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Stability of periodic waves in shallow water

Published online by Cambridge University Press:  29 March 2006

P. J. Bryant
P. J. Bryant
Affiliation:
Mathematics Department, University of Canterbury, Christchurch, New Zealand
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Abstract

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 66 , Issue 1 , 21 October 1974 , pp. 81 - 96
Copyright
© 1974 Cambridge University Press

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References

Benjamin, T. B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. Roy. Soc. A 299, 59.Google Scholar
Benjamin, T. B. 1973 Lectures on nonlinear wave motion. Fluid Mech. Res. Inst., University of Essex, Rep. no. 44.
Benney, D. J. & Roskes, G. J. 1969 Wave instabilities Studies in Appl. Math. 48, 377.Google Scholar
Bona, J. L. & Bryant, P. J. 1973 A mathematical model for long waves generated by wavemakers in nonlinear dispersive systems Proc. Camb. Phil. Soc. 73, 391.Google Scholar
Bryant, P. J. 1973 Periodic waves in shallow water J. Fluid Mech. 59, 625.Google Scholar
Hayes, W. D. 1973 Group velocity and nonlinear dispersive wave propagation. Proc. Roy. Soc. A 332, 199.Google Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves Phil. Mag. 39, 422.Google Scholar
Laitone, E. V. 1962 Limiting conditions for cnoidal and Stokes waves J. Geophys. Res. 67, 1555.Google Scholar
Lamb, H. 1945 Hydrodynamics, Dover.
Madsen, O. S., Mei, C. C. & Savage, R. P. 1970 The evolution of time-periodic long waves of finite amplitude J. Fluid Mech. 44, 195.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves Trans. Camb. Phil. Soc. 8, 441. (See also Papers, vol. 1, p. 197.)Google Scholar
Stokes, G. G. 1880 Mathematical and Physical Papers, vol. 1, p. 314. Cambridge University Press.
Whitham, G. B. 1967 Non-linear dispersion of water waves J. Fluid Mech. 27, 399.Google Scholar
Wilkinson, J. H. & Reinsch, C. 1971 Linear Algebra. Springer.
Zabusky, N. J. 1967 Nonlinear Partial Differential Equations (ed. W. F. Ames). Academic.