Starting from the equations for one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants,

$$R_{\pm}=u \pm \frac{2}{\gamma-1}\left(c-c_{0}\right)$$

are constant on characteristics $C_{\pm}$given by $\frac{d x}{d t}=u \pm c$, where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound and $\gamma$ is the specific heat ratio.

Such a gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest. At time $t=0$ the piston starts moving to the left at a constant speed $V$. Find $u(x, t)$ and $c(x, t)$ in the three regions

$$\begin{aligned} \text { (i) } \quad & c_{0} t \leq x \\ \text { (ii) } \quad a t & \leq x<c_{0} t \\ \text { (iii) }-V t & \leq x<a t \end{aligned}$$

where $a=c_{0}-\frac{1}{2}(\gamma+1) V$. What is the largest value of $V$ for which $c$ is positive throughout region (iii)? What happens if $V$ exceeds this value?