Controllability, returning waves and scattering without reverberation in 3D acoustic dynamic system

The dynamic acoustic scattering system is governed by the wave equation $u_{tt}-Δu+qu=0$ in $\Bbb R^3,\,\,\,-\infty<t<\infty$, with a compactly supported potential $q$ and infinitely distant sources (controls) $f$, which initiate incoming spherical waves $u=u^f(x,t)$ provided $u^f\big|_{|x|<-t,\,\,\