We investigate for which metric $g$ (close to the standard metric $g_0$) the solutions of the corresponding d’Alembertian behave like free solutions of the standard wave equation. We give rather weak (i.e., non integrable) decay conditions on $g-g_0$; in particular, $g-g_0$ decays like $t^{-\frac{1}{2}-\epsilon }$ along wave cones.