  We investigate second order linear wave equations in periodic media, aiming
at the derivation of effective equations in $\R^n$, $n \in \{1, 2, 3\}$.
Standard homogenization theory provides, for the limit of a small periodicity
length $\eps>0$, an effective second order wave equation that describes
solutions on time intervals $[0,T]$. In order to approximate solutions on large
time intervals $[0,T\eps^{-2}]$, one has to use a dispersive, higher order wave
equation. In this work, we provide a well-posed, weakly dispersive effective
equation, and an estimate for errors between the solution of the original
heterogeneous problem and the solution of the dispersive wave equation. We use
Bloch-wave analysis to identify a family of relevant limit models and introduce
an approach to select a well-posed effective model under symmetry assumptions
on the periodic structure. The analytical results are confirmed and illustrated
by numerical tests.
