  The re-scaled Water-Waves equations depend strongly on the ratio epsilon
between the amplitude of the wave and the depth of the water. We investigate in
this paper the convergence as epsilon goes to zero of the free surface Euler
equations to the so called rigid lid model. We first prove that the only
solutions of this model are zero. Due to the conservation of the Hamiltonian,
the solutions of the free surface Euler equations converge weakly to zero, but
not strongly in the general case, as epsilon goes to zero. We then study this
default of convergence. More precisely, we show a strong convergence result of
the solutions of the water waves equations in the Zakharov-Craig-Sulem
formulation to the solutions of the linear water-waves equations. It is then
easy to observe that these latter converge weakly to zero. The simple structure
of this system also allows us to explain the mechanisms of the weak convergence
to zero. Finally, we show that this convergence to the rigid lid model also
holds for the solutions of the Euler equations. To this end we give a new proof
of the equivalence of the free surface Euler equations and of the
Zakharov-Craig-Sulem equation by building an extension of the velocity and
pressure fields.
