Interactions of nonlinear gravity waves and uniform current in Lagrangian system

The particle trajectory on a weakly nonlinear progressive surface wave obliquely interacting with a uniform cur-rent is studied by using an Euler-Lagrange transformation. The third-order asymptotic solution is a periodic bounded function of Lagrangian labels and time, which imply that the entire solution is uniformly-valid. The ex-plicit parametric solution highlights the trajectory of a water particle and mass transport associated with a particle displacement can now be obtained directly in Lagrangian form. The angular frequency and Lagrangian mean lev-el of the particle motion in Lagrangian form differ from those of the Eulerian. The variations in the water particle orbits resulting from the oblique interaction with a steady uniform current of different magnitudes are also investi-gated.