### Abstract

Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ε{lunate} with ε{lunate} > 0 a small parameter, then a time-dependent forced Korteweg-de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ≥ 0 (or F ≥ 1, called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ_{0} of λ. For λ ≥ λ_{0} there are steady solutions, while for 0 ≤ λ < λ_{0} no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ_{0}, the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 < λ < λ_{0} a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.

Original language | English |
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Pages (from-to) | 750-770 |

Number of pages | 21 |

Journal | European Journal of Mechanics, B/Fluids |

Volume | 27 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2008 Nov 1 |

### Keywords

- Forced gravity waves
- Supercritical surface waves

### ASJC Scopus subject areas

- Mechanical Engineering

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## Cite this

*European Journal of Mechanics, B/Fluids*,

*27*(6), 750-770. https://doi.org/10.1016/j.euromechflu.2008.01.006