Receding-piston rarefaction wave
= 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
$$
x=X_p(s)+\left[c_0+\frac{\gamma+1}{2}\dot X_p(s)\right](t-s).
$$