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Nonlinear evolution of surface gravity waves over an uneven bottom

Published online by Cambridge University Press:  26 April 2006

Y. Matsuno
Y. Matsuno
Affiliation:
Department of Physics, Faculty of Liberal Arts, Yamaguchi University, Yamaguchi 753, Japan
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Abstract

A unified theory is developed which describes nonlinear evolution of surface gravity waves propagating over an uneven bottom in the case of two-dimensional incompressible and inviscid fluid of arbitrary depth. Under the assumptions that the bottom of the fluid has a slowly varying profile and the wave steepness is small, a system of approximate nonlinear evolution equations (NEEs) for the surface elevation and the horizontal component of surface velocity is derived on the basis of a systematic perturbation method with respect to the steepness parameter. A single NEE for the surface elevation is also presented. These equations are expressed in terms of original coordinate variables and therefore they have a direct relevance to physical systems. Since the formalism does not rely on the often used assumptions of shallow water and long waves, the NEEs obtained are uniformly valid from shallow water to deep water and have wide applications in various wave phenomena of physical and engineering importance. The shallow- and deep-water limits of the equations are discussed and the results are compared with existing theories. It is found that our theory includes as specific cases almost all approximate theories known at present.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 249 , April 1993 , pp. 121 - 133
Copyright
© 1993 Cambridge University Press

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References

Akylas, T. R. 1989 Highest-order modulation effects on solitary wave envelopes in deep water. J. Fluid Mech. 198, 387– 397.Google Scholar
Akylas, T. R. 1991 Higher-order modulation effects on solitary wave envelopes in deep water. Part 2. Multi-soliton envelopes. J. Fluid Mech. 224, 417– 428.Google Scholar
Brinch-Nielsen, U. & Jonsson, I. G. 1986 Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth. Wave Motion 8, 455– 472.Google Scholar
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430– 466.Google Scholar
Broer, L. J. F. 1975 Approximate equations for long water waves. Appl. Sci. Res. 31, 377– 395.Google Scholar
Broer, L. J. F., Groesen, E. W. C. van & Timmers, J. M. W. 1976 Stable model equations for long water waves. Appl. Sci. Res. 32, 619– 636.Google Scholar
Byatt-Smith, J. G. B. 1971 An integral equation for unsteady surface waves and a comment on the Boussinesq equation. J. Fluid Mech. 49, 625– 633.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183– 230.Google Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105– 114.Google Scholar
Grimshaw, R. 1970 The solitary wave in water of variable depth. J. Fluid Mech. 42, 639– 656.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
Kakutani, T. 1971 Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272– 276.Google Scholar
Lee, S. J., Yates, G. T. & Wu, T. Y. 1989 Experiments and analysis of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569– 593.Google Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 385– 416.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
Matsuno, Y. 1992 Nonlinear evolutions of surface gravity waves on fluid of finite depth. Phys. Rev. Lett. 69, 609– 611.Google Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves (second printing with corrections). World Scientific.
Mei, C. C. & Méhauté, B. L. 1966 Note on the equations of long waves over an uneven bottom. J. Geophys. Res. 71, 393– 400.Google Scholar
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11– 43.Google Scholar
Ono, H. 1972 Wave propagation in an inhomogeneous anharmonic lattice. J. Phys. Soc. Japan 32, 332– 336.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815– 827.Google Scholar
Peregrine, D. H. 1983 Breaking waves on beaches. Ann. Rev. Fluid Mech. 15, 149– 178.Google Scholar
Peregrine, D. H. 1987 Recent developments in the modelling of unsteady and breaking water waves. In Nonlinear Water Waves (IUTAM Symp. Tokyo/Japan August 25–28, 1987) (ed. K. Horikawa & H. Maruo), pp. 17– 27. Springer.
Radder, A. C. 1992 An explicit Hamiltonian formulation of surface waves in water of finite depth. J. Fluid Mech. 237, 435– 455.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553– 578.Google Scholar
Stokes, G. G. 1849 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441– 455.Google Scholar
Su, M. Y. 1982 Evolution of groups of gravity waves with moderate to high steepness. Phys. Fluids 25, 2167– 2174.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Woods, L. C. 1961 The Theory of Subsonic Plane Flow. Cambridge University Press.
Wu, T. Y. 1981 Long waves in ocean and coastal waters. J. Engng Mech. Div. ASCE 107, 501– 522.Google Scholar
Yuen, H. C. & Lake, M. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67– 229.Google Scholar