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The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments

Published online by Cambridge University Press:  29 March 2006

Joseph L. Hammack  and
Harvey Segur
Joseph L. Hammack
Affiliation:
W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena
Harvey Segur
Affiliation:
Department of Mathematics, Clarkson College of Technology, Potsdam, New York
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Abstract

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 2 , 28 August 1974 , pp. 289 - 314
Copyright
© 1974 Cambridge University Press

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References

Ablowitz, M. J. & Newell, A. C. 1973 The decay of the continuous spectrum for solutions of the Korteweg–de Vries equation J. Math. Phys. 14, 1277–1284.Google Scholar
Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Model equations for long waves in nonlinear dispersive systems Phil. Trans. 272, 47–78.Google Scholar
Benjamin, T. B. & Mahony, J. J. 1971 On an invariant property of water waves J. Fluid Mech. 49, 385–389.Google Scholar
Benney, D. J. & Luke, J. C. 1964 On the interaction of permanent waves of finite amplitude J. Math. & Phys. 43, 309–313.Google Scholar
French, J. A. 1969 Wave uplift pressures on horizontal platforms. W. M. Keck Lab. Hydraul. & Water Res., Calif. Inst. Tech. Rep. KH-R-19.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
Hammack, J. L. 1972 Tsunamis – a model of their generation and propagation. W. M. Keck Lab. Hydraul. & Water Res., Calif. Inst. Tech. Rep. KH-R-28.Google Scholar
Hammack, J. L. 1973 A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799.Google Scholar
Hershkowitz, N., Romesser, T. & Montgomery, D. 1972 Multiple soliton production and the Korteweg–de Vries equation Phys. Rev. Lett. 29, 1586–1589.Google Scholar
Ippen, A. T., Kulin, G. & Raza, M. A. 1955 Damping characteristics of the solitary wave. Hydrodyn. Lab. M.I.T. Tech. Rep. no. 16.Google Scholar
Keulegan, G. H. 1948 Gradual damping of solitary waves J. Res. Nat. Bur. Stand. 40, 487–498.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 waves. Phil. Mag. 39 (5), 422–443.Google Scholar
Miura, R. M., Gardner, C. S. & Kruskal, M. D. 1968 Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion J. Math. Phys. 9, 1204–1209.Google Scholar
Segur, H. 1973 The Korteweg–de Vries equation and water waves. Part 1. Solutions of the equation J. Fluid Mech. 59, 721–736.Google Scholar
Whitham, G. B. 1965 Nonlinear dispersive waves. Proc. Roy. Soc. A283, 238–261.Google Scholar
Zabusky, N. J. & Galvin, C. J. 1971 Shallow-water waves, the Korteweg–de Vries equation and solitons J. Fluid Mech. 47, 811–824.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interactions of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.Google Scholar