Long Wave — Short Wave Interaction

  • Dieter Hasselmann


We shall consider the interaction of a long wave (Z1, k1, σ1) with a short wave (Z2, k2, ω2.), σ2 = kg, for irrotational waves, surface tension will be neglected and all waves shall be in deep water. We shall consider the case of strong coupling ε = k2Z1 > l, There are essentially three approaches to this problem.
  1. a)

    Direct treatment of the partial differential equations.

  2. b)

    A treatment of the Fourier transformed equations, that is then essentially a weak interaction treatment.

  3. c)

    A WKBJ treatment which for this case can make use of the fact that a Lagragian formulation exists, so that the theory of adiabatic invariants as developed by Whitham (see Whitham(1974) for an extensive list of references) can be employed.



Short Wave Stationary Observer Wind Wave Dynamical Boundary Condition Mathieu Equation 
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  1. 1.)
    Garrett, C. and Smith, J. (1976): On the interaction between long and short surface waves. J. Phys. Oceanogr. 6 925ADSCrossRefGoogle Scholar
  2. 2.)
    Hasselmann, D.E. (1977a): Wind wave generation by energy and momentum flux to the forced components of a wavefield. (Submitted to J. Fluid Mech.)Google Scholar
  3. 3.)
    Hasselmann, D.E. (1977b): Long wave-short wave interaction - or - weak interaction theory for surface gravity waves - how weak is weak? (in prep.)Google Scholar
  4. 4.)
    Hasselmann, K.(1966): Feynman diagrams and interaction rules for wave-wave scattering processes. Rev. Geophysics 4, 1ADSCrossRefGoogle Scholar
  5. 5.)
    Hasselmann, K. (1967): Nonlinear interactions treated by the methods of theoretical physics. Proc. Roy. Soc.A, 299, 77ADSCrossRefGoogle Scholar
  6. 6.)
    Whitham, G.B. (1967) Variational methods and applications to water waves. Proc. Roy. Soc.A 299, 6ADSMATHCrossRefGoogle Scholar
  7. 7.)
    Whitham, G.B. (1974): Dispersive waves and variational principles. in:Nonlinear Waves. (Ed. Leibovich, S.and Seebass, A.R.) Cornell. (Ithaca and London)Google Scholar

Copyright information

© Plenum Press, New York 1978

Authors and Affiliations

  • Dieter Hasselmann
    • 1
  1. 1.Meteorological InstituteUniversity HamburgHamburg 13Germany

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