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Parametrically excited standing edge waves

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California. San Diego, La Jolla, CA 92093, USA
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Abstract

The resonant excitation of weakly nonlinear, standing edge waves of frequency ω and longshore wavenumber k by a normally incident, non-breaking gravity wave of frequency 2ω and shoreline amplitude a to on a bottom that descends smoothly from a shoreline depth of zero and slope σ to an offshore depth h∞ is calculated for ka [Lt ] σ [Lt ] kh∞ [Lt ] 1. The analysis generalizes hose of Guza & Bowen (1976) and Rockliff (1978), which assume uniform slope and perfect reflection, and culminates in a pair of evolution equations for the slowly varying, quadrature amplitudes (or, equivalently, amplitude and phase) of the edge wave. Weak, linear damping (which implies imperfect reflection) is incorporated, and the resulting fixed points and bifurcation points of the evolution equations are determined. It is shown that the solution for prescribed initial conditions must tend to one of the stable fixed points, which correspond to an edge-wave amplitude of either zero or O((σa/k)½), depending on whether the damping exceeds or is inferior to a certain critical value. The restriction σ [Lt ] kh∞ is relaxed for the special depth profile h/h∞ = 1 −exp (−σx/h∞), for which the inviscid, shallow-water equations admit exact solutions. These solutions serve to validate the asymptotic (σ/kh↓O) approximations for arbitrary depth profiles.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990 , pp. 43 - 57
Copyright
© 1990 Cambridge University Press

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References

Ball, F. K. 1967 Edge waves in an ocean of finite depth. Deep-Sea Res. 14, 79–88.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1954 Tables of Integral Transforms, vol. II, 9.3 (4). McGraw-Hill.
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.
Guza, R. T. & Bowen, A. J. 1976 Finite amplitude edge waves. J. Mar. Res. 34, 269–293.Google Scholar
Miles, J. 1988 Parametrically excited standing cross-waves. J. Fluid Mech. 186, 119–127.Google Scholar
Miles, J. 1989 Edge waves on a gently sloping beach. J. Fluid Mech. 199, 125–131.Google Scholar
Miles, J. 1990a Wave motion in a viscous fluid of variable depth. J. Fluid Mech. 212, 365–372.Google Scholar
Miles, J. 1990b Wave reflection from a gently sloping beach. J. Fluid Mech. 214, 59–66.Google Scholar
Minzoni, A. A. 1976 Nonlinear edge waves and shallow-water theory. J. Fluid Mech. 74, 369–374.Google Scholar
Minzoni, A. A. & Whitham, G. B. 1977 On the excitation of edge waves on beaches. J. Fluid Mech. 79, 273–287.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 1, pp. 764–766. McGraw-Hill.
Rockliff, N. 1978 Finite amplitude effects in free and forced edge waves. Math. Proc. Cambridge Phil. Soc. 83, 463–479.Google Scholar