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Modelling criteria for long water waves

Published online by Cambridge University Press:  12 April 2006

Joseph L. Hammack  and
Harvey Segur
Joseph L. Hammack
Affiliation:
Coastal and Oceanographic Engineering Laboratory, Department of Engineering Sciences, University of Florida, Gainesville Present address: Aeronautical Research Associates of Princeton, Inc., Princeton, New Jersey.
Harvey Segur
Affiliation:
Department of Mathematics, Clarkson College of Technology, Potsdam, New York
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Abstract

Model equations which describe the evolution of long-wave initial data in water of uniform depth are tested to determine explicit criteria for their applicability. We consider linear and nonlinear, dispersive and non-dispersive equations. Separate criteria emerge for the leading wave and trailing oscillations of the evolving wave train. The evolution of the leading wave depends on two parameters: the volume (non-dimensional) of the initial data and an Ursell number based on the amplitude and length of the initial data. The magnitudes of these two parameters determine the appropriate model equation and its time of validity. For the trailing oscillatory waves, a local Ursell number based on the amplitude of the initial data and the local wavelength determines the appropriate model equation. Finally, these modelling criteria are applied to the problem of tsunami propagation. Asymptotic (t → ∞) linear dispersive theory does not appear to be applicable for describing the leading wave of tsunamis. If the length of the initial wave is approximately 100 miles, the leading wave is described by a linear non-dispersive model from the source region until shoaling occurs near the coastline. For smaller lengths (∼ 40 miles) a linear dispersive (but not asymptotic) model is applicable. The longer-period oscillatory waves following the leading wave, which can induce harbour resonance, apparently require a nonlinear dispersive model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 84 , Issue 2 , 30 January 1978 , pp. 359 - 373
Copyright
© 1978 Cambridge University Press

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References

Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. 1974 The inverse scattering transform – Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315.Google Scholar
Ablowitz, M. J. & Segur, H. 1976 Asymptotic solutions of the Korteweg–de Vries equation. Stud. Appl. Math. (to appear).Google Scholar
Airy, G. B. 1845 Tidal waves. Encyclopedia Metropolitana. London.Google Scholar
Benney, D. J. 1966 Long non-linear waves in fluid flows. J. Math. & Phys. 45, 52–63.Google Scholar
Carrier, G. F. 1971 The dynamics of tsunamis. Lectures in Appl. Math., Am. Math. Soc. 13, 157–189.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 1095–1097.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1974 Korteweg–de Vries equation and generalizations. VI. Methods for exact solutions. Comm. Pure Appl. Math. 27, 97–133.Google Scholar
Hammack, J. L. & Segur, H. 1974 The Korteweg–de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65, 289–314.Google Scholar
Hammack, J. L. & Segur, H. 1978 The Korteweg–de Vries equation and water waves. Part 3. Oscillatory waves. J. Fluid Mech. 84, 337–358.Google Scholar
Hwang, L. S. & Divoky, D. 1970 Tsunami generation. J. Geophys. Res. 75, 6802–6817.Google Scholar
Korteweg, D. J. & De vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary wave. Phil. Mag. Ser. 5, 39, 422–443.Google Scholar
Plafker, G. 1969 Tectonics of the March 27, 1964 Alaska earthquake. Geol. Survey Prof. Paper, no. 543–1.Google Scholar
Raichlen, F. 1970 Tsunamis: some laboratory and field observations. Proc. 12th Conf. Coastal Engng, Washington, D.C. pp. 2103–2122.Google Scholar
Schiff, L. I. 1968 Quantum Mechanics, 3rd edn. McGraw-Hill.Google Scholar
Segur, H. 1973 The Korteweg–de Vries equation and water waves. Part 1. Solution of the equation. J. Fluid Mech. 59, 721–736.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685–694.Google Scholar
Van Dorn, W. G. 1964 Source mechanism of the tsunami of March 28, 1964 in Alaska. Proc. 9th Conf. Coastal Engng, Lisbon, pp. 166–190.Google Scholar