“…The first method is based on an adaptation of the one used to derive the inviscid Burgers equations from the Nonlinear Shallow Water system using the Riemann invariants of the latter and requires, as in [15], that initial data are prepared to generate unidirectional waves. We show from it that the Whitham equation can be derived from the water waves equations at the order of precision µε, which is the same order as for the Whitham-Boussinesq equations when considering general initial data (see [10] for a rigorous derivation of the latter equations in the shallow water regime). From this derivation, we prove that solutions of the water waves equations, associated with well-prepared initial data, can be approximated in the shallow water regime, to within order µεt in a time scale of order ε −1 , by a right or left propagating wave solving a Whitham-type equation.…”