Explicit Solutions for a Long-Wave Model with Constant Vorticity

Explicit parametric solutions are found for a nonlinear long-wave model describing steady surface waves propagating on an inviscid fluid of finite depth in the presence of a linear shear current. The exact solutions, along with an explicit parametric form of the pressure and streamfunction give a complete description of the shape of the free surface and the flow in the bulk of the fluid. The explicit solutions are compared to numerical approximations previously given in [1], and to numerical approximations of solutions of the full Euler equations in the same situation [31]. These comparisons show that the long-wave model yields a fairly accurate approximation of the surface profile as given by the Euler equations up to moderate waveheights. The fluid pressure and the flow underneath the surface are also investigated, and it is found that the long-wave model admits critical layer recirculating flow and non-monotone pressure profiles similar to the flow features of the solutions of the full Euler equations.