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Archive for Rational Mechanics and Analysis

, Volume 214, Issue 3, pp 971–1018 | Cite as

Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

  • Vladimir Kozlov
  • Nikolay KuznetsovEmail author
Article

Abstract

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.

Keywords

Vorticity Solitary Wave Dispersion Equation Water Wave Critical Layer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsLinköping UniversityLinköpingSweden
  2. 2.Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt PetersburgRussian Federation

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