Abstract
Weakly nonlinear gravity waves of given wavenumber in a horizontally unbounded two-dimensional domain are expected to undergo slow modulations in space and time. Together with an attendant analysis of the water wave equations, this work gives a mathematical justification of the modulation approximation. It proves that the resulting wavepacket, whose envelope is governed by the cubic nonlinear Schrodinger equation is a solution of the water wave equations to leading order. An upper bound of the remainder is also provided.