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

The gas state at a receding piston reaches vacuum when ${\dot{X}}_p=-2c_0/(\gamma -1)$, because then the sound speed and mass density both vanish. If the piston subsequently recedes faster, a vacuum gap opens between it and the gas boundary, which continues with velocity $-2c_0/(\gamma -1)$.

For $X_p(t)=-t^{1+\alpha }/(1+\alpha )$ with $0<\alpha \ll 1$, a point on the ray $x=(c_0-a)t$, $0<a<(\gamma +1)/2$, receives the characteristic emitted at the exponentially small timeAlthough $s\to 0$, the piston speed $-s^{\alpha }$ has the finite limit $-2a/(\gamma +1)$, producing the centered-rarefaction profile of the singular limit $\alpha =0$.