Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T09:03:23.001Z Has data issue: false hasContentIssue false
  • English
  • Français

On the evolution of a solitary wave in a gradually varying channel

Published online by Cambridge University Press:  19 April 2006

Peter Chang ,
W. K. Melville  and
John W. Miles
Peter Chang
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
W. K. Melville
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, California 92093
Get access Rights & Permissions [Opens in a new window]

Abstract

The adiabatic approximation for a solitary wave in a channel of gradually varying breadth b and uniform depth is tested by experiment and by numerical solution of the generalized Korteweg-de Vries (KdV) equation. The results for a linearly diverging channel show good agreement with the prediction α (dimensionless wave amplitude)$\propto b^{-\frac{2}{3}}$. The experiments and numerical solutions for the linearly converging channel show that the wave growth is well approximated by α ∞ b−½. The discrepancy between the diverging and converging channels is shown to be due to nonlinear effects associated with the choice of the spatial variable as the slow variable in the generalized KdV equation. The measured and computed profiles display the predicted ‘shelves’ of elevation and depression in the converging and diverging channels, respectively.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 95 , Issue 3 , 11 December 1979 , pp. 401 - 414
Copyright
© 1979 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

Chang, P. 1978 The solitary wave in a channel of varying width. M.S. dissertation, University of California, San Diego.
Gazdag, J. 1976 Time-differencing schemes and transform methods. J. Comp. Phys. 20, 196–207.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. Roy. Soc. A 361, 413–446.Google Scholar
Ko, K. & Kuehl, H. H. 1978 Korteweg-de Vries soliton in a slowly varying medium. Phys. Rev. Lett. 40, 233–236.Google Scholar
Meiss, J. D. & Pereira, N. A. 1978 Internal wave solitons. Phys. Fluid 21, 700–702.Google Scholar
Miles, J. W. 1976 Damping of weakly nonlinear shallow-water waves. J. Fluid Mech. 76, 251–257.Google Scholar
Miles, J. W. 1979 On the Korteweg-de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Shuto, N. 1974 Non-linear long waves in a channel of variable section. Coastal Engng in Japan 17, 1–12.Google Scholar
Tappert, F. 1974 Numerical solutions of the Korteweg-de Vries equation and its generalisations by the split-step Fourier method. In Nonlinear Wave Motion (ed. A. C. Newell). Providence, R.I.: American Mathematical Society. Lectures in Applied Mathematics, vol. 15, pp. 215–216.