПИСОК1. Jones D.S. Note on diffraction by an edge. Q. J. Mech. Appl. Math. 1950. Vol. 3, Iss. 4. P. 420—434. DOI: 10.1093/qjmam/3.4.420.2. Jones D.S. A simplifying technique in the solution of a class of diffraction problems. Q. J. Math. 1952. Vol. 3, Iss. 1. P. 189—196. DOI: 10.1093/qmath/3.1.1893. Copson E.T. On an integral equation arising in the theory of diffraction. Q. J. Math. 1946. Vol. os-17, Iss. 1. P. 19—34. DOI: 10.1093/qmath/os-17.1.194. Copson E.T. Diffraction by a plane screen. Proc. R. Soc. Lond. A. 1950. Vol. 202, Iss. 1069. P. 277—284. DOI: 10.1098/rspa.1950.01005. Noble D. Methods based on the Wiener-Hopf technique for the solution of partial diff erential equations. London: Pergamon Press, 1958. 246 p.6. Хестанов Р.Х. Дифракция произвольного поля на полуплоскости. Радиотехника и электроника. 1970. T. 15, No 2. С. 289—297.7. Bertoni H.L., Green A., Felsen L.B. Shadowing an inhomogeneous plane wave by an edge. J. Opt. Soc. Am. 1978. Vol. 68, Iss. 7. P. 983—989. DOI: 10.1364/JOSA.68.0009838. Bouwkamp C.J. On the diff raction of electromagnetic waves by small circular disks and holes. Philips Research Reports. 1950. Vol. 5. P. 401—422.9. Maixner J., Andrejewski W. Strenge theorie der Beugung ebener elektromagnetischer Wellen an der vollkommen leitenden Kreisscheibe und an der kreisförmigen Öffnung im vollkommen leitenden ebenen Schirm. Ann. Phys. 1950. Vol. 442, Iss. 3—4. P. 157—168.10. Nomura Y., Katsura S. Diffraction of electromagnetic waves by circular plate and circular hole. J. Phys. Soc. Jpn. 1955. Vol. 10, Iss. 4. P. 285—304. 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DOI: 10.1002/cpa.3160180119 Видавничий дім «Академперіодика» 2023-06-15 Article Article application/pdf http://rpra-journal.org.ua/index.php/ra/article/view/1391 10.15407/rpra27.03.167 РАДИОФИЗИКА И РАДИОАСТРОНОМИЯ; Vol 27, No 3 (2022); 167 RADIO PHYSICS AND RADIO ASTRONOMY; Vol 27, No 3 (2022); 167 РАДІОФІЗИКА І РАДІОАСТРОНОМІЯ; Vol 27, No 3 (2022); 167 2415-7007 1027-9636 10.15407/rpra27.03 uk http://rpra-journal.org.ua/index.php/ra/article/view/1391/pdf Copyright (c) 2022 RADIO PHYSICS AND RADIO ASTRONOMY |