Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-29T19:58:08.400Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave Trains Associated with a Cascade of Bifurcations of Space-Time Caustics over Elongated Underwater Banks

Published online by Cambridge University Press:  17 September 2013

S. Yu. Dobrokhotov ,
D. A. Lozhnikov  and
V. E. Nazaikinskii
S. Yu. Dobrokhotov*
Affiliation:
A. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences Moscow Institute of Physics and Technology
D. A. Lozhnikov
Affiliation:
A. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences Moscow Institute of Physics and Technology
V. E. Nazaikinskii
Affiliation:
A. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences Moscow Institute of Physics and Technology
*
⋆ Corresponding author. E-mail: dobr@ipmnet.ru
Get access

Abstract

We study the behavior of linear nonstationary shallow water waves generated by an instantaneous localized source as they propagate over and become trapped by elongated underwater banks or ridges. To find the solutions of the corresponding equations, we use an earlier-developed asymptotic approach based on a generalization of Maslov’s canonical operator, which provides a relatively simple and efficient analytic-numerical algorithm for the wave field computation. An analysis of simple examples (where the bank and source shapes are given by certain elementary functions) shows that the appearance and dynamics of trapped wave trains is closely related to a cascade of bifurcations of space-time caustics, the bifurcation parameter being the bank length-to-width ratio.

Keywords

linearized shallow water equationsasymptoticswave front singularity evolutiontrapped nonstationary waveunderwater ridgeunderwater bank
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 1 - 12
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

V. Arnold, A. Varchenko, S. Gussein-Zade. Singularities of differentiable maps. Nauka, Moscow, 1984. (Russian)
V. M. Babich, V. S. Buldyrev. Asymptotic methods in short wave diffraction problems. Nauka, Moscow, 1972. (Russian)
Berry, M. V., ODell, D. H. J.. Ergodicity in wave–wave diffraction. J. Phys. A: Math. Gen., 32 (1999), 3571–3582. CrossRefGoogle Scholar
Berry, M. V., Upstill, C.. Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Opt., 18 (1980), 257–346. CrossRefGoogle Scholar
Dobrokhotov, S. Yu.. Asymptotic behavior of water surface waves trapped by shores and irregularities of the bottom relief. Dokl. Akad. Nauk SSSR, 289:3 (1986), 575–579. English transl., Soviet Phys. Dokl., 31:7 (1986), 537–539. Google Scholar
Dobrokhotov, S. Yu., Nekrasov, R., Tirozzi, B.. Asymptotic solutions of the linear shallow water equations with localized initial data. J. Engng Math., 69:2 (2011), 225–242. CrossRefGoogle Scholar
Dobrokhotov, S., Rouleux, M.. The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory. Asymptotic Analysis, 74:1–2 (2011), 33–73. Google Scholar
Dobrokhotov, S. Yu., Sekerzh-Zenkovich, S. Ya.. A class of exact algebraic localized solutions of the multidimensional wave equation. Mat. Zametki, 88:6 (2010), 942–945. English transl., Math. Notes, 88:6 (2010), 894–897. Google Scholar
Dobrokhotov, S. Yu., Sekerzh-Zenkovich, S. Ya., Tirozzi, B., Tudorovskiy, T. Ya.. Description of tsunami propagation based on the Maslov canonical operator. Dokl. Ross. Akad. Nauk, 409:2 (2006), 171–175. English transl., Russian Acad. Sci. Dokl. Math., 74:1 (2006), 592–596. Google Scholar
Dobrokhotov, S. Yu., Sekerzh-Zenkovich, S. Ya, Tirozzi, B., Volkov, B.. Explicit asymptotics for tsunami waves in framework of the piston model. Russian J. Earth Sci., 8:ES403, 1–12 (2006). CrossRefGoogle Scholar
Dobrokhotov, S. Yu., Ya Sekerzh-Zenkovich, S., Tirozzi, B., Volkov, B.. Asymptotic description of tsunami waves in the framework of the piston model: The general constructions and explicitly solvable models. Fund. Appl. Geophysics, 2 (2009), 15–29. (Russian) Google Scholar
Dobrokhotov, S., Shafarevich, A., Tirozzi, B.. Localized wave and vortical solutions to linear hyperbolic systems and their application to the linear shallow water equations. Russian J. Math. Phys., 15:2 (2008), 192–221. CrossRefGoogle Scholar
Dobrokhotov, S. Yu., 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. Russian J. Math. Phys., 16:2 (2009), 228–245. CrossRefGoogle Scholar
Dobrokhotov, S. Yu., Zhevandrov, P. N.. Nonstandard characteristics and Maslov’s operator method in linear problems on transient water waves. Funktsional. Anal. i Prilozhen., 19:4 (1985), 43–54. English transl., Functional Anal. Appl., 19:4 (1985), 285–295. Google Scholar
Dotsenko, S. F., Sergievskii, B. Yu., Cherkasov, L. V.. Space tsunami waves generated by alternating displacement of the ocean surface. Tsunami Research, 1 (1986), 7–14. (Russian) Google Scholar
Duistermaat, J. J.. Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math., 27 (1974), 207–281. CrossRefGoogle Scholar
M. V. Fedoryuk. Mountain pass method. Moscow, Nauka, 1977.
Garipov, R. M.. Transient waves over an underwater ridge. Dokl. Akad. Nauk SSSR, 161:3 (1965), 547–550. (Russian) Google Scholar
D. A. Indeytsev, N. G. Kuznetsov, O. V. Motygin, Yu. A. Mochalova. Trapped linear waves. St.-Petersburg State University Press, St.-Petersburg, 2007. (Russian)
Keller, J. B.. Surface waves on water of nonuniform depth. J. Fluid Mech., 4 (1958), 607–614. CrossRefGoogle Scholar
P. H. Le Blond, L. A. Mysak. Waves in the ocean. Elsevier, Amsterdam, 1978.
Lozhnikov, D. A.. Analytic-numerical description of asymptotic solutions of a Cauchy problem in a neighbourhood of singularities for a linearized system of shallow water equations. Russian J. Math. Phys., 19 (1958), 607–614. Google Scholar
Matveev, V. S.. The Asymptotic eigenfunctions of the operator ∇D(x, y)∇ corresponding to liouville metrics and waves on water captured by bottom irregularities. Mat. Zametki, 64 (1958), 607–614. English transl., Math. Notes, 64:3 (1998), 357–363. Google Scholar
C. C. Mei. The applied dynamics of ocean surface waves. World Scientific, Singapore, 1989.
E. N. Pelinovskii. Hydrodynamics of tsunami waves. Nizhni Novgorod, 1996. (Russian)
Sekerzh-Zenkovich, S. Ya.. Simple asymptotic solution to the Cauchy–Poisson problem for leading waves. Russian J. Math. Phys., 16 (1958), 607–614. Google Scholar
R. Thom. Structural stability and morphogenesis. Advanced Books Classics. Benjamin, Reading, 1975.
B. R. Vainberg. Asymptotic methods in equations of mathematical physics. Moscow University, Moscow, 1982. English transl., Gordon and Breach, New York, 1989.
S. Wang. The Propagation of the Leading Wave. ASCE Specialty Conference on Coastal Hydrodynamics, University of Delaware, June 29–July 1 (1987), 657–670.