  This paper is devoted to numerical approximations for the wave equation with
a multiscale character. Our approach is formulated in the framework of the
Localized Orthogonal Decomposition (LOD) interpreted as a numerical
homogenization with an $L^2$-projection. We derive explicit convergence rates
of the method in the $L^{\infty}(L^2)$-, $W^{1,\infty}(L^2)$- and
$L^{\infty}(H^1)$-norms without any assumptions on higher order space
regularity or scale-separation. The order of the convergence rates depends on
further graded assumptions on the initial data. We also prove the convergence
of the method in the framework of G-convergence without any structural
assumptions on the initial data, i.e. without assuming that it is
well-prepared. This rigorously justifies the method. Finally, the performance
of the method is demonstrated in numerical experiments.
