Probability Distribution of Surface Elevation in Surf and Swash Zones

Abstract

Three irregular wave tests were conducted on a 1:16 smooth impermeable slope to investigate the detailed cross-shore variations of the probability distributions and statistics of the free surface elevations and middepth horizontal velocities in the shoaling, surf, and swash zones. The exponential gamma distribution with the measured mean, standard deviation, and skewness is shown to be capable of describing the measured probability distributions in a unified manner, although the agreement becomes worse in the lower swash zone. The probability distribution of the free surface elevation whose lower limit is imposed by the beach face in the swash zone becomes exponential with the skewness s= 2 and the standard deviation σ=h¯ with h¯= mean water depth. These upper limits of s and σ/h¯ in the lower swash zone are in qualitative agreement with the data in the region of h¯≳ 0.4 cm in these small-scale tests. The cross-shore variations of the mean (undertow) and standard deviation of the middepth horizontal velocity measured in the shoaling and surf zones can be predicted fairly accurately using the simple relationships derived using linear long-wave theory together with the measured values of h¯ and σ. The adopted distribution and these simple relationships will need to be verified using extensive data on sand beaches of various profiles.