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6.—Smooth Fronted Waves in the Shallow Water Approximation

Published online by Cambridge University Press:  14 February 2012

Alan Jeffrey
Alan Jeffrey
Affiliation:
Department of Engineering Mathematics, University of Newcastle upon Tyne.
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Synopsis

This paper examines the mathematical problem of the propagation of a smooth fronted wavein the context of shallow water theory. Here, a smooth fronted wave will be taken to be one in which the surface slope is continuous across a line in the free-surface, while the second derivative of the surface slope is discontinuous across that same line. This discontinuity line in the surface then plays the role of the wavefront. After establishing that such wavefronts propagate along the characteristics, and deriving the appropriate transport equations, the explicit form is found for the acceleration with respect to distance of the horizontal component of the water velocity of the surface immediately behind the wavefront as a function of position and seabed profile when the wave propagates into still water. The result is then used to prove that in this approximation such a wave cannever break immediately behind the wavefront before the shore line is reached.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 107 - 116
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

1Stoker, J. J., Water Waves. New York: Interscience, 1957.Google Scholar
2Carrier, G. F. and Greenspan, H. P., Water waves of finite amplitude on a sloping beach. J. Fluid Mech., 4, 97–109, 1958.CrossRefGoogle Scholar
3Jeffrey, A. and , Tin, , Saw, Waves over obstacles on a shallow seabed. Proc. Roy. Soc.Edinburgh, 71A, 181–192, 1973.Google Scholar
4Jeffrey, A., The propagation of weak discontinuities in quasilinear hyperbolic systemswith discontinuous coefficients, I: Fundamental theory. J. Appl. Anal., 3, 79–100, 1973.CrossRefGoogle Scholar
5Jeffrey, A., The propagation of weak discontinuities in quasilinear hyperbolic systems with discontinuous coefficients, II: Special cases and application. J. Appl. Anal., 3, 359–375, 1973.Google Scholar
6Mader, C. L., Numerical simulation of tsunamis. J.Phys. Oceanogr., 4, 74–82, 1974.2.0.CO;2>CrossRefGoogle Scholar
7Harlow, F. H. and Amsden, A. A., A simplified MAC technique for incompressible fluid flow calculations. J.Computational Phys., 6, 322–325, 1970.Google Scholar
8Meyer, R. E., On waves of finite amplitude in ducts. Quart. J. Mech. Appl. Math., 5, 257–269, 1952.Google Scholar
9Jeffrey, A., The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables. Arch. Rational Mech. Anal., 14, 27–37, 1963.Google Scholar
10Douglis, A., Existence theorems for hyperbolic systems, Comm. Pure Appl. Math., 5, 119–154, 1952.Google Scholar
11Lax, P. D., Nonlinear hyperbolic equations. Comm. Pure Appl. Math., 6, 231–258, 1953.CrossRefGoogle Scholar
12Stoker, J. J., The formation of breakers and bores. Comm. Pure Appl. Math., 1,1–87, 1948.Google Scholar