  In order to improve the frequency dispersion effects of irrotational shallow
water models in coastal oceanography, several full dispersion versions of
classical models were formally derived in the literature. The idea, coming from
G. Whitham in [21], was to modify them so that their dispersion relation is the
same as the water waves equations. In this paper we construct new shallow water
approximations of the velocity potential then deducing ones on the vertically
averaged horizontal component of the velocity. We make use of them to derive
rigorously from the water waves equations two new Hamiltonian full dispersion
models. This provides for the first time non-trivial precision results
characterizing the order of approximation of the full dispersion models. They
are non-trivial in the sense that they are better than the ones for the
corresponding classical models.
