  In this article, we develop the local Cauchy theory for the gravity water
waves system, for rough initial data which do not decay at infinity. We work in
the context of $L^2$-based uniformly local Sobolev spaces introduced by Kato.
We prove a classical well-posedness result (without loss of derivatives). Our
result implies also a local well-posedness result in H\"older spaces (with loss
of $d/2$ derivatives). As an illustration, we solve a question raised by
Boussinesq on the water waves problem in a canal. We take benefit of an
elementary observation to show that the strategy suggested by Boussinesq does
indeed apply to this setting.
