Comptes Rendus
no. 8
A novel approach for detecting trapped surface waves in a canal with periodic underwater topography
[Une nouvelle approche pour la détection d'ondes de surfaces piégées dans un canal à topographie souterraine périodique]

Présenté par : Jean-Baptiste Leblond

Sergey A. Nazarov1
1 Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr., 61, 199178, St.-Petersburg, Russia
Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 610-615.

Une nouvelle approche est proposée pour détecter les modes piégés dans les canaux périodiques. La condition suffisante obtenue est nouvelle également dans le cas d'un canal droit avec un corps traversant la surface de l'eau.

A new approach is proposed to detect trapped modes in periodic canals. The obtained sufficient condition is new even for a straight canal with a body piercing the water surface as well.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2009.06.029
Keywords: Waves, Surface water-waves, Trapped modes, Localized solutions
Mot clés : Ondes, Ondes de surface marines, Modes piégés, Solutions localisées

Sergey A. Nazarov 1

1 Institute of Mechanical Engineering Problems, V.O., Bol'shoi pr., 61, 199178, St.-Petersburg, Russia 
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Sergey A. Nazarov. A novel approach for detecting trapped surface waves in a canal with periodic underwater topography. Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 610-615. doi : 10.1016/j.crme.2009.06.029. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.06.029/

[1] J.J. Stoker Water Waves: The Mathematical Theory with Applications, John Willey & Sons, Inc., New York, 1992 (Reprint of the 1957 original)

[2] N. Kuznetsov; V. Maz'ya; B. Vainberg Linear Water Waves, Cambridge University Press, Cambridge, 2002

[3] S.A. Nazarov; J. Taskinen On the spectrum of the Steklov problem in a domain with a peak, Vestnik St. Petersburg Univ. Math., Volume 41 (2008) no. 1, pp. 45-52

[4] O.A. Ladyzhenskaya The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985

[5] J. Sanchez-Hubert; E. Sanchez-Palencia Vibration and Coupling of Continuous Systems: Asymptotic Methods, Springer-Verlag, Heidelberg, 1989

[6] I.V. Kamotskii; S.A. Nazarov Elastic waves localized near periodic sets of flaws, Dokl. Ross. Akad. Nauk, Volume 368 (1999) no. 6, pp. 771-773 (English transl.: Dokl. Phys., 44, 10, 1999, pp. 715-717)

[7] R.M. Garipov On the linear theory of gravity waves: the theorem of existence and uniqueness, Arch. Rational Mech. Anal., Volume 24 (1967), pp. 352-362

[8] F. Ursell Mathematical aspects of trapping modes in the theory of surface waves, J. Fluid Mech., Volume 183 (1987), pp. 421-437

[9] A.-S. Bonnet-Ben Dhia; P. Joly Mathematical analysis of guided water-waves, SIAM J. Appl. Math., Volume 53 (1993), pp. 1507-1550

[10] N. Kuznetsov; R. Porter; D.V. Evans; M.J. Simon Uniqueness and trapped modes for surface-piercing cylinders in oblique waves, J. Fluid Mech., Volume 365 (1998), pp. 351-368

[11] C.M. Linton; P. McIver Embedded trapped modes in water waves and acoustics, Wave Motion, Volume 45 (2007), pp. 16-29

[12] I.M. Gel'fand Expansions in eigenfunctions of an equation with periodic coefficients, Dokl. Acad. Nauk SSSR, Volume 73 (1950), pp. 1117-1120

[13] P.A. Kuchment Floquet theory for partial differential equations [in Russian], Uspekhi Mat. Nauk, Volume 37 (1982) no. 4, pp. 3-52 (English transl.: Russian Math. Surveys, 37, 4, 1982, pp. 1-60)

[14] S.A. Nazarov; B.A. Plamenevsky Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, New York, 1994

[15] P. Kuchment Floquet Theory for Partial Differential Equations, Birkhäuser, Basel, 1993

[16] I.C. Gohberg; M.G. Krein Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, 1969

[17] S.A. Nazarov Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR Ser. Mat., Volume 45 (1981) no. 1, pp. 101-112 (English transl.: Math. USSR Izv., 18, 1, 1982, pp. 89-98)

[18] T. Kato Perturbation Theory for Linear Operators, Springer-Verlag, Heidelberg, 1966

[19] O.V. Motygin On trapping of surface water waves by cylindrical bodies in a channel, Wave Motion, Volume 45 (2008), pp. 940-951

[20] S.A. Nazarov Concentration of trapped modes in problems of the linearized theory of water waves, Mat. Sb., Volume 199 (2008) no. 12, pp. 53-78 (English transl.: Sb. Math., 199, 12, 2008, pp. 1783-1807)

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☆ The author gratefully acknowledges the support by N.W.O., the Netherlands Organization for Scientific Research.

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