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

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

are constant on characteristics $C_{\pm}$given by $d x / d t=u \pm c$, where $u(x, t)$ is the speed of the gas, $c(x, t)$ is the local speed of sound, $c_{0}$ is a constant and $\gamma>1$ is the exponent in the adiabatic equation of state for $p(\rho)$.

At time $t=0$ the gas occupies $x>0$ and is at rest at uniform density $\rho_{0}$, pressure $p_{0}$ and sound speed $c_{0}$. For $t>0$, a piston initially at $x=0$ has position $x=X(t)$, where

$$X(t)=-U_{0} t\left(1-\frac{t}{2 t_{0}}\right)$$

and $U_{0}$ and $t_{0}$ are positive constants. For the case $0<U_{0}<2 c_{0} /(\gamma-1)$, sketch the piston path $x=X(t)$ and the $C_{+}$characteristics in $x \geqslant X(t)$ in the $(x, t)$-plane, and find the time and place at which a shock first forms in the gas.

Do likewise for the case $U_{0}>2 c_{0} /(\gamma-1)$.