 Solution

At $t=t_c$, the piston velocity reachesThe simple-wave relation then gives $c=0$ and $\rho =0$. For later times the formal continuation would require negative sound speed, so the gas can no longer follow the piston. By vacuum formation behind a receding piston, a vacuum region opens between $X_p(t)$ and the gas edgeThe earlier rarefaction solution remains to the right of $x_v(t)$, while $X_p(t)<x<x_v(t)$ is vacuum.