Three-dimensional deep-water focusing waves by Irrotational Green–Naghdi equations

Nonlinear interactions and the superposition of various wave groups can generate rogue waves with extreme heights in oceans that significantly affects the ocean dynamics. A comprehensive understanding of these phenomena is essential for accurate wave-force analysis. This study introduces the Irrotational Green–Naghdi (IGN) deep-water equations designed to study, specifically, the propagation and generation of three-dimensional focused waves. The proposed equations employ finite-difference methods for spatial discretization on a Cartesian grid and use the Adams time-stepping scheme for temporal iterations. Discussion is provided on identifying the optimized value of the representative wave-number. The proposed IGN equations are compared with focused wave experimental measurements and second-order wave theory results. These reveal that the selected representative wave-number significantly affects the computational efficiency: an appropriate value enables rapid algorithm convergence with high accuracy, whereas unsuitable values yield slower convergence and reduced efficiency. The wave surface profiles generated by the IGN equations at the focal location exhibit excellent agreement with experimental data, both before and after the focus. In addition, the velocity field along the water depth at the focal time closely matches the experimental velocity field.