Nonlinear free-surface evolution and focusing behavior of directionally spread waves

Accurate prediction of extreme waves is essential for ensuring the reliability of coastal and offshore infrastructures under complex sea conditions. However, a clear spectral interpretation of the harmonic decomposition employed in directional wave focusing is still lacking. This research develops a fully nonlinear numerical model, validates it against laboratory experiments to investigates the nonlinear evolution and focusing behavior of unidirectional, bidirectional, and multidirectional wave groups. It covers distinct nonlinear phenomena in wave focusing and quantifies the amplitudes and spatial distributions of the components up to the fourth order, providing new insights into the nonlinear mechanisms governing wave focusing. A novel higher-order boundary element method (HOBEM) is developed for directionally focused waves, which employs uniformly distributed inner sources on a three-dimensional cylindrical boundary, coupled with a vector-based control strategy for precise phase alignment and improved computational efficiency. The results demonstrate that unidirectional focused waves exhibit stronger energy transfer from the fundamental frequency to high harmonics than multidirectional waves, with growing differences for steeper waves. For bidirectional focused waves, the strength of the energy transfer to higher harmonics weakens with the increasing directional separation angle. Directional spreading reduces phase coherence, thereby altering the manifestation of nonlinear interactions and influencing the resulting focal behaviour. These findings elucidate the nonlinear characteristics of wave focusing in realistic sea states, offering important theoretical insights into the wave interaction mechanisms under extreme ocean conditions.