Abstract
The paper is devoted to the calculation of the reflection and the transmission coefficients, when a plane longitudinal wave is incident on a three-dimensional grating with a periodic array of rectangular cracks in the elastic material. In the one-mode frequency range the problem is reduced to a system of integral equations, which can be solved for various sizes of the cracks to give an explicit representation for the wave field inside the cracked structure, as well as the values of the reflection and the transmission coefficients.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Achenbach, J.D., Li, Z.L.: Reflection and transmission of scalar waves by a periodic array of screens. Wave Motion 8, 225–234 (1986)
Belotserkovsky, S.M., Lifanov, I.K.: Method of Discrete Vortices. CRC Press, Boca Raton, Florida (1992)
Kok, Y-L., Galagher, Jr. N.C., Ziolkovski, R.V.: Dual series solution to the scattering of plane wave from a binary conducting grating. IEEE Trans. Anten. Prop. AP-37, 901–917 (1989)
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)
Miles, J.W.: On Rayleigh scattering by a grating. Wave Motion 4, 285–292 (1982)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vols. 1, 2. Gordon & Breach, Amsterdam (1986)
Remizov, M.Yu, Sumbatyan, M.A.: 3-D one-mode penetration of elastic waves through a doubly periodic array of cracks. Math. Mech. Solids (in press)
Remizov, M.Yu, Sumbatyan, M.A.: Asymptotic analysis in the anti-plane high-frequency diffraction by interface cracks. Appl. Math. Lett. 34(1), 72–75 (2014)
Remizov, M.Yu., Sumbatyan, M.A.: A semi-analytical method of solving problems of the high-frequency diffraction of elastic waves by cracks. J. Appl. Math. Mech. 77(4), 452–456 (2013)
Remizov, M.Yu., Sumbatyan, M.A.: A semi-analytical approach in the high-frequency diffraction by cracks. Mech. Res. Comm. 38(8), 607–609 (2011)
Sumbatyan, M.A.: Low-frequency penetration of acoustic waves through a periodic arbitrary-shaped grating: the three-dimensional problem. Wave Motion 22, 133–144 (1995)
Scarpetta, E., Sumbatyan, M.A.: Explicit analytical results for one-mode oblique penetration into a periodic array of screens. IMA J. Appl. Math. 56, 109–120 (1996)
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., 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)
Shenderov, E.L.: Sound penetration through a rigid screen of finite thickness with apertures. Soviet Phys. Acoust. 16(2) (1970)
Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. Wiley, London (1969)
Acknowledgements
The first author is thankful to the Russian Science Foundation (RSCF), for its support by Project 15-19-10008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Sumbatyan, M.A., Remizov, M.Y. (2017). On the Theory of Acoustic Metamaterials with a Triple-Periodic System of Interior Obstacles. In: Sumbatyan, M. (eds) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials . Advanced Structured Materials, vol 59. Springer, Singapore. https://doi.org/10.1007/978-981-10-3797-9_2
Download citation
DOI: https://doi.org/10.1007/978-981-10-3797-9_2
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3796-2
Online ISBN: 978-981-10-3797-9
eBook Packages: EngineeringEngineering (R0)