## Abstract

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. A Hamiltonian system of Whitham type allowing for two-way wave propagation is also derived. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to different free surface models: the KdV equation, the BBM equation, and the Padé (2,2) model. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than the three considered models.

Original language | British English |
---|---|

Pages (from-to) | 99-107 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 309 |

DOIs | |

State | Published - 19 Aug 2015 |

## Keywords

- Hamiltonian models
- Nonlinear dispersive equations
- Nonlocal equations
- Solitary waves
- Surface waves