  The goal of much research in relativity is to understand gravitational waves
generated by a strong-field dynamical spacetime. Quantities of particular
interest for many calculations are the Weyl scalar $\psi_4$, which is simply
related to the flux of gravitational waves far from the source, and the flux of
energy carried to distant observers, $\dot E$. Conservation laws guarantee
that, in asympotically flat spacetimes, $\psi_4 \propto 1/r$ and $\dot E
\propto 1/r^2$ as $r \to \infty$. Most calculations extract these quantities at
some finite extraction radius. An understanding of finite radius corrections to
$\psi_4$ and $\dot E$ allows us to more accurately infer their asymptotic
values from a computation. In this paper, we show that, if the final state of
the system is a black hole, then the leading correction to $\psi_4$ is ${\cal
O}(1/r^3)$, and that to the energy flux is ${\cal O}(1/r^4)$ --- not ${\cal
O}(1/r^2)$ and ${\cal O}(1/r^3)$ as one might naively guess. Our argument only
relies on the behavior of the curvature scalars for black hole spacetimes.
Using black hole perturbation theory, we calculate the corrections to the
leading falloff, showing that it is quite easy to correct for finite extraction
radius effects.
