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Two-dimensional periodic waves in shallow water

Published online by Cambridge University Press:  26 April 2006

Joe Hammack ,
Norman Scheffner  and
Harvey Segur
Joe Hammack
Affiliation:
Department of Aerospace Engineering, Mechanics and Engineering Sciences, University of Florida, Gainesville, FL 32611, USA
Norman Scheffner
Affiliation:
US Army Engineering Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS 39180, USA
Harvey Segur
Affiliation:
Department of Mathematics, State University of New York, Buffalo, NY 14214, USA Present address: Program in Applied Mathematics, University of Colorado, Boulder, CO 80309, USA.
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Abstract

Experimental data are presented that demonstrate the existence of a family of gravitational water waves that propagate practically without change of form on the surface of shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional and fully periodic, i.e. they are periodic in two spatial directions and in time. The amplitudes of these waves need not be small; their form persists even up to breaking. The waves are easy to generate experimentally, and they are observed to propagate in a stable manner, even when perturbed significantly. The measured waves are described with reasonable accuracy by a family of exact solutions of the Kadomtsev-Petviashvili equation (KP solutions of genus 2) over the entire parameter range of the experiments, including waves well outside the putative range of validity of the KP equation. These genus-2 solutions of the KP equation may be viewed as two-dimensional generalizations of cnoidal waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 567 - 589
Copyright
© 1989 Cambridge University Press

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