Nonlinear surface waves interacting with a linear shear current

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To describe the evolution of fully nonlinear surface gravity waves in a linear shear current, a closed system of exact evolution equations for the free surface elevation and the free surface velocity potential is derived using a conformal mapping technique. Traveling wave solutions of the system are obtained numerically and it is found that the maximum wave amplitude for a positive shear current is much smaller than that in the absence of any shear while the opposite is true for a negative shear current. The new evolution equations are also solved numerically using a pseudo-spectral method to study the Benjamin-Feir instability of a modulated wave train in both positive and negative shear currents. With a fixed wave slope, compared with the irrotational case, the envelope of the modulated wave train grows faster in a positive shear current and slower in a negative shear current.

Original languageEnglish (US)
Pages (from-to)29-36
Number of pages8
JournalMathematics and Computers in Simulation
Issue number1
StatePublished - Sep 2009

All Science Journal Classification (ASJC) codes

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


  • Benjamin-Feir instability
  • Linear shear current
  • Nonlinear surface waves


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