  We consider the asymptotic behaviour of small-amplitude gravity water waves
in a rectangular domain where the water depth is much smaller than the
horizontal scale. The control acts on one lateral boundary, by imposing the
horizontal acceleration of the water along that boundary, as a scalar input
function u. The state z of the system consists of two functions: the water
level $\zeta$ along the top boundary, and its time derivative $\partial$$\zeta$
$\partial$t. We prove that the solution of the water waves system converges to
the solution of the one dimensional wave equation with Neumann boundary
control, when taking the shallowness limit. Our approach is based on a special
change of variables and a scattering semigroup, which provide the possiblity to
apply the Trotter-Kato approximation theorem. Moreover, we use a detailed
analysis of Fourier series for the dimensionless version of the partial
Dirichlet to Neumann and Neumann to Neumann operators introduced in [1].
