Show that, in the standard notation for one-dimensional flow of a perfect gas, the Riemann invariants $u \pm 2\left(c-c_{0}\right) /(\gamma-1)$ are constant on characteristics $C_{\pm}$given by

$$\frac{\mathrm{d} x}{\mathrm{~d} t}=u \pm c$$

Such a gas occupies the region $x>X(t)$ in a semi-infinite tube to the right of a piston at $x=X(t)$. At time $t=0$, the piston and the gas are at rest, $X=0$, and the gas is uniform with $c=c_{0}$. For $t>0$ the piston accelerates smoothly in the positive $x$-direction. Show that, prior to the formation of a shock, the motion of the gas is given parametrically by

$$u(x, t)=\dot{X}(\tau) \quad \text { on } \quad x=X(\tau)+\left[c_{0}+\frac{1}{2}(\gamma+1) \dot{X}(\tau)\right](t-\tau)$$

in a region that should be specified.

For the case $X(t)=\frac{2}{3} c_{0} t^{3} / T^{2}$, where $T>0$ is a constant, show that a shock first forms in the gas when

$$t=\frac{T}{\gamma+1}(3 \gamma+1)^{1 / 2}$$