Abstract
The problem of wave generation on the surface of a liquid layer lying on an elastic half-space is considered. The generation source is located in an elastic medium. The joint system of equations of the theory of elasticity in a half-space and the theory of waves in a liquid is solved. On the basis of the previously obtained simplified solution of the dispersion equation for the water mode, taking into account the influence of the elastic half-space and the integral representation of the displacement of the liquid surface caused by a source of a simple form, analytical formulas are constructed for solving the problem under the assumption of long waves. Comparison of results obtained by analytical formulas and integral representations is carried out.
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REFERENCES
E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (IPF RAN, Nizhny Novgorod, 1996) [in Russian].
H. Kanamori, “Mechanism of tsunami earthquakes,” Phys. Earth Planet. Inter., No. 6, 349–359 (1972).
T. Yamashita and R. Sato, “Generation of tsunami by a fault model,” J. Phys. Earth, No. 22, 415–440 (1974).
B. W. Levin and M. A. Nosov, Physics of Tsunamis, 2nd ed. (Springer, 2016).
G. S. Pod’yapol’skii, “Generation of tsunamis by earthquakes,” in Computational Methods of Tsunami Generation and Propagation (Nauka, Moscow, 1978), pp. 30–87 [in Russian].
P. C. Sabatier, “On water waves produced by ground motions,” J. Fluid Mech., No. 126, 27–58 (1983).
V. K. Gusyakov and L. B. Chubarov, “Numerical simulation of tsunami excitation and propagation in the coastal zone,” Izv. RAN Fiz. Zemli, No. 11, 53–64 (1987)
A. Fragela, “On the problem of the motion of an ideal fluid in an unbounded elastic basin,” Differ. Equations 26, 1013–1021 (1990).
S. Yu. Dobrokhotov, O. L. Tolstova, and I. Yu. Chudinovich, “Waves in a fluid over an elastic bottom. The existence theorem and exact solutions,” Math. Notes 54, 1208–1222 (1993).
N. V. Zvolinskii, I. I. Karpov, I. S. Nikitin, and S. Ya. Sekerzh-Zen’kovich, “Generation of tsunami and Rayleigh waves by a harmonic two-dimensional rotation center,” Izv. Phys. Sol. Earth 30 (9), 773–777 (1995).
R. O. Griniv, S. Yu. Dobrokhotov, and A. A. Shkalikov, “An operator model for the oscillation problem of liquids on an elastic bottom,” Math. Notes 68, 57–70 (2000).
S. Yu. Dobrokhotov, O. L. Tolstova, S. Ya. Sekerzh-Zen’kovich, and C. A. Vargas, “Influence of the elastic base of a basin on the propagation of waves on the water surface,” Russ. J. Math. Phys. 25 (4), 459–469 (2018).
S. Yu. Dobrokhotov, Kh. Kh. Ilyasov, S. Ya. Sekerzh-Zenkovich, and O. L. Tolstova, “Simple Solutions to the Wave Problem on the Surface of a Fluid with the Linear Hydroelastic Model,” Dokl. Phys. 64, 319–324 (2019).
M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solutions of 2-D wave equations with variable velocity and localized right-hand side,” Russ. J. Math. Phys. 17 (1), 66–76 (2010).
Wolfram Mathematica, www.wolfram.com/mathematica/.
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This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00644 А).
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Translated by I. K. Katuev
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Dobrokhotov, S.Y., Ilyasov, K.K. & Tolstova, O.L. Simple Solutions to the Linear Problem of the Generation of Long Waves on the Surface of a Liquid by a Source in an Elastic Foundation Bottom. Mech. Solids 55, 561–572 (2020). https://doi.org/10.3103/S0025654420040032
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DOI: https://doi.org/10.3103/S0025654420040032