## Abstract

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 language | English (US) |
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Pages (from-to) | 29-36 |

Number of pages | 8 |

Journal | Mathematics and Computers in Simulation |

Volume | 80 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2009 |

## All Science Journal Classification (ASJC) codes

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

## Keywords

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