Abstract
The Cauchy problem for the wave equations of Boussinesq type is treated by considering the initial conditions taken from the solution of generalized Cauchy problem for the potential model of tsunami with some “simple” impulsive source under the assumption that the depth of the liquid is constant. The solutions of the problem under consideration are derived in the form of a single integral giving the wave height at every point of observation at any time moment after the pulsed action of the source. The results of comparing the time history of the the height of tsunami waves at different distances from the source for different values of its characteristic radius (these histories are calculated using two solutions, namely, the solution derived here and the solution known for the potential tsunami model) are described. Conclusions concerning the accuracy of the tested solutions are made.
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References
T. Saito, K. Satake, and T. Furumura, “Tsunami Waveform Inversion Including Dispersive Waves: the 2004 Earthquake off Kii Peninsula, Japan,” J. Geophys. Res. 115, B06303, doi:10.1029/2009JB006884, (2010).
S. B. Yoon, C. H. Lim, and J. Choi, “Dispersion-Correction Finite Difference Model for Simulation of Transoceanic Tsunamis,” Terr. Atmos. Ocean. Sci. 18 (1), 31–53 (2007).
S. Krenk, “Dispersion-Correction Explicit Integration of the Wave Equation,” Comp. Math. Appl. Mech. Eng. 191, 975–987 (2001).
G. F. Carrier, “Tsunami Propagation from a Finite Source,” Proc. of 2nd UJNR Tsunami Workshop, NGDC, Hawaii, 101–115 (1991).
S. Ya. Sekerzh-Zen’kovich, “Analytical Study of the Tsunami Potential Model with a Simple Piston-Like Source 1. Exact Solution. Creation of Tsunami,” Russ. J. Math. Phys. 19 (3), 385–393 (2012).
S. S. Voit, “Tsunami Waves,” in Oceanology, Physics of the Ocean, Part 2: Hydrodynamics of the Ocean (Nauka, Moscow, 1978), pp. 229–254 [in Russian].
S. Wang, B. le Mehaute, and Chia-Chi Lu, “Effect of Dispersion on Impulsive Waves,” Marine Geophysical Researches 9 (1), 95–111 (1987).
T. Saito, “Dynamic Tsunami Generation due to Sea-Bottom Deformation: Analytical Representation Based on Linear Potential Theory,” Earth Planets Space 65, 1411–1423 (2013).
I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1 (Fizmatgiz, Moscow, 1959; Academic Press, New York–London, 1964).
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Sekerzh-Zen’kovich, S.Y. Testing of solutions for the Boussinesq wave equations on a solution of a potential tsunami model with “simple” source. Russ. J. Math. Phys. 22, 528–531 (2015). https://doi.org/10.1134/S1061920815040123
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DOI: https://doi.org/10.1134/S1061920815040123