Distributions of envelope and phase in weakly nonlinear random waves

The representation of directional random waves in deep water under locally homogeneous conditions is examined in terms of the effect of second-order nonlinearities on the wave envelope and phase. Theoretical expression describing the joint, marginal, and conditional and distributions of wave envelope and phase are derived systematically, correct to the first order in the root-mean-square surface slope. Some immediate implications of these results are discussed in detail. In particular, it is found that the wave envelope is Rayleigh-distributed, as in the case of linear waves. But, the wave phase and envelope are no longer statistically independent, and the phase distribution is nonuniform over the interval ( 0, 2 Π). As the root-mean-square slope and thus the surface skewness increase, the phase distribution deviates from uniformity progressively, indicating an excess of values near the mean phase Π and corresponding symmetrical deficiencies away from the mean toward 0 and 2Π. Comparisons with four sets of wave data gathered in the Gulf of Mexico during a hurricane provide a favorable confirmation of these theoretical results, and thus reinforce the validity of the second-order random wave model.