Focusing Wave Group Propagating in Finite Depth in the Presence of Surface Current and Vorticity

The kinematics of two-dimensional focusing wave trains on a shearing flow in water of finite depth are investigated analytically. In the absence of waves, the vorticity due to the vertical gradient of the horizontal current velocity is assumed constant. A linear kinematic model based on the spatio-temporal evolution of the frequency is derived predicting the focusing distance and time of a chirped wave packet in the presence of constant vorticity, and surface current. Based on this model, the kinematic behavior of the transient wave packet is analyzed, and described in terms of spreading of the focusing point into a wider area. The effects of bathymetry, vorticity, and surface current are analyzed and discussed. Two main results are obtained: (i) the combined effects of surface current and vorticity, in deep water, are nontrivial, highly depending on the presence of surface current (ii) the effects of bathymetry, in the presence of shear, are also counterintuitive in the presence of vorticity, since significant effects can be observed when considering high values of the depth parameter.