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We establish the full justification of a "Whitham-Green-Naghdi" systemmodeling the propagation of surface gravity waves with bathymetry in theshallow water regime. It is an asymptotic model of the water waves equationswith the same dispersion relation. The model under study is a nonlocalquasilinear symmetrizable hyperbolic system without surface tension. We provethe consistency of the general water waves equations with our system at theorder of precision $O(\mu^2 (\varepsilon + \beta))$, where $\mu$ is the shallowwater parameter, $\varepsilon$ the nonlinearity parameter, and $\beta$ thetopography parameter. Then we prove the long time well-posedness on a timescale $O(\frac{1}{\max\{\varepsilon,\beta\}})$. Lastly, we show the convergenceof the solutions of the Whitham-Green-Naghdi system to the ones of the waterwaves equations on the later time scale.

Long time well-posedness and full justification of a Whitham-Green-Naghdi system