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Nonlinear water waves in channels of arbitrary shape

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
Theodore Y. Wu
Affiliation:
Engineering Science, 104-44. California Institute of Technology, Pasadena. CA 91125. USA
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Abstract

The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg–de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a k-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct k-shape factors will propagate with equal velocity and with their effective wavelengths appearing k times of that in the rectangular channel, for which k = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of k. Based on this observation, the solitary and cnoidal waves in a k-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 211 - 233
Copyright
© 1992 Cambridge University Press

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References

Benjamin, T. B. & Mahony, J. J. 1971 On an invariant property of water waves. J. Fluid Mech. 49, 385–389.Google Scholar
Cai, Y. H. & Shen, M. C. 1985 The development of a solitary wave in a channel of variable elliptical cross section. Phys. Fluids 28, 2352–2356.Google Scholar
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
Daily, J. W. & Stephan, S. C. 1952 The solitary wave In Proc. Third Conf. on Coastal Engineering, pp. 13–30.
David, D., Levi, D. & Winternitz, P. 1987 Integrable nonlinear equations for water waves in straits of varying depth and width. Stud. Appl. Maths 76, 133–168.Google Scholar
Fenton, J. D. 1973 Cnoidal waves and bores in uniform channels of arbitrary cross-section. J. Fluid Mech. 58, 417–434.Google Scholar
Johnson, R. S. 1973 On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Keulegan, G. H. 1948 Gradual damping of solitary waves. J. Res. Natl Bur. Stand. 40, 457–487.Google Scholar
Keulegan, G. H. & Patterson, G. W. 1940 Mathematical theory of irrotational translation waves. J. Res. Natl Bur. Stand. 24, 47–101, PR1272.Google Scholar
Lee, S. J., Yates, G. T. & Wu, T. Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.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. 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
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11–43.Google Scholar
Pederson, G. & Gjevik, B. 1983 Run-up of solitary waves. J. Fluid Mech. 135, 283–299.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Peregrine, D. H. 1968 Long waves in a uniform channel of arbitrary cross-section. J. Fluid Mech. 32, 353–365.Google Scholar
Peters, A. S. 1966 Rotational and irrotational solitary waves in a channel with arbitrary cross section. Commun. Pure Appl. Maths 19, 445–471.Google Scholar
Shen, M. C. 1968 Long waves in a stratified fluid over a channel of arbitrary cross section. Phys. Fluids 11, 1853–1882.Google Scholar
Shen, M. C. 1969 Asymptotic theory of unsteady three-dimensional waves in a channel of arbitrary cross section. SIAM J. Appl. Maths 17, 260–271.Google Scholar
Shen, M. C. & Zhong, X. C. 1981 Derivation of K-dV equations for water waves in a channel with variable cross section. J. Méc. 20, 789–801.Google Scholar
Shuto, N. 1974 Nonlinear long waves in a channel of variable section. Coastal Engng Japan 17, 1–12.Google Scholar
Teng, M. H. 1990 Forced emissions of nonlinear water waves in channels of arbitrary shape. Ph.D. 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.
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
Wu, T. Y. 1987 Generation of upstream-advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–99.Google Scholar
Zelt, J. A. 1986 Tsunamis: the response of harbours with sloping boundaries to long wave excitation. Ph.D. thesis, California Institute of Technology, Pasadena. CA.
Zelt, J. A. & Raichlen, F. 1990 A Lagrangian model for wave-induced harbour oscillations. J. Fluid Mech. 213, 203–225.Google Scholar