  This paper is concerned with space-time homogenization problems for damped
wave equations with spatially periodic oscillating elliptic coefficients and
temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main
results consist of a homogenization theorem, qualitative properties of
homogenized matrices which appear in homogenized equations and a corrector
result for gradients of solutions. In particular, homogenized equations and
cell problems will turn out to deeply depend on the quasi-periodicity as well
as the log ratio of spatial and temporal periods of the coefficients. Even
types of equations will change depending on the log ratio and
quasi-periodicity. Proofs of the main results are based on a (very weak)
space-time two-scale convergence theory.
