Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T05:18:48.732Z Has data issue: false hasContentIssue false
  • English
  • Français

Passage through the critical Froude number for shallow-water waves over a variable bottom

Published online by Cambridge University Press:  26 April 2006

J. Kevorkian  and
J. Yu
J. Kevorkian
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
J. Yu
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the behaviour of shallow (of order δ [Lt ] 1) water waves excited by a small (of order ε [Lt ] 1) amplitude bottom disturbance in the presence of a uniform oncoming flow with either constant or slowly varying Froude number F. When F* ≡ |F − 1|ε−½ [Gt ] 1, the speed and free surface perturbations are of order ν = O(ε); these grow to become of order ε½ if F* = O(1). Therefore, the asymptotic expansions of the solution for ε → 0 depend on the order of F*. These expansions are constructed in a form which remains valid for times of order ν−1; they are then matched to provide results which are also valid for all F. The analytic results exhibit the interesting effects of weak nonlinearities including the steepening of waves and eventual formation of bores if δν−½ [Lt ] 1, the surface rippling due to dispersion if δν−½ = O(1), the strong interaction of waves and the periodic generation of upstreampropagating solitary waves if F* = O(1), etc. All these results are confirmed by numerical integration of the governing equations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 204 , July 1989 , pp. 31 - 56
Copyright
© 1989 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

Akylas, T. R.: 1984 J. Fluid Mech. 141, 455–466.
Cole, S. L.: 1983 Q. Appl. Maths 41, 301–309.
Cole, S. L.: 1985 Wave Motion 7, 579–587.
Copeland, B.: 1977 A description of a Fourier method for Korteweg-de Vries type equations. Tech. Rep. 77–2, The Institute of Applied Mathematics and Statistics, University of British Columbia, Vancouver, B. C.
Frenzen, C. L.: 1982 Multiple variable expansions for interacting weakly nonlinear waves; applications to shallow water surface waves with variable mean shear and bottom topography. Ph.D thesis, University of Washington, Seattle, WA.
Frenzen, C. L. & Kevorkian, J., 1985 Wave Motion 7, 25–42.
Grimshaw, R. H. J. & Smyth, N. 1986 J. Fluid Mech. 169, 429–464.
Houghton, D. D. & Kasahara, A., 1968 Comm. on Pure and Appl. Math. 21, 1–23.
Kevorkian, J.: 1982 Stud. Appl. Maths 66, 95–119.
Kevorkian, J.: 1987 SIAM Rev. 29, 391–461.
Kevorkian, J. & Cole, J. D., 1981 Perturbation Methods in Applied Mathematics. Springer.
Mainardi, F. & LeBlond, P. H., 1987 In Proc. Intl Conf. on the Applications of Multiple Scaling in Mechanics, pp. 239–245. Paris: Masson.
Mei, C. C.: 1986 J. Fluid Mech. 162, 53–67.
Melville, W. K. & Helfrich, K. R., 1987 J. Fluid Mech. 178, 31–57.
Miles, J. W.: 1986 J. Fluid Mech. 162, 489–499.
Smyth, N. F.: 1987 Proc. R. Soc. Lond. A 409, 79–97.
Stoker, J. J.: 1957 Water Waves. Interscience.
Whitham, G. B.: 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Wu, T. Y.: 1987 J. Fluid Mech. 184, 75–99.
Yu, J.: 1988 Passage through the critical Froude number for shallow water waves over a variable bottom. Ph.D. thesis, University of Washington, Seattle, WA.