A general theory of three-dimensional wave groups Part I: The formal derivation

If we know that a wave of given exceptionally large crest-to-trough height occurs at a fixed point at an instant t_o in a random wind-generated sea state, we can predict what happens with a very high probability before and after t_o in an area surrounding . The expressions of the surface displacement and velocity potential in this area are obtained in closed form. They are exact to the first order in a Stokes expansion and hold for nearly arbitrary bandwidth and solid boundary. It will be shown in Part II that these expressions represent either the evolution of a single three-dimensional wave group or the collision of two wave groups, according to the configuration of the solid boundary. The theory was developed in a series of papers starting on 1981. This paper presents the whole theory in a compact form thanks to a radical simplification of the mathematical proof.