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Two-dimensional periodic waves in shallow water. Part 2. Asymmetric waves

Published online by Cambridge University Press:  26 April 2006

Joe Hammack ,
Daryl Mccallister ,
Norman Scheffner  and
Harvey Segur
Joe Hammack
Affiliation:
Departments of Geosciences and Mathematics, Penn State University, University Park, PA 16802, USA
Daryl Mccallister
Affiliation:
Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
Norman Scheffner
Affiliation:
US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS 39180, USA
Harvey Segur
Affiliation:
Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA
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Abstract

We demonstrate experimentally the existence of a family of gravity-induced finiteamplitude water waves that propagate practically without change of form in shallow water of uniform depth. The surface patterns of these waves are genuinely two-dimensional, and periodic. The basic template of a wave is hexagonal, but it need not be symmetric about the direction of propagation, as required in our previous studies (e.g. Hammack et al. 1989). Like the symmetric waves in earlier studies, the asymmetric waves studied here are easy to generate, they seem to be stable to perturbations, and their amplitudes need not be small. The Kadomtsev–Petviashvili (KP) equation is known to describe approximately the evolution of waves in shallow water, and an eight-parameter family of exact solutions of this equation ought to describe almost all spatially periodic waves of permanent form. We present an algorithm to obtain the eight parameters from wave-gauge measurements. The resulting KP solutions are observed to describe the measured waves with reasonable accuracy, even outside the putative range of validity of the KP model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 285 , 25 February 1995 , pp. 95 - 122
Copyright
© 1995 Cambridge University Press

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References

Ablowitz, M. J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform, SIAM.CrossRefGoogle Scholar
Ablowitz, M. J. & Villarroel, J. 1991 Stud. Appl. Maths 85, 195–213.Google Scholar
Boggs, P. T., Byrd, R. H., Rogers, J. E. & Schnabel, R. B. 1992 User's Reference Guide for ODRPACK Version 2.01. NISTIR 92-4834, US Dept of Commerce.CrossRefGoogle Scholar
Dubrovin, B. A. 1981 Russ. Math. Surveys 36, 11–92.Google Scholar
Dubrovin, B. A. 1991 Geometry of Hamiltonian Evolutionary Systems (Lectures at University of Naples), Naples: Bibliopolis.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.CrossRefGoogle Scholar
Hammack, J., Scheffner, N. & Segur, H. 1989 J. Fluid Mech. 209, 567–589.Google Scholar
Hammack, J., Scheffner, N. & Segur, H. 1991 J. Geophys. Res. 96, 4909–4914.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 Sov. Phys. Dokl. 15, 539–541.Google Scholar
Krichever, I. M. 1977 Russ Math. Surveys 32, 185–213.Google Scholar
Peregrine, D. H. 1983 Ann Rev. Fluid Mech. 15, 149–178.Google Scholar
Peregrine, D. H. 1985 Proc. R. Soc. Lond. A 400, 1–18.Google Scholar
Piexoto, M. M. 1962 Topology 1, 101–120.Google Scholar
Scheffner, N. 1988 Misc. Pap. CERC-88-4, Vicksburg, MS.Google Scholar
Segur, H. & Finkel, A. 1985 Stud. Appl. Maths. 73, 183–220.Google Scholar
Stokes, G. G. 1947 Trans. Camb. Phil. Soc. 8, pp. 441–455 (see also Scientific Papers, vol. 1, pp. 197–219, Cambridge University Press).Google Scholar
Wiegel, R. L. 1960 J. Fluid Mech. 7, 273–286.Google Scholar