Non-Linear Space-Time Evolution of Wave Groups With a High Crest

The theory of quasi-determinism, for the mechanics of linear three-dimensional waves, was obtained by Boccotti in the eighties. The first formulation of the theory deals with the largest crest amplitude; the second formulation deals with the largest wave height. The theory was verified in the nineties with some small-scale field experiments. In this paper the first formulation of Boccotti’s theory, valid for the space-time domain, is extended to the second order. The analytical expressions of the non-linear free surface displacement and velocity potential are obtained. Therefore the space-time evolution of a wave group, to the second-order in a Stokes expansion, when a very large crest occurs at a fixed time and location, is investigated. Finally the second-order probability of exceedance of the crest amplitude is obtained, as a function of two deterministic parameters.