  We consider the capillary-gravity water-waves problem of finite depth with a
flat bottom of one or two horizontal dimensions. We derive the modulation
equations of leading and next-to-leading order in the hyperbolic scaling for
three weakly amplitude-modulated plane-wave solutions of the linearized problem
in the absence of quadratic and cubic resonances. We fully justify the derived
system of macroscopic equations in the case of pure gravity waves, i.e. in the
case of zero surface tension, employing the stability of the water-waves
problem on the time-scale $O(1/\ep)$ obtained by Alvarez-Samaniego and Lannes.
