The effect of non-linearities on statistical distributions in the theory of sea waves

The statistical density function is derived for a variable (such as the surface elevation in a random sea) that is ‘weakly non-linear’. In the first approximation the distribution is Gaussian, as is well known. In higher approximations it is shown that the distribution is given by successive sums of a Gram–Charlier series; not quite in the form that has sometimes been used as an empirical fit for observed distributions, but in a modified form due to Edgeworth. It is shown that the cumulants of the distribution are much simpler to calculate than the corresponding moments; and the approximate distributions are in fact derived by inversion of the cumulant-generating function. The theory is applied to random surface waves on water. The third cumulant and hence the skewness of the distribution of surface elevation is evaluated explicity in terms of the directional energy spectrum. It is shown that the skewness λ 3 is generally positive, and positive upper and lower bounds for λ 3 are derived. The theoretical results are compared with some measurements made by Kinsman (1960). It is found that for free, undamped surface waves the skewness of the distribution of surface slopes is of a higher order than the skewness of the surface elevation. Hence the observed skewness of the slopes may be a sensitive indicator of energy transfer and dissipation within the water.