 Receding-piston rarefaction wave

For a piston receding from initially stationary homentropic gas, the right-moving simple wave has $R_{-}=0$, so $c=c_0+(\gamma -1)u/2$. A $C_{+}$ characteristic emitted by the piston at time $s$ carries $u={\dot{X}}_p(s)$ and is the straight line