Hey, so just to clarify the 1/r scaling thing, a radiation field by definition has amplitude scaling as 1/r, since it is thing responsible for carrying energy away. What is weird about gravitational waves is that radiation is associated with changing quadrupole moments rather than changing dipole moments as in electromag. Incidentally, I think there is a slick high-level argument. A gravitational wave has two indices (it is a wavelike perturbation to a matrix describing the shape of space) so it cannot be associated to a vector quantity like a dipole. The quadrupole is also a matrix, so Lorentz invariance (and dimensional analysis) give us the equation Jess wrote down.

https://en.wikipedia.org/wiki/Gravitational_wave

https://en.wikipedia.org/wiki/Sticky_bead_argument

https://en.wikipedia.org/wiki/Quadrupole_formula