Stable three-dimensional waves of nearly permanent form on deep water

Walter Craig, Diane M. Henderson, Maribeth Oscamou, Harvey Segur

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9 Scopus citations


Consider a uniform train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water. One approximate model of the evolution of these waves is a pair of coupled nonlinear Schrödinger equations, which neglects any dissipation of the waves. We show that in this model, such a wave train is linearly unstable to small perturbations in the initial data, because of a Benjamin-Feir-type instability. We also show that when the model of coupled equations is generalized to include appropriate wave damping, the corresponding wave train is linearly stable to perturbations in the initial data. Therefore, according to the damped model, the two-dimensional surface wave patterns studied by Hammack et al. [J.L. Hammack, D.M. Henderson, H. Segur, Progressive waves with persistent, two-dimensional surface patterns in deep water, J. Fluid Mech. 532 (2005) 1-51] are linearly stable in the presence of wave damping.

Original languageEnglish (US)
Pages (from-to)135-144
Number of pages10
JournalMathematics and Computers in Simulation
Issue number2-3
StatePublished - Mar 7 2007

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science
  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


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