Wave diffraction from the truncated hollow wedge: analytical regularization and Wiener-Hopf analysis. (English) Zbl 1498.78020
Summary: The electromagnetic wave diffraction from perfectly conducting truncated wedges is considered on the rigorous level in cylindrical coordinates. An analytical regularization method is developed to obtain mathematically accurate problem solutions. The solution method is based on the unknown field representation through the principal value Kontorovich-Lebedev integral and the eigenfunctions series. We analyze the scattering from the semi-infinite truncated wedge, which consists of two non-parallel and non-intersecting perfectly conducting and infinitely thin half-planes, and develop this technique for analysis of more complicated problems of wave diffraction from the truncated wedge of finite length. The problems are reduced to the infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators and their inverse ones are used to reduce them to the ISLAE of the second kind applied to the analytical regularization procedure. Two versions of the procedure, such as left- and right-sides regularization, are considered. The developed technique is compared with the Wiener-Hopf method. The numerical examples of wave scattering from the truncated wedge, including its well-known geometries as the semi-plane and the slit in the infinite plane, are analyzed.
MSC:
| 78A45 | Diffraction, scattering |
| 78A50 | Antennas, waveguides in optics and electromagnetic theory |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 45F10 | Dual, triple, etc., integral and series equations |
| 47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
Kontorovich-Lebedev integral; truncated wedge; wave diffraction; integral equation; analytical regularization; Wiener-Hopf techniqueReferences:
| [1] | Sommerfeld, A., Mathematical Theory of Diffraction (2003), Boston: Birkhauser, Boston |
| [2] | Bowman, JJ; Bowman, JJ; Senior, TBA; Uslenghi, PLE, The wedge in electromagnetic and acoustic scattering by simple shapes, Electromagnetic and Acoustic Scattering by Simple Shapes, 256-283 (1969), Amsterdam: North-Holland Publishing Co., Amsterdam · Zbl 0181.56502 |
| [3] | Grinberg, GA, Selected Problems in the Mathematical Theory of Electric and Magnetic Phenomena (1948), Moskow-Leningrad: Izd-vo AN, Moskow-Leningrad |
| [4] | Babich, V.M., Lyalinov, M.A., Grikurov, V.E.: Diffraction Theory. The Sommerfeld-Malyuzhinets Technique, Alpha Science International (2008) |
| [5] | Budaev, B.V.: Diffraction by Wedges, vol. 322. CRC Press (1995) · Zbl 0835.73003 |
| [6] | Osipov, A.V., Tretyakov, S.A.: Modern Electromagnetic Scattering Theory with Applications. Wiley (2017) |
| [7] | Osipov, AV; Norris, AN, The malyuzhinets theory for scattering from wedge boundaries: a review, Wave Motion, 29, 4, 313-340 (1999) · Zbl 1074.76611 · doi:10.1016/S0165-2125(98)00042-0 |
| [8] | Nethercote, MA; Assier, RC; Abrahams, ID, Analytical methods for perfect wedge diffraction: a review, Wave Motion, 93 (2020) · Zbl 1524.78039 · doi:10.1016/j.wavemoti.2019.102479 |
| [9] | Rawlins, AD, Diffraction by, or diffusion into, a penetrable wedge, Proc. R. Soc. A: Math., 455, 1987, 2655-2686 (1999) · Zbl 1062.78502 · doi:10.1098/rspa.1999.0421 |
| [10] | Daniele, V.; Lombardi, G., The wiener-hopf solution of the isotropic penetrable wedge problem: diffraction and total field, IEEE Trans. Antennas Propag., 59, 10, 3797-3818 (2011) · Zbl 1369.78174 · doi:10.1109/TAP.2011.2163780 |
| [11] | Tolstoy, I., Exact, explicit solutions for diffraction by hard sound barriers and seamounts, J. Acoust. Soc. Am., 85, 2, 661-669 (1989) · doi:10.1121/1.397592 |
| [12] | Ufimtsev, P.Y.: Fundamentals of the Physical Theory of Diffraction. Wiley (2007) |
| [13] | Hrinchenko, VT; Matsypura, VT, Sound radiation from an open end of the wedge waveguide. i. Method for solution and algorithm for calculation, Acoust. Bull., 2, 4, 32-41 (1999) · Zbl 1073.76635 |
| [14] | Hrinchenko, VT; Matsypura, VT, Sound radiation from an open end of the wedge waveguide. ii. Numerical analysis, Acoust. Bull., 3, 2, 63-71 (2000) |
