3-DIMENSIONAL WATER-WAVES

Citation
Pa. Milewski et Jb. Keller, 3-DIMENSIONAL WATER-WAVES, Studies in applied mathematics, 97(2), 1996, pp. 149-166
Citations number
13
Categorie Soggetti
Mathematics,Mathematics
ISSN journal
00222526
Volume
97
Issue
2
Year of publication
1996
Pages
149 - 166
Database
ISI
SICI code
0022-2526(1996)97:2<149:3W>2.0.ZU;2-7
Abstract
We study the evolution of small-amplitude water waves when the fluid m otion is three dimensional. An isotropic pseudodifferential equation t hat governs the evolution of the free surface of a fluid with arbitrar y, uniform depth is derived. It is shown to reduce to the Benney-Luke equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviash vili (KP) equation, and to the nonlinear shallow water theory in the a ppropriate limits. We compute, numerically, doubly periodic solutions to this equation. In the weakly two-dimensional long wave limit, the c omputed patterns and nonlinear dispersion relations agree well with th ose 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, Sc heffner, and Segur. In the fully two-dimensional long wave case, the s olutions deviate considerably from those of KP, indicating the limitat ion of that equation. In the finite depth case, both resonant and nonr esonant traveling wave patterns are obtained.