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Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow

Published online by Cambridge University Press:  18 November 2014

Thomas J. Bridges
Thomas J. Bridges*
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
*
†Email address for correspondence: [email protected]
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Abstract

A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective – from the viewpoint of hydraulics – on how and why key shallow water models like the Korteweg–de Vries equation and Kadomtsev–Petviashvili equations arise in the theory of water waves.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Shallow water flows Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , R1
Copyright
© 2014 Cambridge University Press 

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References

Abbott, M. B. 1979 Computational Hydraulics. Pitman.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137–185.CrossRefGoogle Scholar
Bridges, T. J. 2013 A universal form for the emergence of the Korteweg–de Vries equation. Proc. R. Soc. Lond. A 469, 20120707.Google Scholar
Bridges, T. J. & Donaldson, N. M. 2006 Secondary criticality of water waves. Part 1: Definition, bifurcation and solitary waves. J. Fluid Mech. 565, 381–417.Google Scholar
Bridges, T. J. & Donaldson, N. M. 2007 Reappraisal of criticality for two-layer flows and its role in the generation of internal solitary waves. Phys. Fluids 19, 072111.Google Scholar
Dingemans, M. W. 1997 Wave Propagation over Uneven Bottoms. Part 2: Nonlinear Wave Propagation. World Scientific.Google Scholar
Grimshaw, R. H. J. 2005 Korteweg–de Vries equation. In Nonlinear Waves in Fluids: Recent Advances and Modern Applications (ed. Grimshaw, R. H. J.), pp. 1–28. Springer.CrossRefGoogle Scholar
Henderson, F. M. 1966 Open Channel Flow. MacMillan.Google Scholar
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.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, 422–443.CrossRefGoogle Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. World Scientific.Google Scholar
Osborne, A. R. 2010 Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic Press, Elsevier.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar