The Focusing of Uni-Directional Gaussian Wave-Groups in Finite Depth: An Approximate NLSE Based Approach

The non-linear changes to a NewWave type wave-group are helpful in developing our understanding of the non-linear interactions which can lead to the formation of freak waves. In addition, Gaussian wave-groups are used in model tests where it is useful to have a simple model for their non-linear dynamics. This paper derives a simple analytical model to describe the nonlinear changes to a wave-group as it focuses. This paper is an extension to finite depth of the theory developed for deep water in Adcock & Taylor (2009) (Proc. Roy. Soc. A 465(2110)). The model is derived using the conserved quantities of the cubic nonlinear Schrodinger equation (NLSE). In deep water there are substantial changes to the group shape and spectrum as the wave-group focuses, and the characteristics of these changes are governed by the Benjamin-Feir Index. However, in finite depth the characteristics of the non-linear interactions change, reducing the non-linear changes to the group shape. The analytical model is validated against simulations using the NLSE and against full potential flow solutions using a QALE-FEM numerical scheme. We also compare its predictions against experiments in a physical wavetank. We find that the NLSE, and thus analytical theories derived from it, capture the dominant physics in the evolution of narrowbanded wave-groups.