The irrotational progressive gravity waves propagating on uniform currents in Lagrangian analysis and experiments Part 1. Theoretical analysis

A third-order Lagrangian solutions is found directly by the full Lagrangian parametric governing equations, with the parameters of identifying each water particle which is obtained in mass conservation from taking the wavelength-averaged level of the vertical displacement of water particles along the direction of the wave propagation and is just the position of particle in the original still water, for the irrotational free surface progressive gravity waves propagating on uniform current in the three dimension. It is found that the effect of wave-current interaction is occurred in Lagrangian velocity potential, so that the pressure is not affected by the uniform current in the wave-current field. The drift velocity, Lagrangian mean level of particle and its motion period longer than the wave period which arc exponentially decreasing with the depth of particle's level and the pathlines of particle and the streakline are then all presented, these are to be excluded in Eulerian solution. Furthermore, the forms of the helical pathlines of particles varying with the direction and speed of the uniform current are appeared to be turned from the wave velocity and current velocity component in co-direction to in anti-direction, even become into a closed orbit. The special case when the wave-current field is at the steady motion is also interpreted its consequential charateristics. The present solution can be reduced to that of the pure progressive waves as no current exists.