Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T22:13:22.087Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Korteweg—de Vries equation for a gradually varying channel

Published online by Cambridge University Press:  19 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla
Get access Rights & Permissions [Opens in a new window]

Abstract

Two integral invariants of Shuto's (1974) generalization of the Korteweg—de Vries equation for a unidirectional wave in a channel of gradually varying breadth b and depth d are derived. The second-order (in amplitude) invariant measures energy, as expected, but the first-order invariant measures mass divided by b½d¼; accordingly, mass is conserved only if either the mean free-surface displacement vanishes or bd½ is constant. This difficulty is associated with the reflected wave that is excited by the channel variation but neglected in the KdV approximation. The total mass flux is resolved into a primary (KdV) flux and a residual flux that is proportional to the mean displacement of the primary wave. The reflected wave associated with the residual flux is constructed by neglecting both nonlinearity and dispersion (even though both are significant for the primary wave). The results are applied to a slowly varying cnoidal wave, which is fully determined by conservation of mass and energy and the known results for a uniform channel, and to a slowly varying solitary wave, for which mass is not conserved and both trailing and reflected residuals are excited. The development of the Boussinesq equations for a gradually varying channel and their reduction to Shuto's equation are sketched in an appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 91 , Issue 1 , 9 March 1979 , pp. 181 - 190
Copyright
© 1979 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

Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Washington: Nat. Bur. Stand.
Boussinesq, J. 1872 Théorie des ondes des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108.Google Scholar
Iwagagi, Y. & Sakai, T. 1969 Studies on cnoidal waves (seventh report) — experiments on wave shoaling. Dis. Pre. Res. Inst. Annals, Kyoto Univ. no. 12 B, pp. 569–583 (in Japanese).Google Scholar
Johnson, R. S. 1973 On the asymptotic solution of the Korteweg—de Vries equation with slowly varying coefficients. J. Fluid Mech. 60, 813–824.Google Scholar
Karpman, V. I. & Maslov, E. M. 1977 A perturbation theory for the Korteweg—de Vries equation. Phys. Lett. A 60, 307–308.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. Roy. Soc. A 361, 413–446.Google Scholar
Ko, K. & Kuehl, H. H. 1978 Korteweg—de Vries soliton in a slowly varying medium. Phys. Rev. Lett. 40, 233–236.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Maxon, S. & Viecelli, J. 1974 Spherical solitons. Phys. Rev. Lett. 32, 4–6.Google Scholar
Miles, J. W. 1977a Diffraction of solitary waves. Z. angew. Math. Phys. 28, 889–902.Google Scholar
Miles, J. W. 1977b Note on a solitary wave in a slowly varying channel. J. Fluid Mech. 80, 149–152.Google Scholar
Miles, J. W. 1978 An axisymmetric Boussinesq wave. J. Fluid Mech. 84, 181–191.Google Scholar
Ostrovskiy, L. A. & Pelinovskiy, E. N. 1970 Wave transformation on the surface of a fluid of variable depth. Akad. Nauk SSSR, Izv. Atmos. Ocean. Phys. 6, 552–555.Google Scholar
Ostrovskiy, L. A. & Pelinovskiy, E. N. 1975 Refraction of nonlinear sea waves in a coastal zone. Akad. Nauk SSSR, Izv. Atmos. Ocean. Phys. 11, 37–41.Google Scholar
Ostrovskiy, L. A. & Shrira, V. I. 1976 Instability and self-refraction of solitons. Sov. Phys. J. Exp. Theor. Phys. 44, 738–743.Google Scholar
Peregrine, D. H. 1967 Long waves on a beach. J. Fluid Mech. 27, 815–827.Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. 1, 257–279. (See Papers 1, 251–271.)Google Scholar
Saeki, H., Takagi, K. & Ozaki, A. 1971 Study on the transformation of the solitary wave (2). Proc. 18th Conf. Coastal Engng in Japan pp. 49–53 (in Japanese).
Shuto, N. 1973 Shoaling and deformation of non-linear long waves. Coastal Engng in Japan 16, 1–12.Google Scholar
Shuto, N. 1974 Nonlinear waves in a channel of variable section. Coastal Engng in Japan 17, 1–12.Google Scholar
Svendsen, I. A. & Brink-Kjaer, O. 1972 Shoaling of cnoidal waves. Proc. 13th Coastal Engng Conf., Vancouver vol. 1, pp. 365–383.
Svendsen, I. A. & Hansen, J. B. 1978 On the deformation of periodic long waves over a gently sloping bottom. J. Fluid Mech. 87, 433–448.Google Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. Roy. Soc. A 299, 6–25.Google Scholar