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We consider the wave equation on a manifold $(\Omega,g)$ of dimension $d\geq2$ with smooth strictly convex boundary $\partial\Omega\neq\emptyset$, withDirichlet boundary conditions. We construct a sharp local in time parametrixand then proceed to obtain dispersion estimates: our fixed time decay rate forthe Green function exhibits a $t^{1/4}$ loss with respect to the boundary lesscase. We precisely describe where and when these losses occur and relate themto swallowtail type singularities in the wave front set, proving that our decayis optimal. Moreover, we derive better than expected Strichartz estimates,balancing lossy long time estimates at a given incidence with short time oneswith no loss: for $d=3$, it heuristically means that, on average the decay lossis only $t^{1/6}$.

Dispersion for the wave equation inside strictly convex domains II: the general case