537.8.029.6

сингулярне інтегральне рівняння
поліном Чебишева
тонке проникливе включення
поперечник розсіяння
поляризована електромагнітна хвиля
singular integral equation
Chebishev polinomial
thin penetrating inclusions
diameter of dispersion
polarized electromagnetic wave

Запропоновано математичну модель взаємодії електромагнітного поля – поляризованої плоскої хвилі з системою тонких діелектричних циліндричних включень у вигляді системи інтегродиференціальних сингулярних інтегральних рівнянь, визначених на середній лінії поперечного перетину розсіювачів. З використанням квадратурних формул інтерполяційного типу за вузлами, що співпадають з коренями полінома Чебишева, числову реалізацію математичної моделі зведено до розв’язання системи лінійних алгебраїчних рівнянь. Наведено результати числового дослідження поперечника розсіяння двох включень.
Using the method of integral equations the mathematical model of interwork between the electromagnetic field of H-polarized plane wave and the system of thin dielectric or little conducting cylindrical embedding has been built. For its building the system of two-dimensional integral equations defined on the basis transverse section diffusers was taken. The direct implementation of the available methods for their numerous solving faces numerical difficulties caused by the little thickness of diffuser and are connected with calculus of singular and quasisingular two-dimensional integrals. Taking into consideration the fact that inside the diffuser the field changes its thickness unsufficeintly the mathematical model of interwork between the electromagnetic field of H-polarized plane wave with the system of thin dielectric cylindrical inclusions in the form of the system of integrodifferential singular integral equations defined on the centre line of the diffusers transverse section has been proposed. They were obtained by averaging of the system of integral equations defined the diffusers transverse section area. Using the quadrature formula of interpolatory type by the crosspoints which coincide with the Chebyshev polynomial, roots the numerical realization of mathematical model is redused to the solving of the system of linear algebraic equations. To test the proposed mathematical model the case of whole and cross-cut plate and half-ring shape has been proposed. The research of the influence of the distance between the two parts of infusions and the direction of plane wave propagation on the field in the far-field zone was conducted. The built mathematical model allows conducting research of the field scattered characteristics scattered by the arbitrarily-spaced cylindrical open coverings, the maximum thickness of which is considerably less than the length, the latter being considerably bigger or the same as the length of the explorative electromagnetic wave.

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11. Bleszynski, E. Surface-integral equations for electromagnetic scattering from impenetrable and penetrable sheets [Text] / E. Bleszynski, M. Bleszynski, T. Jaroszewicz // IEEE Antennas and Propagation Magazine. – 1993. – 35, No 6. – P.14–25.
12. Bouchitte, G. On the concept of a perfectly conducting material and of a perfectly conducting and infinitely thin screen [Text] / G. Bouchitte, R. Petit // Radio Science. –1989. – 24, No1 – P.13–26.
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15. Karlsson, A. Approximate boundary conditions for thin structures [Text] / A. Karlsson // IEEE Trans. Antennas and Propagation. – 2009. – 57, No1. – P.144–148.
16. Lerer, A.M. Full-wave analysis of three-dimensional planar structures [Text] / A.M. Lerer, A.G. Schuchinsky // IEEE Trans. Microwave Theory and Techniques. – 1993. – 41, No11. – P.2002–2014.
17. Integral equation analysis of plane wave scattering from multilayered resistive strip gratings [Text] / A. Matsushima, T.L. Zinenko, H. Minami, Y. Okuno // J. Electromagnetic Waves and Application. – 1998. – 12. – P.1449–1469.
18. Mitzner, K.M. Effective boundary conditions for reflection and transmission by an absorbing shell of arbitrary shape [Text] / K.M. Mitzner // IEEE Trans. Antennas and Propagation. –1968. –16. – R.706–712.
19. Nazarchuk, Z.T. Singular integral equations in diffraction theory [Text] / Z.T. Nazarchuk. – Lviv: Karpenko Physico-Mechanical Institute, 1994. – 210 p.
20. Nazarchuk, Z.T. Mathematical modeling of electromagnetic scattering from a thin penetrable target [Text] / Z.T. Nazarchuk, K. Kobayashi // Progress in Electromagnetic Research. – 2005. – 55. – P.95–116.
21. Nosich, A.I. Scattering and absorption of E- and H-polarized plane waves by a circularly curved resistive strip [Text] / A.I. Nosich, Y. Okuno, T. Shiraishi // Radio Science. – 1996. – 31, No6. – R.1733–1742.
22. Richmond, J.H. Scattering by a dielectric cylinder of arbitrary cross-section shape [Text] / J.H. Richmond // IEEE Trans. Antennas and Propagation. – 1965. – 13, No3. – P.334–341.
23. Richmond, J.H. TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape [Text] / J.H. Richmond // IEEE Trans. Antennas and Propagation. – 1966. – 14, No4. – P.460–464.
24. Schmidt, K. Asymptotic boundary element methods for thin conducting sheets in two dimensions [Text] / K. Schmidt, R. Hiptmair // IEEE Trans. Magn. – 2014. – 50, No2. – P.469–472.
25. Senior, B.A. Combined resistive and conductive sheets [Text] / B.A. Senior // IEEE Trans. Antennas and Propagation. – 1985. – 33. – R.577–579.
26. Senior, B.A. Approximate boundary conditions in electromagnetics [Text] / B.A. Senior, J. Volakis. – London: The Institution of Electrical Engineers, 1995. – 353 p.
27. Tsalamengas, J.L. Direct and efficient solutions of integral equations for scattering from strips and slots [Text] / J.L. Tsalamengas, J.G. Fikioris, B.T. Babili // J. Applied Physics. – 1989. – 66, No 7. – P.69–80.
28. Zinenko, T.L. Scattering and absorption of electromagnetic plane waves by a multilayered resistive strip grating embedded in a dielectric slab [Text] / T.L. Zinenko, A. Matsushima, Y. Okuno // Trans. IEICE Electronics. – 1999. – E82-C, No12. – P.2255–2264.
29. Zinenko, T.L. Plane wave scattering and absorption by resistive-strip and dielectric-strip periodic gratings [Text] / T.L. Zinenko, A.I. Nosich, Y. Okuno // IEEE Trans. Antennas and Propagation. – 1998. – 46, No10. – P.1498–1505.