  We consider wave equations with time-independent coefficients that have
$C^{1,1}$ regularity in space. We show that, for nontrivial ranges of $p$ and
$s$, the standard inhomogeneous initial value problem for the wave equation is
well posed in Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ over the
Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral
operators introduced recently by the authors and Portal, following work of
Smith. In spatial dimensions $n = 2$ and $n=3$, this includes the full range $1
< p < \infty$. As a corollary, we obtain the optimal fixed-time $L^{p}$
regularity for such equations, generalizing work of Seeger, Sogge and Stein in
the case of smooth coefficients.
