  In this paper we propose a multiscale method for the acoustic wave equation
in highly oscillatory media. We use a higher-order extension of the localized
orthogonal decomposition method combined with a higher-order time stepping
scheme and present rigorous a-priori error estimates in the energy-induced
norm. We find that in the very general setting without additional assumptions
on the coefficient beyond boundedness, arbitrary orders of convergence cannot
be expected but that increasing the polynomial degree may still considerably
reduce the size of the error. Under additional regularity assumptions, higher
orders can be obtained as well. Numerical examples are presented that confirm
the theoretical results.
