Starting from the equations for the 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 $d x / d t=u \pm c$, where $u(x, t)$ is the velocity of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma$ is the ratio of specific heats.

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 with $c=c_{0}$. At time $t=0$ the piston starts moving to the left at a constant velocity $V$. Find $u(x, t)$ and $c(x, t)$ in the three regions

$$\begin{array}{cc} \text { (i) } & c_{0} t \leqslant x \\ \text { (ii) } & a t \leqslant x \leqslant c_{0} t \\ \text { (iii) } & -V t \leqslant x \leqslant a t, \end{array}$$

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?