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Mat. Zametki, 2014, Volume 95, Issue 3, Pages 359–375 (Mi mz10426)  

This article is cited in 2 scientific papers (total in 2 papers)

Homogenization in the Problem of Long Water Waves over a Bottom Site with Fast Oscillations

V. V. Grushina, S. Yu. Dobrokhotovbc

a National Research University "Higher School of Economics"
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, Moscow
c Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moskovskaya obl.

Abstract: The system of equations of gravity surface waves is considered in the case where the basin's bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom's oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors' works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.

Keywords: surface waves, homogenization, asymptotic methods, small parameter, adiabatic approximation, rapidly oscillating function.

DOI: https://doi.org/10.4213/mzm10426

Full text: PDF file (608 kB)
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English version:
Mathematical Notes, 2014, 95:3, 324–337

Bibliographic databases:

Document Type: Article
UDC: 517.9
Received: 07.07.2013
Revised: 08.11.2013

Citation: V. V. Grushin, S. Yu. Dobrokhotov, “Homogenization in the Problem of Long Water Waves over a Bottom Site with Fast Oscillations”, Mat. Zametki, 95:3 (2014), 359–375; Math. Notes, 95:3 (2014), 324–337

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Dobrokhotov S.Yu. Grushin V.V. Sergeev S.A. Tirozzi B., “Asymptotic theory of linear water waves in a domain with nonuniform bottom with rapidly oscillating sections”, Russ. J. Math. Phys., 23:4 (2016), 455–474  crossref  mathscinet  zmath  isi  elib  scopus
    2. Karaeva D.A. Karaev A.D. Nazaikinskii V.E., “Homogenization Method in the Problem of Long Wave Propagation From a Localized Source in a Basin Over An Uneven Bottom”, Differ. Equ., 54:8 (2018), 1057–1072  crossref  isi  scopus
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