Abstract
In the present paper the author study the propagation of a plane wave through a doubly-periodic infinite array of identical obstacles of elliptic shape. The symmetry of the geometry allows us to reduce the problem to a certain single layer, where a special form of the Green’s function leads to a basic boundary integral equation (BIE) for this diffraction problem. The BIE is studied in the one-mode frequency range. Then the author construct an appropriate numerical method, to solve this integral equation, which allows us to evaluate the wave properties of the periodic structure including the reflection and transmission coefficients versus frequency parameter.
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Shenderov, E.L.: Propagation of sound through a screen of arbitrary wave thickness with gaps. Sov. Phys. Acoust. 16(1), 115–131 (1970)
Achenbach, J.D., Li, Z.L.: Reflection and transmission of scalar waves by a periodic array of screens. Wave Motion 8, 225–234 (1986)
Miles, J.W.: On Rayleigh scattering by a grating. Wave Motion 4, 285–292 (1982)
Zarrillo, G., Aguiar, K.: Closed-form low frequency solutions for electromagnetic waves through a frequency selective surface. IEEE Trans. Antennas Propag. AP-35, 1406–1417 (1988)
Scarpetta, E., Sumbatyan, M.A.: Explicit analytical results for one-mode normal reflection and transmission by a periodic array of screens. J. Math. Anal. Appl. 195, 736–749 (1995)
Scarpetta, E., Sumbatyan, M.A.: On wave propagation in elastic solids with a doubly periodic array of cracks. Wave Motion 25, 61–72 (1997)
Scarpetta, E., Sumbatyan, M.A.: On the oblique wave penetration in elastic solids with a doubly periodic array of cracks. Q. Appl. Math. 58, 239–250 (2000)
Scarpetta, E., Sumbatyan, M.A.: Wave propagation through elastic solids with a periodic array of arbitrarily shaped defects. J. Math. Comput. Model. 37, 19–28 (2003)
Scarpetta, E., Tibullo, V.: Explicit results for scattering parameters in three-dimensional wave propagation through a doubly periodic system of arbitrary openings. Acta Mech. 185, 1–9 (2006)
Scarpetta, E., Tibullo, V.: On the three-dimensionl wave propagation through cascading screens having a periodic system of arbitrary openings. Int. J. Eng. Sci. 46, 105–111 (2008)
Angel, Y.C., Achenbach, J.D.: Harmonic waves in an elastic solid containing a doubly periodic array of cracks. Wave Motion 9, 377–385 (1987)
Scarpetta, E.: In-plane problem for wave propagation through elastic solids with a periodic array of cracks. Acta Mech. 154, 179–187 (2002)
Angel, Y.C., Bolshakov, A.: In-plane waves in an elastic solid containing a cracked slab region. Wave Motion 31, 297–315 (2000)
Mykhaskiv, V.V., Zhbadynskyi, I.Ya., Zhang, Ch.: Dynamic stresses due to time-harmonic elastic wave incidence on doubly periodic array of penny-shaped cracks. J. Math. Sci. 203, 114–122 (2014)
Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289(5485), 1734–1736 (2000)
Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47, 610–617 (2009)
Craster, R.V., Guenneau, S.: Acoustic Metamaterials. Springer Series in Materials Science, vol. 166. Springer, Dordrecht (2013)
Deymier, P.A.: Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid-State Sciences. Springer, Berlin (2013)
Yang, Ch., Achenbach, J.D.: Time domain scattering of elastic waves by a cavity, represented by radiation from equivalent body forces. Int. J. Eng. Sci. 115, 43–50 (2017)
Remizov, M.Yu., Sumbatyan, M.A.: 3-D one-mode penetration of elastic waves through a doubly periodic array of cracks. Math. Mech. Solids 23(4), 636–650 (2018)
Remizov, M.Yu., Sumbatyan, M.A.: On 3D theory of acoustic metamaterials with a triple-periodic system of interior obstacles. Proc. Natl. Acad. Sci. Armen. 70(4), Mechanics 35–49 (2017)
Twersky, V.: Multiple scattering of sound by a periodic line of obstacles. J. Acoust. Soc. Am. 53, 96–112 (1973)
Jones, D.S.: Acoustic and Electromagnetic Waves. Clarendon Press, Oxford (1986)
Acknowledgements
The author expresses his gratitude to Professor M.A. Sumbatyan, Southern Federal University, Russia, for valuable comments. He would also like to notice that this work has been performed in frames of the project 9.5794.2017/8.9 under support of the Russian Ministry for Education and Science.
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Remizov, M.Y. (2019). The BIE Method in the Problem of Wave Propagation Through an Infinite Doubly-Periodic Array of Elliptic Obstacles. In: Altenbach, H., Belyaev, A., Eremeyev, V., Krivtsov, A., Porubov, A. (eds) Dynamical Processes in Generalized Continua and Structures. Advanced Structured Materials, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-030-11665-1_24
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DOI: https://doi.org/10.1007/978-3-030-11665-1_24
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