Skip to main content
    • Aa
    • Aa

The Benjamin-Feir instability of a deep-water Stokes wavepacket in the presence of a non-uniform medium

  • Marius Gerber (a1)

The influence of a non-uniform medium on the Benjamin-Feir instability of weakly nonlinear deep-water waves has been investigated, and an approach via a suitable nonlinear Schrödinger equation was adopted. For the derivation of the relevant cubic Schrödinger equation. the approach of Yuen & Lake (1975) was followed and an applicable dispersion relation and energy equation was derived by the averaged Lagrangian technique. With the assumption that the lengthscale of current variation is much greater than the lengthscale of the wavepacket, a cubic Schrödinger equation with slowly varying coefficients is obtained. Three different examples of non-uniform media are treated: (i) waves on a current with variation along the stream; (ii) waves on a shear current; and (iii) long deep-water gravity waves interacting with shorter waves.

Hide All
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417430.
Djordjevic, V. D. & Redekopp, L. G. 1978 On the development of a packet of surface gravity waves moving over an uneven bottom. Z. angew. Math. Phys. 29, 950962.
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, 105114.
Janssen, P. A. E. M. 1983 On a fourth-order envelope equation for deep-water waves. J. Fluid Mech. 126, 111.
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 4974.
Lighthill, M. J. 1965 Contributions to the theory of waves in non-linear dispersive systems. J. Inst. Maths. Applics. 1, 269306.
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489505.
Longuet-Higgins, M. S. & Stewart, R. W. 1961 The changes in amplitude of short gravity waves on steady non-uniform currents. J. Fluid Mech. 10, 529549.
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves: a physical discussion, with applications. Deep-Sea Res. 11, 529562.
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395397.
Michell, J. H. 1893 The highest waves in water. Phil. Mag. 36, 430437.
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 10117.
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.
Phillips, O. M. 1981 Wave interactions-the evolution of an idea. J. Fluid Mech. 106, 215227.
Smith, R. 1976 Giant waves. J. Fluid Mech. 77, 417431.
Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin-Feir resonance mechanisms. Proc. R. Soc. Lond. A 362, 2741.
Turpin, F-M., Benmoussa, C. & Mei, C. C. 1983 Effects of slowly varying depth and current on the evolution of a Stokes wavepacket. J. Fluid Mech. 132, 123.
West, B. J. 1981 Deep water gravity waves (weak interaction theory). Lecture Notes in Physics, vol. 146. Springer.
Whitham, G. B. 1960 A note on group velocity. J. Fluid Mech. 9, 347352.
Whitham, G. B. 1962 Mass, momentum and energy flux in water waves. J. Fluid Mech. 12, 135147.
Whitham, G. B. 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22, 273283.
Whitham, G. B. 1974 Linear and Non-Linear Waves. Wiley-Interscience.
Yuen, H. C. & Lake, B. M. 1975 Non-linear deep water waves: theory and experiment. Phys. Fluids 18, 956960.
Yuen, H. C. & Lake, B. M. 1978 Non-linear wave concepts applied to deep water waves. In Solitons in Action, pp. 89–126. Academic.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 28 *
Loading metrics...

Abstract views

Total abstract views: 135 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th October 2017. This data will be updated every 24 hours.