Random breaking waves: A closed-form solution for planar beaches

Abstract

Based on the assumption that in the surf zone, random waves behave as a collection of individual regular waves, a closed-form transformation of random variable is performed to yield the probability density function for wave height across a beach of uniform slope. Starting from a Rayleigh distribution well seaward of the surf zone, the transformation is accomplished by using linear wave theory for shoaling and an analytical solution of a model for decay of regular waves due to breaking. Comparisons of the solution to histograms from the DUCK'85 field experiment demonstrate the model's ability to reproduce salient changes in shape of the histogram as the surf zone is traversed. Both data and model indicate that relative position in the surf zone, denoted by the local proportion of waves that are breaking, and bottom slope are the parameters which have the greatest effect on the shape of the probability density function. General expressions for characteristic wave heights (e.g. root-mean-square wave height) are also derived, and their transformation across the surf zone is found to depend distinctly on beach slope and mean wave steepness behavior that has been previously reported as trends in laboratory and field data. Although some facets not included in the model can be important, such as nonlinear shoaling or a distribution in wave period, the closed-form solution made possible by this simpler formulation inherently contains much of the behavior observed in field data, and thereby serves to edify the problem of random breaking waves in the surf zone.