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Cnoidal waves and bores in uniform channels of arbitrary cross-section

Published online by Cambridge University Press:  29 March 2006

J. D. Fenton
J. D. Fenton
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Present address: Department of Civil Engineering, Imperial College, London.
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Abstract

A steady nonlinear dispersive wave theory is developed in terms of three important invariants of channel flow: discharge, energy, and momentum flux. As such, the work is an extension of Benjamin & Lighthill's approach for rectangular channels.

Considering the differential equation obtained, we examine the behaviour of flows and wave systems in arbitrary channels for changes of energy and momentum. In particular, the bore problem is studied, and previous approaches to this problem, using linear wave theory, are seen to be invalid. The present theory describes several phenomena of open-channel flow, explains a scatter in previously obtained experimental results, and enables simple design recommendations to be made for channels in which stationary or moving bores are expected.

While this work does describe the variation of physical quantities across the channel section, there are some important three-dimensional phenomena, noted experimentally, which remain unexplained.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 58 , Issue 3 , 8 May 1973 , pp. 417 - 434
Copyright
© 1973 Cambridge University Press

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References

Benet, F. & Cunge, J. A. 1971 Analysis of experiments on secondary undulations caused by surge waves in trapezoidal channels. J. Hyd. Res. 9, 11–33.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy, Soc. A 224, 448–460.Google Scholar
Escoffier, F. F. & Boyd, M. B. 1962 Stability aspects of flow in open channels. J. Hyd. Div. Proc. A.S.C.E. 88, 145–166.Google Scholar
Favre, H. 1935 Etude Thiorique et Expirimentale des Ondes de Translation dans les Canaux Découverts. Paris: Dunod.
Kelland, P. 1839 On the theory of waves. Tram. Roy. Soc. Edin. 14, 497–545.Google Scholar
Kortewec, 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. 39 (5), 422–443.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.
Lemoine, R. 1948 Sur les ondes positives de translation dans les canaux et sur le ressaut ondulé de faible amplitude. Houille Blanche, 3, 183–185.Google Scholar
Peregrine, D. H. 1968 Long waves in a uniform channel of arbitrary cross-section. J. Fluid Mech. 32, 353–365.Google Scholar
Peregrine, D. H. 1969 Solitary waves in trapezoidal channels. J. Fluid Mech. 35, 1–6.Google Scholar
Peters, A. S. 1966 Rotational and irrotational solitary waves in a channel with arbitrary cross-section. Commun. Pure Appl. Math. 19, 445–471.Google Scholar
Preissmann, A. & Cunce, J. A. 1967 Low-amplitude undulating hydraulic jump in trapezoidal canals. J. Hyd. Res. 5, 263–279.Google Scholar
Rayleich, Lord 1876 On waves. Phil. Mag. 1 (5), 257–279. (See also Papers, vol. 1, pp. 251–271. Cambridge University Press.)Google Scholar
Sandover, J. A. & Taylor, C. 1962 Cnoidal waves and bores. Houille Blanche, 17, 443–455.Google Scholar
Shen, M. C. 1968 Long waves in a stratified fluid over a channel of arbitrary cross section. Phys. Fluids, 11, 1853–1862.Google Scholar
Shen, M. C. 1969 Asymptotic theory of unsteady three-dimensional waves in a channel of arbitrary cross section. SIAM J. Appl. Math. 17, 260–271.Google Scholar