Hostname: page-component-7bb8b95d7b-s9k8s Total loading time: 0 Render date: 2024-09-17T12:04:20.403Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of long water waves in variable channels

Published online by Cambridge University Press:  26 April 2006

Michelle H. Teng  and
Theodore Y. Wu
Michelle H. Teng
Affiliation:
Engineering Science, 104-44, California Institute of Technology, Pasadena, CA 91125, USA Present address: Department of Civil Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA.
Theodore Y. Wu
Affiliation:
Engineering Science, 104-44, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 266 , 10 May 1994 , pp. 303 - 317
Copyright
© 1994 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., Melville, W. K. & Miles, J. W. 1979 On the evolution of a solitary wave in a gradually varying channel. J. Fluid Mech. 95, 401–414.Google Scholar
Choi, H.-S., Bai, K. J., Kim, J.-W. & Cho, I.-H. 1990 Nonlinear free surface waves due to a ship moving near the critical speed in a shallow water. In 18th Symp. on Naval Hydrodynamics, Aug. 12–24, Ann Arbor, MI. pp. 173–190. Washington: National Academy Press.
Chwang, A. T. & Wu, T. Y. 1976 Cylindrical solitary waves. In Waves on Water of Variable Depts. (ed. D. G. Provis & R. Radok), Lecture Notes in Physics, vol. 64, pp. 80–90. Springer.
Ertekin, R. C. & Wehausen, J. V. 1987 Some solition calculations. In Proc. 16th Symp. on Naval Hydrodynamics. (ed. W. C. Webster), pp. 167–184. Washington: National Academy Press.
Kirby, J. T. & Vengayil, P. 1988 Nonresonant and resonant reflection of long waves in varying channels. J. Geophy. Res. 93, 10782–10796.Google Scholar
Liu, P. L.-F., Yoon, S. B. & Kirby, J. T. 1985 Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech. 153, 185–201.Google Scholar
Madsen, O. S. & Mei, C. C. 1969 The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39, 781–791.Google Scholar
Mathew, J. & Akylas, T. R. 1990 On three-dimensional long water waves in a channel with sloping sidewalls. J. Fluid Mech. 215, 289–307.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
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Schember, H. R. 1982 A new model for three-dimensional nonlinear dispersive long waves. PhD thesis, California Institute of Technology, Pasadena, CA.
Shuto, N. 1974 Nonlinear long waves in a channel of variable section. Coastal Engng in Japan 17, 1–12.Google Scholar
Teng, M. H. 1990 Forced emissions of nonlinear water waves in channels of arbitrary shape. PhD thesis, California Institute of Technology, Pasadena, CA.
Teng, M. H. & Wu, T. Y. 1990 Generation and propagation of nonlinear water waves in a channel with variable cross section. In Engineering Science, Fluid Dynamics. pp. 87–108. World Scientific.
Teng, M. H. & Wu, T. Y. 1991 Run-up of tsunami waves in coastal water of variable depth and width. In Proc. of Intl Workshop of Technology for Hong Kong's Infrastructure Development. pp. 529–544. Commercial Press (Hong Kong) Ltd.
Teng, M. H. & Wu, T. Y. 1992 Nonlinear water waves in channels of arbitrary shape. J. Fluid Mech. 242, 211–233 (referred to herein as TW1).Google Scholar
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 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE, 107, 501–522.Google Scholar