  We study decay rates for the energy of solutions of the damped wave equation
on the torus. We consider dampings invariant in one direction and bounded above
and below by multiples of $x^{\beta}$ near the boundary of the support and show
decay at rate $1/t^{\frac{\beta+2}{\beta+3}}$. In the case where $W$ vanishes
exactly like $x^{\beta}$ this result is optimal by work of the second author.
The proof uses a version of the Morawetz multiplier method.
