Maximum-entropy probability distribution of wind wave free-surface elevation

The probability density function of the surface elevation of a non-Gaussian random wave field is obtained. The derivation is based on the maximum entropy (information) principle with the first four statistical moments of the surface elevation used as constraints. The density function is found by the use of the Lagrangian multipliers method and it is shown that only two of four Lagrangian multipliers are independent. The applied method of numerical solution is described in detail and the useful nomograms that give the Lagrangian multipliers as functions of skewness and kurtosis are calculated and incorporated in the paper. For slightly nonlinear waves the approximate maximum-entropy probability distribution is developed. The condition of the existence of this approximate distribution agrees with the empirical criterion for small deviations from the Gaussian distribution of random water waves. The theoretical results compare well with field experiment data of Ochi and Wang (1984), even in the strongly non-Gaussian case.