text: In deeper water, Stokes waves are unstable.  This was shown by T. Brooke Benjamin and Jim E. Feir in 1967. The Benjamin–Feir instability is a side-band or modulational instability, with the side-band modulations propagating in the same direction as the carrier wave; waves become unstable on deeper water for a relative depth  (with k the wavenumber and h the mean water depth). The Benjamin–Feir instability can be described with the nonlinear Schrödinger equation, by inserting a Stokes wave with side bands. Subsequently, with a more refined analysis, it has been shown – theoretically and experimentally – that the Stokes wave and its side bands exhibit Fermi–Pasta–Ulam–Tsingou recurrence: a cyclic alternation between modulation and demodulation.
  In 1978 Longuet-Higgins, by means of numerical modelling of fully non-linear waves and modulations (propagating in the carrier wave direction), presented a detailed analysis of the region of instability in deep water: both for superharmonics (for perturbations at the spatial scales smaller than the wavelength )  and subharmonics (for perturbations at the spatial scales larger than ).
  With increase of Stokes wave's amplitude, new modes of superharmonic instability appear. Appearance of a new branch of instability happens when the energy of the wave passes extremum. Detailed analysis of the mechanism of appearance of the new branches of instability has shown that their behavior follows closely a simple law, which allows to find with a good accuracy instability growth rates for all known and predicted branches.
  In Longuet-Higgins studies of two-dimensional wave motion, as well as the subsequent studies of three-dimensional modulations by McLean et al., new types of instabilities were found – these are associated with resonant wave interactions between five (or more) wave components.