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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: nikolay.g.kuznetsov@gmail.com
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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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