  Decay rates for the energy of solutions of the damped wave equation on the
torus are studied. In particular, damping invariant in one direction and equal
to a sum of squares of nonnegative functions with a particular number of
derivatives of regularity is considered. For such damping energy decays at rate
$1/t^{2/3}$. If additional regularity is assumed the decay rate improves. When
such a damping is smooth the energy decays at $1/t^{4/5-\delta}$. The proof
uses a positive commutator argument and relies on a pseudodifferential calculus
for low regularity symbols.
