Particle trajectories in a weakly nonlinear long-wave model on a shear flow

Abstract

This work centers on the numerical examination of particle trajectories associated with the propagation of long water waves of small but finite amplitude on a background shear flow over a flat bottom. Taking into consideration the assumption that the nonlinear and dispersive effects are small and of the same magnitude, the Boussinesq-type equations for two-dimensional water waves on a background flow with constant vorticity are derived. Restricting attention to waves propagating in a single direction, we find a new version of the Benjamin-Bona-Mahony (BBM) equation which takes into account the effect of vorticity. In order to investigate the particle trajectories of the flow, an approximate velocity field associated with the derivation of the BBM equation over a shear flow is obtained. Several cases of particle paths under surface waves profiles such as solitary waves and periodic traveling waves are examined.