The free surface Euler equations or water wave equations model the evolution of a non viscous fluid under the gravity force, over a seabed and under a free surface that separates the fluid from the air. These equations capture all the motion of the fluid and are too complicated in general if one wants to study a specific physical phenomenon. When dealing with the propagation of large oceanic currents, oceanographers reasonably assume that the water surface is flat (still-water level) and use the Euler equations in a flat strip as a model. The goal of this talk is to rigorously derive this asymptotic regime. We will see that it will lead to a singular limit : the free surface tends to 0 but the time scale also has to tend +\infty. The small parameter involved in this regime is the ratio between the amplitude of the typical water waves (that are very small for wave currents) and the typical water depth. First we will explain how we can get an existence time that is uniform with respect to the small parameter, and secondly, how the physical quantities involved (free surface, velocity) converge when the small parameter goes to 0. We will see that we can only hope a weak convergence and we will carefully study the lack of strong convergence.