| [15] | Plonus, M., Electromagnetic radiation from a cylindrically capped bi-wedge, IEEE Trans. Antennas Propag., 10, 2, 206-210 (1962) · doi:10.1109/TAP.1962.1137841 |
| [16] | Polycarpou, A.C., Christou, M.A., Todorov, M.D., Christov, C.I.: Spectral formulation for the solution of full-wave scattering from a conducting wedge tipped with a corrugated cylinder. In: Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings. AIP (2011) |
| [17] | Kim, JJ; Eom, HJ; Hwang, KC, Electromagnetic scattering from a slotted conducting wedge, IEEE Trans. Antennas Propag., 58, 1, 222-226 (2010) · Zbl 1369.78230 · doi:10.1109/TAP.2009.2027454 |
| [18] | Forouzmand, A.; Yakovlev, AB, Electromagnetic cloaking of a finite conducting wedge with a nanostructured graphene metasurface, IEEE Trans. Antennas Propag., 63, 5, 2191-2202 (2015) · doi:10.1109/TAP.2015.2407412 |
| [19] | Weisleib, YV, Electromagnetic wave diffraction by the finite wedge, Radiotechnika and Electronica, 15, 8, 1568-1579 (1970) |
| [20] | Belichenko, V.P.: Finite integral transformation and factorization methods for electro-dynamics and electrostatic problems. In: Mathematical methods for electrodynamics boundary value problems. Izd. Tomsk. Univ. (1990) |
| [21] | Kuryliak, DB, Wave diffraction from the PEC finite wedge, J. Eng. Math., 134, 1, 1-25 (2022) · Zbl 1492.78009 · doi:10.1007/s10665-022-10222-x |
| [22] | Kuryliak, D., Diffraction by semi-infinite cone formed with electric and magnetic surfaces: analytical regularization and wiener-hopf techniques, J. Eng. Math., 115, 1, 43-65 (2019) · Zbl 1426.78017 · doi:10.1007/s10665-019-09991-9 |
| [23] | Kuryliak, D.: Some diffraction problems involving conical geometries and their rigorous analysis. In: 2018 IEEE 17th International Conference on Mathematical Methods in Electromagnetic Theory (MMET). IEEE (2018) |
| [24] | Kuryliak, DB; Sharabura, OM, Wave diffraction from the biconical section in the semi-infinite conical region, Math. Methods Appl. Sci., 43, 4, 1565-1581 (2019) · Zbl 1463.78004 · doi:10.1002/mma.5983 |
| [25] | Kuryliak, D.; Lysechko, V., Plane wave diffraction from a finite soft cone at oblique incidence, J. Sound Vib., 438, 309-323 (2019) · doi:10.1016/j.jsv.2018.09.032 |
| [26] | Kuryliak, D.; Lysechko, V., Scattering of the plane acoustic wave from a finite hollow rigid cone at oblique incidence, ZAMM J. Appl. Math. Mech., 99, 2, e201800127 (2018) · Zbl 07805128 · doi:10.1002/zamm.201800127 |
| [27] | Kuryliak, DB; Lysechko, VO, Acoustic plane wave diffraction from a truncated semi-infinite cone in axial irradiation, J. Sound Vib., 409, 8, 81-93 (2017) · doi:10.1016/j.jsv.2017.07.035 |
| [28] | Popov, GY, Exact Solutions of Some Boundary Problems of Deformable Solid Mechanic (2013), Odessa: Astroprint, Odessa |
| [29] | Vorovich, II; Aleksandrov, VM; Babeshko, VA, Nonclasical Mixed Problems of Elasticity Theory (1974), Moscow: Nauka, Moscow |
| [30] | Shestopalov, VP; Kirilenko, AA; Masalov, SA, Convolution-Type Matrix Equations in the Theory of Diffraction (1984), Kyiv: Naukova Dumka, Kyiv |
| [31] | Veliev, E.I., Veremey, V.V.: Numerical-analytical approach for the solution to the wave scattering by polygonal cylinders and flat strip structures. In: Analytical and Numerical Methods in Electromagnetic Wave Theory. Science House Co., Ltd (1993) |
| [32] | Chumachenko, VP, Domain-product technique solution for the problem of electromagnetic scattering from multiangular composite cylinders, IEEE Trans. Antennas Propag., 51, 10, 2845-2851 (2003) · doi:10.1109/TAP.2003.816310 |
| [33] | Hönl, H., Maue, A.W., Westpfahl, K.: Theorie der beugung. In: Handbuch der Physik. Springer, Berlin (1961) |
| [34] | Rawlins, AD, Plane-wave diffraction by a rational wedge, Proc. R. Soc. Lond. A. Math. Phys. Sci., 411, 1841, 265-283 (1987) · Zbl 0633.76082 |
| [35] | Gradshtein, IS; Ryzhik, IM, Tables of Integrals, Series, and Products (1963), Moscow: Gosudarstvennoe Izdatelstvo Fiziko-Matematiceskoj Literatury, Moscow |
| [36] | Noble B (1958) Methods based on the wiener-hopf technique for the solution of partial differential equations. In: International Series of Monographs on Pure and Applied Mathematics. vol. 7. Pergamon Press, p. 246 · Zbl 0082.32101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
