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Generation of upstream advancing solitons by moving disturbances

Wu, T. Yao-Tsu (1987) Generation of upstream advancing solitons by moving disturbances. Journal of Fluid Mechanics, 184 . pp. 75-99. ISSN 0022-1120. http://resolver.caltech.edu/CaltechAUTHORS:WUTjfm87

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Abstract

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon. To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, T[sub]S, and the scaled amplitude [alpha] of the solitons so generated are related by the formula T[sub]S = const [alpha]^-3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.


Item Type:Article
Additional Information:"Reprinted with the permission of Cambridge University Press." (Received 27 March 1986 and in revised form 14 April 1987). This work was jointly sponsored by ONR Contract N00014-85-K-0536, NR655-005 and NSF Grant MSM-8118429, A03, and was presented at the Symposium on Fluid Mechanics in the Spirit of G. I. Taylor held in Cambridge in March 1986. The paper was submitted to the Journal of Fluid Mechanics for publication in the special Symposium volume (number 173, December 1986), but being delayed by the need for revision was not ready in time for publication in that volume. I am grateful to the referees for a number of enlightening comments and for pointing out a few publications of geophysical context. I am also indebted to George Yates and Jinlin Zhu for helpful discussions and for their valuable assistance in obtaining the numerical results presented here. The numerical calculations were done on the CRAY-1 at the Naval Research Laboratory (sponsored by the Office of Naval Research) and on the CRAY X-MP/48 at San Diego Supercomputer Center (operated by the National Science Foundation).
Record Number:CaltechAUTHORS:WUTjfm87
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ID Code:348
Collection:CaltechAUTHORS
Deposited By: Theodore Yao-tsu Wu
Deposited On:05 Jun 2005
Last Modified:26 Dec 2012 08:39

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