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

Walter Craig, Diane Marie Henderson, Maribeth Oscamou, Harvey Segur

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

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
Volume74
Issue number2-3
DOIs
StatePublished - Mar 7 2007

Fingerprint

Water
Three-dimensional
Linearly
Surface Waves
Surface waves
Damping
Approximate Model
Form
Small Perturbations
Nonlinear equations
Damped
Dissipation
Nonlinear Equations
Unstable
Model
Perturbation
Fluid
Fluids

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)
  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics

Cite this

Craig, Walter ; Henderson, Diane Marie ; Oscamou, Maribeth ; Segur, Harvey. / Stable three-dimensional waves of nearly permanent form on deep water. In: Mathematics and Computers in Simulation. 2007 ; Vol. 74, No. 2-3. pp. 135-144.
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Stable three-dimensional waves of nearly permanent form on deep water. / Craig, Walter; Henderson, Diane Marie; Oscamou, Maribeth; Segur, Harvey.

In: Mathematics and Computers in Simulation, Vol. 74, No. 2-3, 07.03.2007, p. 135-144.

Research output: Contribution to journalArticle

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AB - 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.

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