The statistical distribution of swash maxima on natural beaches

Cartwright and Longuet-Higgins (1956) describe the statistical distribution of maxima that would result from the linear superposition of random, Gaussian waves. The distribution function depends solely upon the relative width of the power spectrum and root-mean-square value of the process time series. Runup field data from three experiments are presented to determine the extent to which the distribution of swash maxima can be approximated using the Cartwright and Longuet-Higgins probability density function. The model is found to be satisfactory for describing various distribution statistics including the average maxima, the proportion of negative maxima, and the elevation at which one third of the swash maxima are exceeded. However, systematic discrepancies that scale as a function of time series skewness are observed in the statistics describing the upper tail of the distributions. Although we conclude that the linear model is incapable of delineating these apparent nonlinearities in the swash time series, the extent of the deviation can be estimated empirically for the purpose of constraining nonlinear models and nearshore engineering design.