The Formal Derivation for the Second-Order Interaction between Random Wave Groups and a Vertical Wall

An analytical solution for the reflection of random non-linear wave groups with a high crest is presented. The non-linear wave field of random wave groups in front of a vertical wall is, firstly, determined by extending to the second-order the ‘Quasi-Determinism’ theory, formulated by Boccotti in the eighties, which gives the mechanics of linear random wave groups. The analytical second-order expressions of free surface displacement and of velocity potential in front of a vertical wall are derived. Then, we investigate how the second-order effects modify the space-time groups generated when a high crest occurs in front of a vertical wall (wave reflection). In particular, if the highest crest height occurs at the point y0 at the instant t0, the analytical expression of the free surface displacement is analysed at any point y0+Y, on the wall, or in front of it, at any instant t0+T, as function of the wave spectrum. The linear predictions are then compared with the non-linear ones, showing as the second-order effects modify the wave profile.