  This paper is concerned with a two dimensional Whitham-Boussinesq system
modelling surface waves of an inviscid incompressible fluid layer. We prove
that the associated Cauchy problem is well-posed for initial data of low
regularity, with existence time of scale $\mathcal O(1/\sqrt{\epsilon})$, where
$\epsilon>0$ is a shallowness parameter measuring the ratio of the amplitude of
the wave to the mean depth of the fluid. The key ingredients in the proof are
frequency loacalised dispersive and Strichartz estimates that depend on
$\epsilon$ as well as bilinear estimates in some Strichartz norms.
