Comptes Rendus
no. 7
On two-dimensional water waves in a canal
[Sur les ondes de surface bidimensionnelles dans un canal]

Présenté par : Évariste Sanchez-Palencia

Vladimir Kozlov1  ; Nikolay Kuznetsov2
1 Department of Mathematics, Linköping University, 581 83 Linköping, Sweden
2 Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol'shoy pr. 61, St. Petersburg 199178, Russia
Comptes Rendus. Mécanique, Volume 331 (2003) no. 7, pp. 489-494.

Cette Note porte sur un problème aux valeurs propres avec le paramètre spectral dans la condition aux limites. Le problème pour l'équation de Laplace bidimensionnelle décrit les ondes de surface libres, harmoniques dans le temps, dans un canal limité supérieurement par une surface libre horizontale. On démontre l'existence d'un domaine tel que au moins une des fonctions propres a une ligne nodale avec les deux extrémités sur la surface libre. Kuttler avait utilisé essentiellement la non - existence de telles lignes nodales dans sa démonstration du caractère simple de la valeur propre de ballottement dans le cas bidimensionnel. Nous proposons donc un nouveau principe variationnel pour prouver ce fait.

This Note deals with an eigenvalue problem that contains a spectral parameter in a boundary condition. The problem for the two-dimensional Laplace equation describes free, time-harmonic water waves in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is shown that there exists a domain for which at least one of eigenfunctions has a nodal line with both ends on the free surface. Since Kuttler essentially used the non-existence of such nodal lines in his proof of simplicity of the fundamental sloshing eigenvalue in the two-dimensional case, we propose a new variational principle for demonstrating this latter fact.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-0721(03)00105-0
Keywords: Fluid mechanics, Sloshing problem, Nodal line, Fundamental eigenvalue, Variational principle
Mot clés : Mécanique des fluides, Oscillations libres d'une fluide, Courbe nodale, Valeur propre fondamenale, Principe variationnelle
Vladimir Kozlov 1 ; Nikolay Kuznetsov 2

1 Department of Mathematics, Linköping University, 581 83 Linköping, Sweden
2 Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol'shoy pr. 61, St. Petersburg 199178, Russia 
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Vladimir Kozlov; Nikolay Kuznetsov. On two-dimensional water waves in a canal. Comptes Rendus. Mécanique, Volume 331 (2003) no. 7, pp. 489-494. doi : 10.1016/S1631-0721(03)00105-0. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(03)00105-0/

[1] D.W. Fox; J.R. Kuttler Sloshing frequencies, Z. Angew. Math. Phys., Volume 34 (1983), pp. 668-696

[2] J.R. Kuttler A nodal line theorem for the sloshing problem, SIAM J. Math. Anal., Volume 15 (1984), pp. 1234-1237

[3] R. Courant; D. Hilbert, Methods of Mathematical Physics, 1, Interscience, 1953

[4] N. Kuznetsov; V. Maz'ya; B. Vainberg Linear Water Waves: A Mathematical Approach, Cambridge University Press, 2002

[5] M. McIver An example of non-uniqueness in the two-dimensional linear water wave problem, J. Fluid Mech., Volume 315 (1996), pp. 257-266

[6] I.S. Gradshteyn; I.M. Ryzhik Table of Intergals, Series, and Products, Academic Press, 1980

[7] F. John On the motion of floating bodies, II, Comm. Pure Appl. Math., Volume 3 (1950), pp. 45-101

[8] H. Lamb Hydrodynamics, Cambridge University Press, 1932

[9] N.G. Kuznetsov; O.V. Motygin The Steklov problem in a half-plane: the dependence of eigenvalues on a piecewise constant coefficient, Zap. Sem. POMI, Volume 297 (2003), pp. 162-190 (in Russian)

[10] A.M.J. Davis Short surface waves in a canal: dependence of frequency on curvature, Proc. Roy. Soc. London Ser. A, Volume 313 (1969), pp. 249-260

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