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On the Benjamin–Lighthill conjecture for water waves with vorticity

Published online by Cambridge University Press:  24 July 2017

V. Kozlov ,
N. Kuznetsov Open the ORCID record for N. Kuznetsov [Opens in a new window]  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, V.O., Bol’shoy pr. 61, St Petersburg 199178, Russia
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

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 961 - 1001
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© 2017 Cambridge University Press 

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