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We analyze a homogenization limit for the linear wave equation of secondorder. The spatial operator is assumed to be of divergence form with anoscillatory coefficient matrix $a^\varepsilon$ that is periodic withcharacteristic length scale $\varepsilon$; no spatial symmetry properties areimposed. 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 effectsare observed. In this contribution we present a well-posed weakly dispersiveequation with homogeneous coefficients such that its solutions $w^\varepsilon$describe $u^\varepsilon$ well on time intervals $(0,T\varepsilon^{-2})$. Moreprecisely, 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 allcoefficients in the effective models. We additionally provide an$\varepsilon$-independent equation of third order that describes dispersionalong rays and we present numerical examples.

Dispersive homogenized models and coefficient formulas for waves in general periodic media