  We describe wave decay rates associated to embedded resonances and spectral
thresholds for waveguides and manifolds with infinite cylindrical ends. We show
that if the cut-off resolvent is polynomially bounded at high energies, as is
the case in certain favorable geometries, then there is an associated
asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave
equation on compact sets as $t \to \infty$. In the most general such case we
have $k_0=1$, and under an additional assumption on the infinite ends we have
$k_0 = \infty$. If we localize the solutions to the wave equation in frequency
as well as in space, then our results hold for quite general waveguides and
manifolds with infinite cylindrical ends.
  To treat problems with and without boundary in a unified way, we introduce a
black box framework analogous to the Euclidean one of Sj\"ostrand and Zworski.
We study the resolvent, generalized eigenfunctions, spectral measure, and
spectral thresholds in this framework, providing a new approach to some mostly
well-known results in the scattering theory of manifolds with cylindrical ends.
