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Mat. Sb., 2008, Volume 199, Number 12, Pages 53–78 (Mi msb3939)  

This article is cited in 27 scientific papers (total in 27 papers)

Concentration of trapped modes in problems of the linearized theory of water waves

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: Problems of the linearized theory of waves on the surface of an ideal fluid filling a half-space or an infinite 3D-canyon are considered. Families of submerged or surface-piercing bodies parametrized by a characteristic linear size $h>0$ are found that have the following property: for each $d>0$ and each positive integer $N$ there exists $h(d,N)>0$ such that for $h\in(0,h(d,N)]$ the interval $[0,d]$ of the continuous spectrum of the corresponding problem contains at least $N$ eigenvalues corresponding to trapped modes, that is, to solutions of the homogeneous problem that decay exponentially at infinity and possess finite energy.
Bibliography: 38 titles.

DOI: https://doi.org/10.4213/sm3939

Full text: PDF file (734 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2008, 199:12, 1783–1807

Bibliographic databases:

UDC: 517.958:531.327
MSC: Primary 76B15; Secondary 35Q35
Received: 28.08.2007 and 17.09.2008

Citation: S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Mat. Sb., 199:12 (2008), 53–78; Sb. Math., 199:12 (2008), 1783–1807

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. A. Nazarov, “Simple method for finding trapped modes in problems of the linear theory of surface waves”, Dokl. Math., 80:3 (2009), 914–917  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. Nazarov S.A., Videman J.H., “A sufficient condition for the existence of trapped modes for oblique waves in a two-layer fluid”, Proc. R. Soc. A, 465:2112 (2009), 3799–3816  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Nazarov S.A., “A novel approach for detecting trapped surface waves in a canal with periodic underwater topography”, Comptes Rendus Mécanique, 337:8 (2009), 610–615  crossref  adsnasa  isi  elib  scopus
    4. S. A. Nazarov, “Sufficient conditions of the existence of trapped modes in problems of the linear theory of surface waves”, J. Math. Sci. (N. Y.), 167:5 (2010), 713–725  mathnet  crossref  elib
    5. S. A. Nazarov, “The point spectrum of water-wave problem in intersecting canals”, J. Math. Sci. (N. Y.), 175:6 (2011), 685–697  mathnet  crossref
    6. S. A. Nazarov, “On the asymptotics of an eigenvalue of a waveguide with thin shielding obstacle and Wood's anomalies”, J. Math. Sci. (N. Y.), 178:3 (2011), 292–312  mathnet  crossref
    7. Cardone G., Durante T., Nazarov S.A., “Water-waves modes trapped in a canal by a near-surface rough body”, ZAMM Z. Angew. Math. Mech., 90:12 (2010), 983–1004  crossref  mathscinet  zmath  isi  elib  scopus
    8. Nazarov S.A., Videman J.H., “Existence of edge waves along three-dimensional periodic structures”, J. Fluid Mech., 659 (2010), 225–246  crossref  mathscinet  zmath  isi  elib  scopus
    9. S. A. Nazarov, “Formation of gaps in the spectrum of the problem of waves on the surface of a periodic channel”, Comput. Math. Math. Phys., 50:6 (2010), 1038–1054  mathnet  crossref  mathscinet  adsnasa  isi  elib
    10. S. A. Nazarov, “Localization of surface waves by small perturbations of the boundary of a semisubmerged body”, J. Appl. Industr. Math., 6:2 (2012), 216–223  mathnet  crossref  mathscinet
    11. Nazarov S.A., “Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies”, J. Math. Sci., 175:3 (2011), 309–348  crossref  mathscinet  zmath  elib  scopus
    12. J. H. Videman, V. Chiado' Piat, S. A. Nazarov, “Asymptotics of frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer”, J. Math. Sci. (N. Y.), 185:4 (2012), 536–553  mathnet  crossref  mathscinet
    13. S. A. Nazarov, J. Taskinen, “Double-sided estimates for eigenfrequencies in the John problem for freely floating body”, J. Math. Sci. (N. Y.), 185:5 (2012), 707–720  mathnet  crossref  mathscinet
    14. Nazarov S.A., Videman J.H., “Trapping of water waves by freely floating structures in a channel”, Proc. R. Soc. A, 467:2136 (2011), 3613–3632  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. Nazarov S.A., “A body traps as many water-wave modes in a symmetric channel as it wishes”, Russ. J. Math. Phys., 18:2 (2011), 183–194  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    16. S. A. Nazarov, “Concentration of frequencies of trapped waves in problems on freely floating bodies”, Sb. Math., 203:9 (2012), 1269–1294  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. S. A. Nazarov, “Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions”, Comput. Math. Math. Phys., 52:11 (2012), 1574–1589  mathnet  crossref  mathscinet  isi  elib  elib
    18. Cal F.S., Dias G.S.A., Videman J.H., “Existence of trapped modes along periodic structures in a two-layer fluid”, Quart. J. Mech. Appl. Math., 65:2 (2012), 273–292  crossref  mathscinet  zmath  isi  elib  scopus
    19. Kamotski I.V. Maz'ya V.G., “On the linear water wave problem in the presence of a critically submerged body”, SIAM J. Math. Anal., 44:6 (2012), 4222–4249  crossref  mathscinet  zmath  isi  elib  scopus
    20. Piat V.Ch., Nazarov S.A., Ruotsalainen K., “Spectral gaps for water waves above a corrugated bottom”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 469:2149 (2013), 20120545  crossref  mathscinet  zmath  isi  elib  scopus
    21. Nazarov S.A., Taskinen J., Videman J.H., “Asymptotic Behavior of Trapped Modes in Two-Layer Fluids”, Wave Motion, 50:2 (2013), 111–126  crossref  mathscinet  zmath  isi  elib  scopus
    22. Nazarov S.A., Taskinen J., “Localization Estimates for Eigenfrequencies of Waves Trapped by a Freely Floating Body in a Channel”, SIAM J. Math. Anal., 45:4 (2013), 2523–2545  crossref  mathscinet  zmath  isi  elib  scopus
    23. Kamotski I., Mazya V., “Estimate for a Solution to the Water Wave Problem in the Presence of a Submerged Body”, Russ. J. Math. Phys., 20:4 (2013), 453–467  crossref  mathscinet  zmath  isi  elib  scopus
    24. Nazarov S.A., Taskinen J., “Properties of the Spectrum in the John Problem on a Freely Floating Submerged Body in a Finite Basin”, Differ. Equ., 49:12 (2013), 1544–1559  crossref  mathscinet  zmath  isi  elib  scopus
    25. S. A. Nazarov, “Asymptotic expansions of eigenvalues of the Steklov problem in singularly perturbed domains”, St. Petersburg Math. J., 26:2 (2015), 273–318  mathnet  crossref  mathscinet  isi  elib
    26. S. A. Nazarov, “Modeling of a Singularly Perturbed Spectral Problem by Means of Self-Adjoint Extensions of the Operators of the Limit Problems”, Funct. Anal. Appl., 49:1 (2015), 25–39  mathnet  crossref  crossref  zmath  isi  elib
    27. Durante T., “Accumulation Effect For Water-Waves Mode Trapped in a Canal”, Proceedings of the International Conference of Numerical Analysis and Applied Mathematics 2014 (Icnaam-2014), AIP Conference Proceedings, 1648, eds. Simos T., Tsitouras C., Amer Inst Physics, 2015, UNSP 390007  crossref  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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