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Generation of upstream advancing solitons by moving disturbances

Published online by Cambridge University Press:  21 April 2006

T. Yao-Tsu Wu
T. Yao-Tsu Wu
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
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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, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 184 , November 1987 , pp. 75 - 99
Copyright
© 1987 Cambridge University Press

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References

Akylas, T. R. 1984 J. Fluid Mech. 141, 455–466.
Baines, P. G. 1977 J. Fluid Mech. 82, 147–159.
Baines, P. G. 1984 J. Fluid Mech. 146, 127–167.
Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Phil. Trans. R. Soc. Lond. A 272, 47–78.
Benjamin, T. B. & Lighthill, M. J. 1954 Proc. R. Soc. Lond. A 224, 448–460.
Binnie, A. M. & Orkney, J. C. 1955 Proc. R. Soc. Lond. A 230, 237–246.
Chu, C. K., Xiang, L. W. & Baransky, Y. 1983 Communs Pure Appl. Maths 36, 495–504.
Cole, S. L. 1985 Wave Motion 7, 579–587.
Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. & Morris, H. S. 1982 Solitons and Nonlinear Wave Equations. Academic.
Ertekin, R. C. 1984 Soliton generation by moving disturbances in shallow water: Theory computation and experiments. Ph.D. Thesis, University of California, Berkeley.
Ertekin, R. C., Webster, W. C. & Wehausen, J. V. 1985 In Proc. 15th Symp. on Naval Hydrodynamics, pp. 347–364. Washington D.C.: National Academy Press.
Ertekin, R. C., Webster, W. C. & Wehausen, J. V. 1986 J. Fluid Mech. 169, 275–292.
Favre, H. 1935 Ondes de Translation. Paris: Dunod.
Fornberg, B. & Whitham, G. B. 1978 Phil. Trans. R. Soc. Lond. A 289, 373–402.
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. 1967 Phys. Rev. Lett. 19, 1095–1097.
Grimshaw, R. H. J. & Smyth, N. F. 1986 J. Fluid Mech. 169, 429–464.
Huang, D. D., Sibul, O. J., Webster, W. C., Wehausen, J. V., Wu, D. M. & Wu, T. Y. 1982 In Proc. Conf. on Behavior of Ships in Restricted Waters, vol. II, pp. 26–1 to 26–10. Varna: Bulgarian Ship Hydrodynamics Centre.
Katsis, C. & Akylas, T. R. 1987 Phy. Fluids 30, 297–301.
Keller, J. B. 1985 Phil. Trans. R. Soc. Lond. A 315, 367–377.
Lax, P. D. 1968 Communs Pure Appl. Maths 21, 467–490.
Lee, S. J. 1985 Generation of long water waves by moving disturbances. Ph.D. Thesis, California Institute of Technology, Pasadena, CA.
Malanotte-Rozzoli, P. 1984 J. Phys. Oceanogr. 14, 1032–1046.
Mei, C. C. 1986 J. Fluid Mech. 162, 53–67.
Miles, J. W. 1980 Ann. Rev. Fluid Mech. 12, 11–43.
Patoine, A. & Warn, T. 1982 J. Atmos. Sci. 39, 1018–1025.
Peregrine, D. H. 1966 J. Fluid Mech. 25, 321–330.
Smyth, N. F. 1986 Modulation theory solution for resonant flow over topography. Department of Math. Rep. 3. University of Melbourne, Victoria, Australia.
Sun, M.-G. 1985 The evolution of waves created by a ship in a shallow canal. In The 60th Anniv. Volume-Zhongshan University, Mechanics Essays (in Chinese), pp. 17–25. China: Guangzhow.
Thews, J. G. & Landweber, L. 1935 U.S. Experimental Model Basin Rep. 408. Washington, D.C.: Navy Yard.
Thews, J. G. & Landweber, L. 1936 U.S. Experimental Model Basin Rep. 414. Washington, D.C.: Navy Yard.
Whitham, G. B. 1965 Proc. R. Soc. Lond. A 283, 238–261.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Wu, T. Y. 1979 Tsunamis - Proc. National Science Foundation Workshop (May 7–9, 1979), pp. 110–149. Pasadena: Tetra Tech. Inc.
Wu, T. Y. 1981 J. Engng Mech. Div. ASCE 107, 501–522.
Wu, T. Y. 1985 Keynote Lecture: On the generation of solitons. Symp. on Fluid Mechanics Honoring Prof. C. S. Yih. 22–24 July 1985. Michigan University, Ann Arbor, MI.
Wu, T. Y. 1986 On generation of solitary waves by moving disturbances. First Intern. Workshop on Water Waves and Floating Bodies. Feb. 16–19, 1986. MIT, Cambridge, MA.
Wu, T. Y. & Wu, D. M. 1982 In Proc. 14th Symp. on Naval Hydrodynamics, pp. 103–125. Washington, D.C.: National Academy of Sciences.
Wu, D. M. & Wu, T. Y. 1987 Precursor solitons generated by three-dimensional disturbances moving in a channel. To be presented at IUTAM Symp. on Non-linear Water Waves. August 25–28, 1987, Tokyo, Japan.
Zhu, J. 1986 Internal solitons generated by moving disturbances. Ph.D. Thesis. California Institute of Technology, Pasadena, CA.
Zhu, J., Wu, T. Y. & Yates, G. T. 1986 Generation of internal runaway solitons by moving disturbances. 16th Symp. on Naval Hydrodynamics, July 14–18, 1986, University of California, Berkeley, CA.
Zhu, J., Wu, T. Y. & Yates, G. T. 1987 Internal solitary waves generated by moving disturbances. Third Intl Symp. on Stratified Flows, February 3–5, 1987, California Institute of California, Pasadena, CA.