Joint Distribution of Large Wave Heights and Associated Periods

Abstract

The joint probability density of zero upcrossing wave heights and periods is investigated over the range of large wave heights. The density of interest is expressed as the product of the marginal density of large wave heights with the conditional density of associated periods. Existing theoretical approximations to these are first reviewed briefly and then compared with empirical results simulated from a generalized form of the Pierson‐Moskowitz spectrum. Comparisons indicate that the marginal probability density of large wave heights is described accurately by an asymptotic expression, as was previously suggested in the literature by the writer. The conditional density of associated wave periods is approximately Gaussian, as predicted by the Longuet‐Higgins approximation in 1975. However, the conditional mean period (standard deviation) tends to be larger (smaller) than that implied by the earlier theory in a manner dependent on the spectrum bandwidth. The theoretical expression describing the joint probability density is thus modified to exhibit these properties, and is then checked with simulated and actual wave data, obtaining fairly good agreement.