  This article examines the accuracy for large times of asymptotic expansions
from periodic homogenization of wave equations. As usual, $\epsilon$ denotes
the small period of the coefficients in the wave equation. We first prove that
the standard two scale asymptotic expansion provides an accurate approximation
of the exact solution for times $t$ of order $\epsilon^{-2+\delta}$ for any
$\delta>0$. Second, for longer times, we show that a different algorithm, that
is called criminal because it mixes different powers of $\epsilon$, yields an
approximation of the exact solution with error $O(\epsilon^N)$ for times
$\epsilon^{-N}$ with $N$ as large as one likes. The criminal algorithm involves
high order homogenized equations that, in the context of the wave equation,
were first proposed by Santosa and Symes and analyzed by Lamacz. The high order
homogenized equations yield dispersive corrections for moderate wave numbers.
We give a systematic analysis for all time scales and all high order corrective
terms.
