  Localized energy estimates have become a fundamental tool when studying wave
equations in the presence of asymptotically at background geometry. Trapped
rays necessitate a loss when compared to the estimate on Minkowski space. A
loss of regularity is a common way to incorporate such. When trapping is
sufficiently weak, a logarithmic loss of regularity suffices. Here, by studying
a warped product manifold introduced by Christianson and Wunsch, we encounter
the first explicit example of a situation where an estimate with an algebraic
loss of regularity exists and this loss is sharp. Due to the global-in-time
nature of the estimate for the wave equation, the situation is more complicated
than for the Schr\"{o}dinger equation. An initial estimate with sub-optimal
loss is first obtained, where extra care is required due to the low frequency
contributions. An improved estimate is then established using energy
functionals that are inspired by WKB analysis. Finally, it is shown that the
loss cannot be improved by any power by saturating the estimate with a
quasimode.
