  Motivated by the desire to test modified gravity theories exhibiting the
Vainshtein mechanism, we solve in various physically relevant limits, the
retarded Galileon Green's function (for the cubic theory) about a background
sourced by a massive spherically symmetric static body. The static limit of our
result will aid us, in a forthcoming paper, in understanding the impact of
Galileon fields on the problem of motion in the solar system. In this paper, we
employ this retarded Green's function to investigate the emission of Galileon
radiation generated by the motion of matter lying deep within the Vainshtein
radius r_v of the central object: acoustic waves vibrating on its surface, and
the motion of compact bodies gravitationally bound to it. If \lambda is the
typical wavelength of the emitted radiation, and r_0 is the typical distance of
the source from the central mass, with r_0 << r_v, then, compared to its
non-interacting massless scalar counterpart, we find that the Galileon
radiation rate is suppressed by the ratio (r_v/\lambda)^{-3/2} at the monopole
and dipole orders at high frequencies r_v/\lambda >> 1. However, at high enough
multipole order, the radiation rate is enhanced by powers of r_v/r_0. At low
frequencies r_v/\lambda << 1, and when the motion is non-relativistic, Galileon
waves yield a comparable rate for the monopole and dipole terms, and are
amplified by powers of the ratio r_v/r_0 for the higher multipoles.
