  We obtain the second-order asymptotics for the radiation field of spherically
symmetric solutions to the wave equation on spherically symmetric and
asymptotically flat backgrounds including the Schwarzschild and sub-extremal
Reissner-Nordstrom families of black holes. These terms appear as logarithmic
corrections to the leading-order asymptotic terms which were rigorously derived
in our previous work. Such corrections were heuristically and numerically
derived in the physics literature in the case of a non-vanishing Newman-Penrose
constant. In this case, our results provide a rigorous confirmation of the
existence of these corrections. On the other hand, the precise logarithmic
corrections for compactly supported initial data (and hence with a vanishing
Newman-Penrose constant) explicitly obtained here appear to be new.
