Weakly Nonlinear Statistics Of High Random Waves

It is well known that in a Gaussian sea state for an infinitely narrow spectrum the crest height and
the trough depth follow the same Rayleigh distribution, because of linearity of the first order Stokes
expansion solution. For spectra of finite bandwidth, Boccotti obtained, as a corollary of his first
formulation of the theory of quasideterminism (which is exact to the first order in a Stokes
expansion), that the crest height and the trough depth still follow asymptotically the Rayleigh law
for high waves in Gaussian sea states. In this paper we extend the theory of quasideterminism of
Boccotti to the second-order, deriving new wave crest and wave trough distributions that take into
account nonlinear effects and are valid for finite bandwidth of the spectrum in deep water. Nonlinear
Monte Carlo simulations validate our theoretical predictions and comparisons with experimental
data and the recent model of Forristall are finally presented.