  In a generic spacetime a massless field propagates not just on the surface of
the forward lightcone of a source, but in its interior. This
inside-the-lightcone "tail radiation" is often described as having "scattered"
off the spacetime curvature. In this work, we study the propagation of such
tail radiation for a compact, static, spherically symmetric weak-field (i.e.
low density) mass distribution that is well off the line-of-sight (LOS) between
a source and an observer, and that is coupled to the radiation only
gravitationally. For such perturbers, there are four distinct epochs in the
observed radiation: the light-cone piece; the subsequent early-time tail --
ending at the first time that a signal moving at the speed of light could
travel from the source to a point in the perturber thence to the observer; the
subsequent middle-time tail; and the late-time tail, beginning at the last time
that a signal could make such a journey. For massless scalar and vector (eg.
electromagnetic radiation), we revisit the previously studied early and
late-time tail, and perform the first full examination of the middle-time tail.
Studying shorter wavelengths and generic perturbers well off the LOS, we find
that the late-time tail carries a small fraction of the energy received by the
observer; however, the total middle-time tail contains much more energy. We
also note that whereas the middle-time tail appears to the observer to emanate
from the perturber -- as one might expect for radiation "scattered" from the
gravitational perturbation -- the late-time tail appears to come back from the
source. We speculate on the potential utility of this middle-time tail for
detecting or probing a wide variety of perturbations to the spacetime geometry
including dark matter candidates and dark matter halos.
