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Steady water waves with vorticity: an analysis of the dispersion equation

Published online by Cambridge University Press:  16 June 2014

V. Kozlov ,
N. Kuznetsov  and
E. Lokharu
V. Kozlov
Affiliation:
Department of Mathematics, Linköping University, S-581 83, Linköping, Sweden
N. Kuznetsov*
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, VO, Bol’shoy pr. 61, St Petersburg 199178, Russian Federation
E. Lokharu
Affiliation:
Department of Mathematics, Linköping University, S-581 83, Linköping, Sweden
*
†Email address for correspondence: [email protected]
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Abstract

Two-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , R3
Copyright
© 2014 Cambridge University Press 

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References

Benjamin, T. B. 1995 Verification of the Benjamin–Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356.Google Scholar
Coddington, E. A. & Levinson, N. 1955 Theory of Ordinary Differential Equations. McGraw-Hill.Google Scholar
Constantin, A. 2012 Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Commun. Pure Appl. Anal. 11, 1397–1406.Google Scholar
Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481–527; (see also C. R. Acad. Sci. Paris, Ser. I, 335 (2002), 797–800).Google Scholar
Constantin, A. & Strauss, W. 2011 Periodic travelling gravity water waves with discontinuous vorticity. Arch. Rat. Mech. Anal. 202, 133–175.Google Scholar
Constantin, A. & Varvaruca, E. 2011 Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Rat. Mech. Anal. 199, 33–67.Google Scholar
Crandall, M. & Rabinowitz, P. 1971 Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340.CrossRefGoogle Scholar
Doole, S. H. 1998 The pressure head and flowforce parameter space for waves with constant vorticity. Q. J. Mech. Appl. Maths 51, 61–71.Google Scholar
Dubreil-Jacotin, M.-L. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl. 13, 217–291.Google Scholar
Ehrnström, M., Escher, J. & Villari, G. 2012 Steady water waves with multiple critical layers: interior dynamics. J. Math. Fluid Mech. 14, 407–419.Google Scholar
Ehrnström, M., Escher, J. & Wahlén, E. 2011 Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456.Google Scholar
Henry, D. 2013a Steady periodic waves bifurcating for fixed-depth rotational flows. Q. Appl. Maths 71, 455–487.CrossRefGoogle Scholar
Henry, D. 2013b Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface. Nonlinear Anal. Real World Appl. 14, 1034–1043.Google Scholar
Keady, G. & Norbury, J. 1978 Waves and conjugate streams with vorticity. Mathematika 25, 129–150.Google Scholar
Kozlov, V. & Kuznetsov, N. 2011 Steady free-surface vortical flows parallel to the horizontal bottom. Q. J. Mech. Appl. Maths 64, 371–399.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2014 Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents. Arch. Rat. Mech. Math. Anal. (submitted). Preprint available online at http://arXiv.org/abs/1207.5181.Google Scholar
Lavrentiev, M. & Shabat, B. 1980 Effets Hydrodynamiques et Modèles Mathématiques. Mir Publishers.Google Scholar
Martin, C. 2014 Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete Continuous Dyn. Syst. Ser. A 34, 3109–3123.Google Scholar
Martin, C. & Matioc, B.-V. 2014a Existence of capillary–gravity water waves with piecewise constant vorticity. J. Differ. Equ. 256, 3086–3114.Google Scholar
Martin, C. & Matioc, B.-V. 2014b Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results. J. Nonlinear Sci.; doi:10.1007/s00332-014-9201-1.CrossRefGoogle Scholar
Martin, C. & Matioc, B.-V. 2014c Capillary–gravity water waves with discontinuous vorticity: existence and regularity results. Commun. Math. Phys.; doi:10.1007/s00220-014-1918-z.Google Scholar
Strauss, W. 2010 Steady water waves. Bull. Am. Math. Soc. 47, 671–694.Google Scholar
Swan, C., Cummins, I. & James, R. 2001 An experimental study of two-dimensional surface water waves propagating in depth-varying currents. J. Fluid Mech. 428, 273–304.Google Scholar
Thomas, G. P. 1990 Wave–current interactions: an experimental and numerical study. J. Fluid Mech. 216, 505–536.Google Scholar
Wahlén, E. 2006 Steady periodic capillary–gravity waves with vorticity. SIAM J. Math. Anal. 38, 921–943.Google Scholar
Wahlén, E. 2009 Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483.Google Scholar