### Abstract

We study the evolution of small-amplitude water waves when the fluid motion is three dimensional. An isotropic pseudodifferential equation that governs the evolution of the free surface of a fluid with arbitrary, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Kortewegde Vries (KdV) equation, the Kadomtsev-Petviashvili (KP) equation, and to the nonlinear shallow water theory in the appropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the computed patterns and nonlinear dispersion relations agree well with those of the doubly periodic theta function solutions to the KP equation. These solutions correspond to traveling hexagonal wave patterns, and they have been compared with experimental measurements by Hammack, Scheffner, and Segur. In the fully two-dimensional long wave case, the solutions deviate considerably from those of KP, indicating the limitation of that equation. In the finite depth case, both resonant and nonresonant traveling wave patterns are obtained.

Original language | English |
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Pages (from-to) | 149-166 |

Number of pages | 18 |

Journal | Studies in Applied Mathematics |

Volume | 97 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 1996 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Studies in Applied Mathematics*,

*97*(2), 149-166. https://doi.org/10.1002/sapm1996972149