Abstract
The Cauchy problem for the equations of motion of a homogeneous transversally isotropic elastic medium is considered. For its solution, a short-wavelength asymptotic expansion is constructed, which is also applicable near specific directions. The resonance set, i.e., the set of points at which the ray expansion cannot be used is described.
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References
L. A. Molotkov, “On a inner source in transversely isotropic elastic medium,” Mathematical Problems in the Theory of Wave Propagation, Part 29, Zap. Nauchn. Sem. POMI 264, 238–249 (2000).
M. M. Popov, “SH waves in homogeneous transversely isotropic media generated by a concentrated force”, Mathematical Problems in the Theory of Wave Propagation, Part 29, Zap. Nauchn. Sem. POMI 264, 285–298 (2000).
M. M. Popov, “Asymptotics of the wave field near the axis of symmetry of a transversally isotropic homogeneous medium,” Mathematical Problems in the Theory of Wave Propagation, Part 30, Zap. Nauchn. Semin. POMI 275, 199–211 (2001).
L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Butterworth-Heinemann, Oxford; 1986 Nauka, Moscow, 1987).
G. I. Petrashen’, Wave Propagation in Anisotropic Elastic Media (Nauka, Leningrad, 1980) [in Russian].
Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Nauka, Moscow, 1980; Springer-Verlag, Heidelberg, 2011).
V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, The Space-Time Ray Method (Leningr. Gos. Univ., Leningrad, 1985; Cambridge Univ. Press, New York 1998).
V. P. Maslow and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (Nauka, Moscow, 1976; Reidel, Dordrecht, 1981).
B. R. Vainberg, Asymptotic Methods in Partial Differential Equations (Mosk. Gos. Univ., Moscow, 1982) [in Russian].
V. V. Kucherenko, “Asymptotics of solutions to the systems \(A(x,ih\frac{\partial } {{\partial x}})u = 0 \) as h → 0 in the case of characteristics of variable multiplicity,” Izv. Akad. Nauk SSSR Mat. 58(3), 625–650 (1974).
F. R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959; Fizmatgiz, Moscow, 1967).
R. Courant and D. Hilbert, Methoden der mathematischen Physik (Springer-Verlag, Berlin, 1924; Gostekhizdat, Moscow, 1945), Vol. 2.
M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
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Original Russian Text © I.N. Shchitov, 2014, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2014, Vol. 54, No. 10, pp. 1608–1617.
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Shchitov, I.N. Propagation and interaction of short waves in a homogeneous transversally isotropic elastic medium. Comput. Math. and Math. Phys. 54, 1550–1559 (2014). https://doi.org/10.1134/S096554251410011X
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DOI: https://doi.org/10.1134/S096554251410011X