Expected Structure of Extreme Waves in a Gaussian Sea. Part I: Theory and SWADE Buoy Measurements

This paper is concerned with the expected configuration in space and time surrounding extremely high crests in a random wave field, or, equivalently, the mean configuration averaged over realizations of extreme events. A simple, approximate theory is presented that predicts that the mean configuration ζ¯(x + r, t + τ) surrounding a crest at (x, t) that is higher than γσ (where σ is the overall rms surface displacement and γ ≫ 1), when normalized by ζ¯(x,t) for ζ > γσ, is the space-time autocorrelation function ρ(r, t) = ¯ζ(x, t)ζ(x + r, t + τ)/ ζ¯2 for the entire wave field. This extends and simplifies an earlier result due to Boccotti and is consistent with a precise calculation of the one-dimensional case with r = 0, involving the time history of measurements at a single point. The results are compared with buoy data obtained during the Surface Wave Dynamics Experiment and the agreement is found to be remarkably good.