High-Order Spectral Modeling of Nonlinear Wave Loading on Vertical-Wall Structures with Improved Incident-Wave Boundary Treatment

Accurate prediction of nonlinear wave–structure interaction is essential for the safe design of coastal structures. In this study, a fully nonlinear high-order spectral numerical wave tank is developed to investigate nonlinear wave interaction with a vertical wall. The incident-wave boundary is introduced through an additional velocity potential, with the incident-wave kinematics prescribed from corresponding nonlinear analytical wave solutions. The model is validated against the Fourier solution, demonstrating good accuracy in predicting free-surface elevation, pressure distribution, and resultant wave force. Numerical results show that wave nonlinearity significantly modifies both the standing-wave field and the wall loading. Under strongly nonlinear conditions, negative pressure develops near the lower part of the wall during the crest phase, giving rise to a characteristic saddle-shaped force history. Water depth further modulates this nonlinear mechanism by altering both the force magnitude and the pressure distribution along the wall. For focused wave groups, the force response is strongly affected by the focusing type, wave steepness, and spectral bandwidth. A narrower bandwidth maintains stronger phase coherence over a longer portion of the wave group, leading to slightly larger focused extrema and more pronounced amplification of adjacent wave and force cycles. These findings highlight the importance of nonlinear pressure effects and spectral characteristics in predicting extreme wave loads on vertical-wall coastal structures.