Journal of Fluid Mechanics

Two-dimensional periodic waves in shallow water

Joe  Hammack a1, Norman  Scheffner a2 and Harvey  Segur a3p1
a1 Department of Aerospace Engineering, Mechanics and Engineering Sciences, University of Florida, Gainesville, FL 32611, USA
a2 US Army Engineering Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS 39180, USA
a3 Department of Mathematics, State University of New York, Buffalo, NY 14214, USA

Article author query
hammack j   [Google Scholar] 
scheffner n   [Google Scholar] 
segur h   [Google Scholar] 


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.

(Published Online April 26 2006)
(Received August 31 1988)
(Revised June 6 1989)

p1 Present address: Program in Applied Mathematics, University of Colorado, Boulder, CO 80309, USA.

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