  We justify rigorously an Isobe-Kakinuma model for water waves as a higher
order shallow water approximation in the case of a flat bottom. It is known
that the full water wave equations are approximated by the shallow water
equations with an error of order $O(\delta^2)$, where $\delta$ is a small
nondimensional parameter defined as the ratio of the mean depth to the typical
wavelength. The Green-Naghdi equations are known as higher order approximate
equations to the water wave equations with an error of order $O(\delta^4)$. In
this paper we show that the Isobe-Kakinuma model is a much higher order
approximation to the water wave equations with an error of order $O(\delta^6)$.
