Particle trajectories for waves on a linear shear current

Within the Lagrangian reference frame we present a third-order solution for water particles in a two-dimensional wave and shear current interaction flow. For the case of linear shear current, parametric solutions of particle trajectory are obtained in the Lagrangian coordinates. In particular, the Lagrangian wave frequency and the Lagrangian mean level of particle motion can be obtained which are different from those in an Eulerian description. For the case of negative vorticity, when compared with the pure wave case, particle trajectory traveling in the positive xx-direction is found to be larger, and faster, while the opposite is true when traveling in the negative xx direction. Moreover, the influence of a negative vorticity is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of a positive vorticity.