Lagrangian Solution for an Irrotational Progressive Water Wave Propagating on a Uniform Current

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.