  We derive the Whitham equations from the water waves equations in the shallow
water regime using two different methods, thus obtaining a direct and rigorous
link between these two models. The first one is based on the construction of
approximate Riemann invariants for a Whitham-Boussinesq system and is adapted
to unidirectional waves. The second one is based on a generalisation of
Birkhoff's normal form algorithm for almost smooth Hamiltonians and is adapted
to bidirectional propagation. In both cases we clarify the improved accuracy on
the fully dispersive Whitham model with respect to the long wave Korteweg-de
Vries approximation.
