Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T06:40:14.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Long waves on a beach

Published online by Cambridge University Press:  28 March 2006

D. H. Peregrine
D. H. Peregrine
Affiliation:
Department of Mathematics, University of Bristol
Get access Rights & Permissions [Opens in a new window]

Abstract

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 27 , Issue 4 , 20 March 1967 , pp. 815 - 827
Copyright
© 1967 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dean, R. G. 1964 Long wave modification by linear transitions Proc. Am. Soc. Civ. Engng J. Waterways and Harbours Div. 90, 1–29.Google Scholar
Ippen, A. T. & Kulin, G. 1954 The shoaling and breaking of the solitary wave. Proc. 5th Conf. on Coastal Engineering, pp. 27–47.Google Scholar
Kajiura, K. 1961 On the partial reflection of water waves passing over a bottom of variable depth. Inter. Union of Geodesy and Geophys. Monograph 24, Tsunami Symposia, pp. 206–30.
Keller, J. B. 1948 The solitary wave and periodic waves in shallow water Commun. Appl. Maths. 1, 323–39.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with applications to ‘surf beats’. J. Fluid Mech. 13, 481–504.Google Scholar
Mei, C. C. & Le Méhauté, B. 1966 Note on the equations of long waves over an uneven bottom J. Geophys. Res. 71, 393–400.Google Scholar
Munk, W. H. 1949a Surf beats Trans. Am. Geophys. Un. 30, 849–54.Google Scholar
Munk, W. H. 1949b The solitary wave theory and its application to surf problems Ann. New York Acad. Sci. 51, 376–424.Google Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore J. Fluid Mech. 25, 321–31.Google Scholar
Rayleigh, Lord. 1894 The Theory of Sound. Reprinted 1945, New York: Dover.
Stoker, J. J. 1957 Water Waves. New York: Interscience.
Tucker, M. J. 1950 Surf beats: sea waves of 1 to 5 minutes period. Proc. Roy. Soc. A 202, 565–73.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves Proc. Camb. Phil. Soc. 49, 685–94.Google Scholar