  We analyze a homogenization limit for the linear wave equation of second
order. The spatial operator is assumed to be of divergence form with an
oscillatory coefficient matrix $a^\varepsilon$ that is periodic with
characteristic length scale $\varepsilon$; no spatial symmetry properties are
imposed. Classical homogenization theory allows to describe solutions
$u^\varepsilon$ well by a non-dispersive wave equation on fixed time intervals
$(0,T)$. Instead, when larger time intervals are considered, dispersive effects
are observed. In this contribution we present a well-posed weakly dispersive
equation with homogeneous coefficients such that its solutions $w^\varepsilon$
describe $u^\varepsilon$ well on time intervals $(0,T\varepsilon^{-2})$. More
precisely, we provide a norm and uniform error estimates of the form $\|
u^\varepsilon(t) - w^\varepsilon(t) \| \le C\varepsilon$ for $t\in
(0,T\varepsilon^{-2})$. They are accompanied by computable formulas for all
coefficients in the effective models. We additionally provide an
$\varepsilon$-independent equation of third order that describes dispersion
along rays and we present numerical examples.
