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Solitary wave, soliton and shelf evolution over variable depth

Published online by Cambridge University Press:  26 April 2006

R. S. Johnson
R. S. Johnson
Affiliation:
Department of Mathematics and Statistics, The University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK
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Abstract

The familiar problem of the propagation of surface waves over variable depth is reconsidered. The surface wave is taken to be a slowly evolving nonlinear wave (governed by the Korteweg–de Vries equation) and the depth is also assumed to be slowly varying; the fluid is stationary in its undisturbed state. Two cases are addressed: the first is where the scale of the depth variation is longer than that on which the wave evolves, and the second is where it is shorter (but still long). The first case corresponds to that discussed by a number of previous authors, and is the problem which has been approached through the perturbation of the inverse scattering transform method, a route not followed here. Our more direct methods reveal a new element in the solution: a perturbation of the primary wave, initiated by the depth change, which arises at the same order as the left-going shelf. The resulting leading-order mass balance is described, with more detail than hitherto (made possible by the use of a special depth variation). The second case is briefly presented using the same approach, and some important similarities are noted.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 276 , 10 October 1994, pp. 125 - 138
Copyright
© 1994 Cambridge University Press

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References

Boussinesq, J. 1871 Théorie de l’intumescence liquid appelée onde solitaire on de translation, se propageant dans un canal rectangulaire. C. R. Acad. Sci. (Paris) 72, 755–759.Google Scholar
Boussinesq, J. 1872 Théorie des ondes des remons qui se propagent le long d’un rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des sensiblement pareilles de las surface au fond. J. Math. Pures Appl. 17, 55–108.Google Scholar
Candler, S. & Johnson, R. S. 1981 On the asymptotic solution of the perturbed KdV equation using the inverse scattering transform. Phys. Lett. 86A, 337–340.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
Green, G. 1837 On the motion of waves in a variable canal of small depth and width. Camb. Trans. VI (Papers p. 225).Google Scholar
Grimshaw, R. 1970 The solitary wave in water of variable depth. J. Fluid Mech. 42, 639–656.Google Scholar
Grimshaw, R. 1971 The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611–622.Google Scholar
Johnson, R. S. 1972 Some numerical solutions of a variable-coefficient Korteweg–de Vries equation (with applications to solitary wave development on a shelf). J. Fluid Mech. 54, 81–91.CrossRefGoogle Scholar
Johnson, R. S. 1973a Asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients. J. Fluid Mech. 60, 813–825.Google Scholar
Johnson, R. S. 1973b On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Kakutani, T. 1971 Effects of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272.Google Scholar
Karpman, V. I. & Maslow, E. M. 1979 A perturbation theory for the Korteweg–de Vries equation. Phys. Lett. 60A, 307–308.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles and oscillators in slowly varying media: a singular perturbation theory. Proc. R. Soc. Lond. A 361, 413–446.Google Scholar
Knickerbocker, C. J. & Newell, A. C. 1980 Shelves and the Korteweg–de Vries equation. J. Fluid Mech. 98, 803–818.Google Scholar
Knickerbocker, C. J. & Newell, A. C. 1985 Reflections from solitary waves in channels of decreasing depth. J. Fluid Mech. 153, 1–16.Google Scholar
Korteweg, D. J. & Vries, G. de 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
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Leibovich, S. & Randall, J. D. 1973 Amplification and decay of long nonlinear wave. J. Fluid Mech. 58, 481–493.Google Scholar
Miles, J. W. 1979 On the Korteweg–de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Newell, A. C. 1978 Soliton perturbation and nonlinear focussing, symposium on nonlinear structure and dynamics in condensed matter. In Solid State Physics, vol. 8, pp. 52–68. Oxford University Press.
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. 1(5), 257–279.Google Scholar
Russell, J. S. 1844 Report on waves. Rep. 14th Meet. Brit. Assoc. Adv. Sci., York, pp. 311–390. London: John Murray.
Tappert, F. D. & Zabusky, N. J. 1971 Gradient-induced fission of solitons. Phys. Rev. Lett. 27, 1774–1776.Google Scholar