  We construct solutions with prescribed radiation fields for wave equations
with polynomially decaying sources close to the lightcone. In this setting,
which is motivated by semi-linear wave equations satisfying the weak null
condition, solutions to the forward problem have a logarithmic leading order
term on the lightcone and non-trivial homogeneous asymptotics in the interior
of the lightcone. The backward scattering solutions we construct are given to
second order by explicit asymptotic solutions in the wave zone, and in the
interior of the light cone which satisfy novel matching conditions. In the
process we find novel compatibility conditions for the scattering data at null
infinity.
  We also relate the asymptotics of the radiation field towards space-like
infinity to explicit homogeneous solutions in the exterior of the light cone.
This is the setting of slowly polynomially decaying data corresponding to mass,
charge and angular momentum in applications. We show that homogeneous data of
degree minus one and minus two for the wave equation results in the same
logarithmic terms on the lightcone and homogeneous asymptotics in the interior
as for the equations with sources close to the lightcone. The proof requires a
delicate analysis of the forward solution close to the light cone and uses the
invertibility of the Funk transform.
