Lagrangian experiment and solution for progressive gravity waves on a sloping bottom

A third-order asymptotic solution in Lagrangian description for nonlinear water wave propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter ε and the bottom slope α to the third order of perturbation. 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 experiment are conducted to investigate the particle trajectories of nonlinear water wave propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.