We prove global existence and modified scattering in physical space for solutions of the one dimensional water wave equation, with smooth, small and rapidly decaying Cauchy data. The proof relies on the one hand on the use of the "good unknown" of Alinhac and normal forms to establish L2 estimates. On the other hand, L∞ bounds are obtained combining Klainerman vector fields and semi-classical analysis to deduce from the PDE an ODE allowing one to get the asymptotics of the solution. (Joint work with Thomas Alazard).