A short comment on the effect of a shear layer on nonlinear water waves

In this paper we study nonlinear periodic deep water waves propagating on a background shear current,which decays exponentially with depth.We extend the study of Cheng,Cang and Liao(2009) by introducing a second parameter which measures the depth of the shear current.A high-order convergent analytical series solution is obtained by the homotopy analysis method(HAM).A detailed analysis of the impact of the depth parameter is given.We find that increasing this parameter so that the shear current is thinner reduces the wave phase speed,smoothes the wave crest,sharpens the trough,and enlarges the maximum wave height for the case of propagating waves on an opposing current;while it produces the opposite effect on an aiding current.