×

Method for the analysis of long water waves taking into account reflection from a gently sloping beach. (English. Russian original) Zbl 1440.76014

J. Appl. Math. Mech. 81, No. 1, 21-28 (2017); translation from Prikl. Mat. Mekh. 81, No. 1, 33-44 (2017).
Summary: Algorithms and their software implementation are described that allow calculating the front propagation for a long wave (for example, a tsunami wave) described in the approximation of the linearized shallow water equations with reflection from a shallow beach taken into account. The algorithms are based on a construction of asymptotic solutions of the Cauchy problem for hyperbolic equations with degeneracy on the boundary earlier proposed with the participation of one of the authors. The results of a numerical experiment are presented.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76-04 Software, source code, etc. for problems pertaining to fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Stoker, J. J., Water Waves: The Mathematical Theory with Applications (1958), Wiley: Wiley New York · Zbl 0812.76002
[2] Mei, C. C., The Applied Dynamics of Ocean Surface Waves (1989), World Scientific: World Scientific Singapore · Zbl 0991.76003
[3] Pelinovsky, E. N., Hydrodynamics of Tsunami Waves (1996), IPF Ross Akad Nauk: IPF Ross Akad Nauk Nizhny Novgorod · Zbl 0925.76129
[4] Oleinik, O. A.; Radkevich, E. V., Second order equations with non-negative characteristic form, (Advances in Science and Technology Ser Math Math Anal 1969 (1971), VINITI: VINITI Moscow), 7-252
[5] Carrier, G. F.; Greenspan, H. P., Water waves of finite amplitude on a sloping beach, J. Fluid Mech, 4, 1, 97-109 (1958) · Zbl 0080.19504
[6] Shokin, Y.u. I.; Chubarov, L. B.; Marchuk, A. G.; Simonov, K. V., Computational Experiment in the Tsunami Problem (1989), Nauka: Nauka Novosibirsk · Zbl 0712.76005
[7] Pelinovsky, E. N.; Mazova, R. K.h, Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles, Natural Hazards, 6, 3, 227-249 (1992)
[8] Dobrokhotov, S. Y.u.; Sinitsyn, S. O.; Tirozzi, B., Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. I. The Cauchy problem, Russ J Math Phys, 14, 1, 28-56 (2007) · Zbl 1146.35055
[9] Didenkulova, I. I.; Pelinovsky, E. N., Run-up of long waves on a beach: The influence of the incident wave form, Oceanology, 48, 1, 1-6 (2008)
[10] Dobrokhotov, S. Y.u.; Tirozzi, B., Localized solutions of one-dimensional non-linear shallow-water equations with \(c = \sqrt{x}\) velocity, Russ Math Surveys, 65, 1, 177-179 (2010) · Zbl 1196.35161
[11] Maslov, V. P., Théorie des perturbations et méthodes asymptotiques (1972), Dunod: Dunod Paris · Zbl 0247.47010
[12] Maslov, V. P.; Fedoryuk, M. V., Semi-Classical Approximation in Quantum Mechanics (1981), Reidel: Reidel Dordrecht · Zbl 0458.58001
[13] Maslov, V. P., Operational Methods (1976), Mir: Mir Moscow · Zbl 0449.47002
[14] Kabanikhin, S. I.; Krivorot’ko, O. I., An algorithm for computing wavefront amplitudes and inverse problems (tsunami, electrodynamics, acoustics, and viscoelasticity), Dokl Math, 93, 1, 103-107 (2016) · Zbl 1348.86005
[15] Kabanikhin, S. I.; Krivorot’ko, O. I., A numerical algorithm for computing tsunami wave amplitudes, Numer Anal Appl, 9, 2, 118-128 (2016) · Zbl 1349.86022
[16] Dobrokhotov, S. Y.u.; Shafarevich, A. I.; Tirozzi, B., Representations of rapidly decaying functions by the Maslov canonical operator, Math Notes, 82, 5, 713-717 (2007) · Zbl 1344.58014
[17] Dobrokhotov, S. Y.u.; Shafarevich, A. I.; Tirozzi, B., Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations, Russ J Math Phys, 15, 2, 192-221 (2008) · Zbl 1180.35336
[18] Dobrokhotov, S. Y.u.; Tirozzi, B.; Vargas, C. A., Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations, Russ J Math Phys, 16, 2, 228-245 (2009) · Zbl 1178.35301
[19] Nazaikinskii, V. E., Phase space geometry for a wave equation degenerating on the boundary of the domain, Math Notes, 92, 1, 144-148 (2012) · Zbl 1308.58017
[20] Vukašinac, T.; Zhevandrov, P., Geometric asymptotics for a degenerate hyperbolic equation, Russ J Math Phys, 9, 3, 371-381 (2002) · Zbl 1104.35309
[21] Dobrokhotov, S. Y.u.; Nazaikinskii, V. E.; Tirozzi, B., Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I, Russ J Math Phys, 17, 4, 434-447 (2010) · Zbl 1387.35404
[22] Dobrokhotov, S. Y.u.; Nazaikinskii, V. E.; Tirozzi, B., Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data, St Petersburg Math J, 22, 6, 895-911 (2011) · Zbl 1230.35057
[23] Nazaikinskii, V. E., The Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary, Math Notes, 96, 2, 248-260 (2014) · Zbl 1330.53108
[24] Nazaikinskii, V. E., Maslov’s canonical operator for degenerate hyperbolic equations, Russ J Math Phys, 21, 2, 289-290 (2014) · Zbl 1316.35207
[25] Sekerzh-Zen’kovich, S. Y.a., Simple asymptotic solution of the Cauchy-Poisson problem for head waves, Russ J Math Phys, 16, 2, 315-322 (2009) · Zbl 1179.35248
[26] Dobrokhotov, S. Y.u.; Nazaikinskii, V. E.; Tirozzi, B., Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data, Russ J Math Phys, 20, 4, 389-401 (2013) · Zbl 1320.35180
[27] Zav’yalov, Y.u. S.; Kvasov, B. I.; Miroshnichenko, V. L., Spline Function Methods (1980), Nauka: Nauka Moscow · Zbl 0524.65007
[28] Bogachev, KYu., Practical Work on a Computer. Approximation Methods for Functions (2002), TsPI Mekh Mat Fak: TsPI Mekh Mat Fak Moscow
[30] Dobrokhotov, S. Y.u.; Tirozzi, B.; Tolchennikov, A. A., Asymptotics of shallow water equations on the sphere, Russ J Math Phys, 21, 4, 430-449 (2014) · Zbl 1311.76012
[31] Sahal, A.; Roger, J.; Allgeyer, S., The tsunami triggered by the 21 May 2003 Boumerdès-Zemmouri (Algeria) earthquake: Field investigations on the French Mediterranean coast and tsunami modelling, Nat Hazards Earth Syst Sci, 9, 1823-1834 (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.