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In order to improve the frequency dispersion effects of irrotational shallowwater models in coastal oceanography, several full dispersion versions ofclassical models were formally derived in the literature. The idea, coming fromG. Whitham in [21], was to modify them so that their dispersion relation is thesame as the water waves equations. In this paper we construct new shallow waterapproximations of the velocity potential then deducing ones on the verticallyaveraged horizontal component of the velocity. We make use of them to deriverigorously from the water waves equations two new Hamiltonian full dispersionmodels. This provides for the first time non-trivial precision resultscharacterizing the order of approximation of the full dispersion models. Theyare non-trivial in the sense that they are better than the ones for thecorresponding classical models.

Rigorous derivation from the water waves equations of some full dispersion shallow water models