volume | PIER Journals

Abstract

The electromagnetic wave diffraction from the modified cone formed by a circular truncated cone whose aperture is closed by a spherical cap is considered. The problem is reduced to the solution of the mixed boundary value problem for the Helmholtz equation. The axially symmetric version of the problem, where the cone is excited by a radial electric dipole (E-polarization wave diffraction problem), is analyzed. A new approach to the solution is proposed. The solution includes the application of the Kontorovich-Lebedev integral transformation, the nonstandard procedure for derivation of the Wiener-Hopf equation and its reduction to the set of linear algebraic equations of the second kind. Their solution ensures the fulfillment of all the necessary conditions including the edge condition. The approximate equation for the sharp truncated cone terminated by the spherical cap is analyzed. The low frequency approximation as well as the transition to the plane which incorporates the hemispheric cavity is analysed. The numerical calculation results are presented.

1. Northover, F. H., "The diffraction of electromagnetic waves around a finite, perfectly conducting cone Pt. 1. The mathematical solution," Journal of Mathematical Analysis and Applications, Vol. 10, 37-49, 1965.

2. Northover, F. H., "The diffraction of electric waves around a finite, perfectly conducting cone Pt. 2. The field singularities," Journal of Mathematical Analysis and Applications, Vol. 10, 50-69, 1965.

3. Syed, A., "The diffraction of arbitrary electromagnetic field by a finite perfectly conducting cone," Journal of Natural Sciences and Mathematics, Vol. 2, No. 1, 85-114, 1981.

4. Daniele, V. and R. Zich, The Wiener-Hopf Method in Electromagnetics, ISMB Series, SCITECH Publishing, 2014.

5. Mittra, R. and S.-W. Lee, Analytical Techniques in the Theory of Guided Waves, Macmillan, 1971.

6. Kobayashi, K., "Some diffraction problems involving modified Wiener-Hopf geometries," Analytical and Numerical Methods in Electromagnetic Wave Theory, M. Hashimoto, M. Idemen, O. A. Tretyakov, Eds., Science House Co., Ltd., Tokyo, 1993.

7. Demir, A., A. Buyukaksoy, and B. Polat, "Diffraction of plane waves by a rigid circular cylindrical cavity with an acoustic absorbing internal surface," Zeitschrift f¨ur Angewandte Mathematik und Mechanik, Vol. 82, No. 9, 619-629, 2002.

8. Kuryliak, D. B., K. Kobayashi, S. Koshikawa, and Z. T. Nazarchuk, "Wiener-Hopf analysis of the diffraction by circular waveguide cavities," Journal of the Institute of Science and Engineering, Vol. 10, 45-52, Tokyo (Japan), 2005.

9. Kobayashi, K. and S. Koshikawa, "Diffraction by a parallel-plate waveguide cavity with a thick planar termination," IEICE Transactions on Electronics, Vol. E76-C, No. 1, 42-158, 1993.

10. Kuryliak, D. B., S. Koshikawa, K. Kobayashi, and Z. T. Nazarchuk, "Wiener-Hopf analysis of the axial symmetric wave diffraction problem for a circular waveguide cavity," International Workshop on Direct and Inverse Wave Scattering, 2-67-2-81, Gebze (Turkey), 2000.

11. Bazer, J. and S. Karp, "Potential flow through a conical pipe with an application to diffraction theory," Research Report No. EM-66, Courant Institute of Mathematical Sciences, New York University in New York, 1954.

12. Vaisleib, Y. V., "Axially-symmetric illumination of the perfectly conducting finite cone," Proceedings of Educational Institutes of Communications, Vol. 35, 58-66, 1967.

13. Pridmore-Brown, D. C., "A Wiener-Hopf solution of a radiation problem in conical geometry," Journal of Mathematical Physics, Vol. 47, 79-94, 1968.

14. Kuryliak, D. B. and Z. T. Nazarchuk, "Analytical-numerical Methods in the Theory of Wave Diffraction on Conical and Wedge-shaped Surfaces,", 2006 (in Ukrainian).

15. Kuryliak, D. B., S. Koshikawa, K. Kobayashi, and Z. T. Nazarchuk, "Diffraction by a truncated, semi-infinite cone: Comparison of the Wiener-Hopf technique and semi-inversion methods," PIERS Proceedings, Vol. 321, Hawaii, October 2003.

16. Noble, B., Method Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, 1958.

17. Belichenko, V. P., "Finite integral transformation and factorization methods for electro-dynamics and electrostatic problems," Mathematical Methods for Electrodynamics Boundary Value Problems, V. P. Belichenko, G. G. Goshin, A. G. Dmitrienko, et al., Eds., Izd. Tomsk. Univ., Tomsk, 1990 (in Russian).

18. Naylor, D., "On a finite Lebedev transform," Journal of Mathematics and Mechanics, Vol. 12, No. 3, 375-383, 1963.

19. Kuryliak, D. B., "Axially-symmetric field of electric dipole over truncated cone. I comparison between mode-matching technique and integral transformation method," Radio Physics and Radio Astronomy, Vol. 4, No. 2, 121-128, 1999 (in Russian).

20. Kuryliak, D. B., "Axially-symmetric field of electric dipole over truncated cone. II. Numerical Modeling," Radio Physics and Radio Astronomy, Vol. 5, No. 3, 284-290, 2000 (in Russian).

21. Kuryliak, D. B., "Dual series equation of the associate Legendre functions for conical and spherical regions and its application for the scalar diffraction problems," Reports of the National Academy of Sciences of Ukraine, No. 10, 70-78, 2000 (in Russian).

22. Kuryliak, D. B. and Z. T. Nazarchuk, "Dual series equations for the wave diffraction by conical edge," Reports of the National Academy of Sciences of Ukraine, No. 11, 103-111, 2000.

23. Kuryliak, D. B. and Z. T. Nazarchuk, "Convolution type operators for wave diffraction by conical surfaces," Radio Science, Vol. 43, No. 4, 2008, doi: 10.1029/2007RS003792.

24. Kuryliak, D. and V. Lysechko, "Acoustic plane wave diffraction from a truncated semi-infinite cone in axial irradiation," Journal of Sound and Vibration, Vol. 409, 81-93, 2017.

25. Kuryliak, D. B., Z. T. Nazarchuk, and O. B. Trishchuk, "Axially-symmetric TM-waves diffraction by sphere-conical cavity," Progress In Electromagnetics Research B, Vol. 73, 1-16, 2017.

26. Hobson, E., Theory of Spherical and Ellipsoidal Harmonics, Izdatelstvo Inostrannoy Literaturi, 1952.

27. Gradshtein, I. S. and I. M. Ryzhik, Tables of Integrals, Series, and Products, Gosudarstvennoe Izdatelstvo Fiziko-Matematiceskoj Literatury, 1963.

28. Vinogradov, S. S., P. D. Smith, and E. D. Vinogradova, Canonical Problems in Scattering and Potential Theory Part II: Acoustic and Electromagnetic Diffraction by Canonical Structures, Chapman and Hall/CRC, 2002.

Professor Dozyslav B. Kuryliak

Professor Kazuya Kobayashi

Dr. Zinoviy Theodorovych Nazarchuk