  In this paper we study the Leray weak solutions of the incompressible Navier
Stokes equation in an exterior domain.We describe, in particular, an hyperbolic
version of the so called artificial compressibility method investigated by
J.L.Lions and Temam. The convergence of these type of approximation show in
general a lack of strong convergence due to the presence of acoustic waves. In
this paper we face this difficulty by taking care of the dispersive nature of
these waves by means of the Strichartz estimates or waves equations satisfied
by the pressure. We actually decompose the pressure in different acoustic
components, each one of them satisfies a specific initial boundary value
problem. The strong convergence analysis of the velocity field will be achieved
by using the associated Leray-Hodge decomposition.
