Waves of space-time from a collapsing compact object

We study the partial time dependent collapse of a spherically symmetric compact object with initial mass $M_1+M_2$ and final mass $M_2$ and the waves of space-time emitted during the collapse via back-reaction effects. We obtain exact analytical solutions for the waves of space-time in an example in which $M_1=M_2=(M_1+M_2)/2$. The wavelengths of the space-time emitted waves during the collapse have the cut (we use natural units $c=\hbar=1$): $\lambda < (2/b)$, $(1/b)$-being the time scale that describes the decay of the compact object.