Skip to main content Accessibility help
×
Home

The solitary wave in water of variable depth

  • R. Grimshaw (a1)

Abstract

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

Copyright

References

Hide All
Boussinesq, J. 1871 Comptes Rendus, 72, 755759.
Boussinesq, J. 1872 Journal of Mathematics, Liouville, 17, 55108.
Bretherton, F. P. 1968 Proc. Roy. Soc. A 302, 555576.
Daily, J. W. & Stephan, S. C. 1952 M.I.T. Hydrodynamics Laboratory Staff Publication, S-17.
Green, G. 1837 Trans. Camb. Phil. Soc. (Republished in 1871 Math. Papers, pp. 225–230.)
Greenspan, H. 1958 J. Fluid Mech. 4, 330334.
Hoogstraten, H. W. 1968 J. Eng. Math. 2, 249274.
Ippen, A. T. & Kulin, G. 1955 M.I.T. Hydrodynamics Laboratory Tech. Report, no. 15.
Keller, H. B., Levine, D. A. & Whitham, G. B. 1960 J. Fluid Mech. 7, 302316.
Keller, J. B. 1958 J. Fluid Mech. 4, 607612.
Kuzmak, G. E. 1959 P.M.M. 23, 515526. (Translated in 1959 Appl. Math. Mech. 23, 730–744.)
Longuet-Higgins, M. H. & Stewart, R. W. 1964 Deep Sea Res. 11, 529562.
Luke, J. C. 1966a Proc. Roy. Soc. A 292, 403412.
Luke, J. C. 1966b J. Fluid Mech. 27, 395398.
Mccowan, J. 1894 Phil. Mag. 38, 351358.
Mei, C. C. & Le Méhauté 1966 J. Geophysical Research, 71, 393400.
Peregrine, D. H. 1967 J. Fluid Mech. 27, 815827.
Rayleigh, Lord 1911 Phil. Mag. 21, 177195.
Russell, J. Scott 1837 Report on Waves. Meeting of the British Association for the Advancement of Science, Liverpool, 417496.
Ursell, F. 1953 Proc. Camb. Phil. Soc. 49, 685694.
Whitham, G. B. 1965a Proc. Roy. Soc. A 283, 238261.
Whitham, G. B. 1965b J. Fluid Mech. 22, 273283.
Whitham, G. B. 1967 Proc. Roy. Soc. A 299, 625.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed