  We study the decay of global energy for the wave equation with H\"older
continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact
waveguides with star-shaped cross-sections. We show there is sharp
$t^{-1/2}$-decay when the damping is uniformly bounded from below on the
cylindrical wall of product cylinders where the Geometric Control Condition is
violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay
when the damping is uniformly bounded from below on the cylindrical wall.
