  In this paper, we consider the classical wave equation with time-dependent,
spatially multiscale coefficients. We propose a fully discrete computational
multiscale method in the spirit of the localized orthogonal decomposition in
space with a backward Euler scheme in time. We show optimal convergence rates
in space and time beyond the assumptions of spatial periodicity or scale
separation of the coefficients. Further, we propose an adaptive update strategy
for the time-dependent multiscale basis. Numerical experiments illustrate the
theoretical results and showcase the practicability of the adaptive update
strategy.
