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 languageEnglish
Pages (from-to)149-166
Number of pages18
JournalStudies in Applied Mathematics
Volume97
Issue number2
DOIs
Publication statusPublished - 1 Jan 1996

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Water waves
Water Waves
Three-dimensional
Kadomtsev-Petviashvili Equation
Nonlinear Dispersion
Fluid
Fluids
Shallow Water
Theta Functions
Dispersion Relation
Periodic Functions
Korteweg-de Vries Equation
Hexagon
Traveling Wave
Free Surface
Periodic Solution
Motion
Arbitrary
Water

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Three-dimensional water waves. / Milewski, Paul A.; Keller, Joseph B.

In: Studies in Applied Mathematics, Vol. 97, No. 2, 01.01.1996, p. 149-166.

Research output: Contribution to journalArticle

Milewski, Paul A. ; Keller, Joseph B. / Three-dimensional water waves. In: Studies in Applied Mathematics. 1996 ; Vol. 97, No. 2. pp. 149-166.
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