For the radially oscillating ball as analyzed in Sec. 16.5.3, verify that the radiation reaction acceleration removes energy from the ball, plus the fluid loaded onto it, at the same rate as the sound waves carry energy away. See Ex. 27.12 for the analogous gravitational-wave result.

Data from Exercises 27.12.

Burke’s radiation-reaction potential (27.64) produces a force per unit volume −ρ∇Φ^react on its nearly Newtonian source. If we multiply this force per unit volume by the velocity v = dx/dt of the source’s material, we obtain thereby a rate of change of energy per unit volume. Correspondingly, the net rate of change of the system’s mass-energy must be

Show that, when averaged over a few gravitational-wave periods, this formula agrees with the rate of change of mass (27.61) that we derived in Ex. 27.11 by integrating the outgoing waves’ energy flux.

dM dt - SpV == pv. VoreactdVx. (27.69)