A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter 3 and the bottom slope a to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

Experiments are conducted to measure the motion properties of water particle for the progressive water wave propagation in the presence of following and adverse uniform currents. The experimental data are used to validate the fifth-order Lagrangian solution from Chen and Chen. The experimental results show that the measured data of the particle motion properties such as the b line (denoted as the line connecting the positions of consecutive particles of the same b label), the particle velocity, the particle transport velocity (drift velocity), the particle trajectory, the particle motion period, and the Lagrangian mean level are in close agreement with those of the fifth-order Lagrangian solution. The study also shows that the particle label could adopt the position coordinates of the particle as if it were in still water. The motion of the b line oscillates like wave motion: its wavelength is equal to the progressive wavelength and its wave velocity obeys the Doppler effect so the sum of the velocities of the progressive wave and current, the particle motion period, the Lagrangian mean level, and the particle transport velocity less current velocity are the same as for the case of pure progressive waves.For following currents, the shape of particle trajectory depends on the horizontal particle velocity at the trajectory trough. For adverse currents, the shape of particle trajectory depends on the horizontal particle velocity at the trajectory crest.For a description of the flow motion, the Lagrangian solution could be more effective and precise than the Eulerian solution.

This paper presents a theoretical solution for the dynamic stability of the ocean current turbine system developed in Taiwan. This system is tethered to the sea floor and uses the Kuroshio Current to produce electricity. To maintain the performance of the turbine system in the presence of the Kuroshio Current, the stability of the surfaced turbine needs to be considered. The proposed system is composed of a turbine, a buoyance platform, a traction rope, and a mooring foundation. The two-dimensional theoretical solutions treat the turbine as a rigid body with a movable structure that is moored with two cables. In this model, the gravity, buoyancy, and drag force generated by the wave on the turbine structure are considered. In addition, an analytical solution is proposed for the general system. Finally, the effects of the wave on the pitch motion and dynamical stability of the ocean current turbine system are investigated.

The purpose of this study is to perform a numerical simulation of caisson breakwater stability concerning the effect of wave overtopping under extreme waves. A numerical model, which solves two-dimensional Reynolds-averaged Navier–Stokes equations with the k−ε turbulence closure and uses the volume of fluid method for surface capturing, is validated with the laboratory observations. The numerical model is shown to accurately predict the measured free-surface profiles and the wave pressures around a caisson breakwater. Considering the dynamic loading on caisson breakwaters during overtopping waves, not only landward force and lift force but also the seaward force are calculated. Model results suggest that the forces induced by the wave overtopping on the back side of vertical breakwater and the phase lag of surface elevations have to be considered for calculating the breakwater stability. The numerical results also show that the failure of sliding is more dangerous than the failure of overturning in the vertical breakwater. Under extreme waves with more than 100 year return period, the caisson breakwater is sliding unstable, whereas it is safe in overturning stability. The influence of wave overtopping on the stability analysis is dominated by the force on the rear side of the caisson and the phase difference on the two ends of caisson. For the case of extreme conditions, if the impulse force happens at the moment of the minimum of load in the rear side, the safety factor might decrease significantly and the failure of sliding might cause breakwater damage. This paper demonstrates the potential stability failure of coastal structures under extreme sea states and provides adapted formulations of safety factors in dynamic form to involve the influence of overtopping waves.