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

Sharp Polynomial Decay for Waves Damped from the Boundary in Cylindrical Waveguides