  We generalize work of Oh & Zumbrun and Serre on spectral stability of
spatially periodic traveling waves of systems of viscous conservation laws from
the one-dimensional to the multi-dimensional setting. Specifically, we extend
to multi-dimensions the connection observed by Serre between the linearized
dispersion relation near zero frequency of the linearized equations about the
wave and the homogenized system obtained by slow modulation (WKB)
approximation. This may be regarded as partial justification of the WKB
expansion; an immediate consequence is that hyperbolicity of the
multi-dimensional homogenized system is a necessary condition for stability of
the wave. As pointed out by Oh & Zumbrun in one dimension, description of the
low-frequency dispersion relation is also a first step in the determination of
time-asymptotic behavior.
