  We consider small amplitude wave packet-like solutions to the 3D inviscid
incompressible irrotational infinite depth water wave problem neglecting
surface tension. Formal multiscale calculations suggest that the modulation of
such a solution is described by a profile traveling at group velocity and
governed by a hyperbolic cubic nonlinear Schr\"odinger equation. In this paper
we show that, given wave packet initial data, the corresponding solution exists
and retains the form of a wave packet on natural NLS time scales. Moreover, we
give rigorous error estimates between the true and formal solutions on the
appropriate time scale in Sobolev spaces using the energy method. The proof
proceeds by directly applying modulational analysis to the formulation of the
3D water wave problem developed by Sijue Wu.
