A nonlinear Schrödinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depth-uniform currents and constant vorticity. Next, using the equation, the properties of the modulational instability of gravity waves on linear shear currents are investigated. It is showed that shear currents significantly modify the modulational instability properties of weakly nonlinear waves. Furthermore, the influence of linear shear currents on Peregrine breather which can be seen as a prototype of freak waves is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contain more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. Additionally, if the water depth is deep enough, shear currents have negligible effect on the characteristics of Peregrine breathers.

Influences of topographic variations of the offshore fringing reef on the harbor oscillations excited by incident N-waves with different amplitudes and waveform types are studied for the first time. Both the propagation of the N-waves over the reef and the subsequently-induced harbor oscillations are simulated by a Boussinesq-type numerical model, FUNWAVE-TVD. The present study concentrates on revealing the influences of the plane reef-face slope, the reef-face profile shape and the lagoon width on the maximum runup, the wave energy distribution and the total wave energy within the harbor. It shows that both the wave energy distribution uniformity and the total wave energy gradually increase with decreasing reef-face slope. The profile shape of the reef face suffering leading-elevation N-waves (LEN waves) has a negligible impact on the wave energy distribution uniformity, while for leading-depression N-waves (LDN waves), the latter gradually decreases with the mean water depth over the reef face. The total wave energy always first increases and then decreases with the mean water depth over the reef face. In general, the total wave energy first sharply decreases and then slightly increases with the lagoon width, regardless of the reef-face width and the incident waveform type. The maximum runup subjected to the LEN waves decreases monotonously with the lagoon width. However, for the LDN waves, its changing trend with the lagoon width relies on the incident wave amplitude.

No abstract