Abstract
Within the geometric theory of diffraction, the problem of the propagation of ultrasonic waves through array of obstacles in an infinite two-dimensional elastic medium is investigated. A tonal impulse of a time-harmonic longitudinal or transverse plane elastic high-frequency wave of several wave-lengths is introduced through the array of obstacles, and in a certain domain inside the elastic medium the through-transmitted wave with arbitrary reflections and transformations is received. Some integral representations for displacement in the reflected waves are written out on the basis of the Kirchhoff physical diffraction theory. With the use of an asymptotic estimate of multiple diffraction integrals by the multidimensional stationary phase method we have written out explicitly the geometric-theory approximation for displacements in the multiply reflected and transformed waves.
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References
Liu, Z., Zhang, X., Mao, Y., et al.: Locally resonant sonic materials. Science 289, 1734–1736 (2000)
Boyev, N.V., Sumbatyan, M.A.: Ray tracing method for a high-frequency propagation of the ultrasonic wave through a triple-periodic array of spheres. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Advanced Structured Materials, vol. 59. Springer Series, pp. 173–187 (2017)
Babich, V.M., Buldyrev, V.S.: Asymptotic methods in short-wavelength diffraction theory. Alpha Sci. Int. (2009)
McNamara, D.A., Pistotius, C.W.I.: Introduction to the Uniform Geometrical Theory of Diffraction. Artech House, Microwave Library (1990)
Boyev, N.V.: Short-wave diffraction of elastic waves by voids in an elastic medium with double reflections and transformations. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, Advanced Structured Materials, vol. 59. Springer Series, pp. 91–106 (2017)
Sumbatyan, M.A., Boyev, N.V.: High-frequency diffraction by nonconvex obstacles (Part 1). J. Acoust. Soc. Am. 95(5), 2347–2353 (1994)
Kupradze, V.D.: Potential methods in the theory of elasticity. In: Israel Program for Scientific Translations. (1965)
Novatsky, V.: Theory of Elasticity. Mir, Moscow (1975). (in Russian)
Borovikov, V.A., Borovikov, V.A., Kinber, B.Y., Kinber, B.E.: Geometrical Theory of Diffraction. The Institution of Electrical Engineers, London (1994)
Achenbach, J.D., Gautesen, A.K., McMaken, H.: Ray Methods for Waves in elastic solids: with applications to scattering by cracks. Pittman, New York (1982)
Hoenl, H., Maue, A., Westpfahl, K.: Diffraction Theory. Mir, Moscow (1964). (in Russian)
Brekhovskikh, L.: Waves in Layered Media. Academic Press, New York (2012)
Grinchenko, V.T., Meleshko, V.V.: Harmonic Oscillations and Waves in Elastic Bodies. Naukova Dumka, Kiev (1981). (in Russian)
Fedorjuk, M.V.: Saddle Point Method. Nauka, Moscow (1977). (in Russian)
Acknowledgements
The present work is performed within the framework of the Project no. 15-19-10008-P of the Russian Science Foundation (RSCF).
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Boyev, N.V. (2019). Diffraction of the High-Frequency Waves by Arrays of Obstacles in the Two-Dimensional Elastic Medium, with Multiple Reflections and Transformations. In: Sumbatyan, M. (eds) Wave Dynamics, Mechanics and Physics of Microstructured Metamaterials. Advanced Structured Materials, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-030-17470-5_5
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DOI: https://doi.org/10.1007/978-3-030-17470-5_5
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