Starting from the equations for one-dimensional unsteady flow of an inviscid compressible fluid, show that it is possible to find Riemann invariants $u \pm Q$ that 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 fluid and $c(x, t)$ is the local speed of sound. Show that $Q=2\left(c-c_{0}\right) /(\gamma-1)$ for the case of a perfect gas with adiabatic equation of state $p=p_{0}\left(\rho / \rho_{0}\right)^{\gamma}$, where $p_{0}, \rho_{0}$ and $\gamma$ are constants, $\gamma>1$ and $c=c_{0}$ when $\rho=\rho_{0}$.

Such a gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas is initially uniform and at rest with density $\rho_{0}$. At $t=0$ the piston starts moving to the left at a constant speed $V$. Assuming that the gas keeps up with the piston, find $u(x, t)$ and $c(x, t)$ in each of the three distinct regions that are defined by families of $C_{+}$characteristics.

Now assume that the gas does not keep up with the piston. Show that the gas particle at $x=x_{0}$ when $t=0$ follows a trajectory given, for $t>x_{0} / c_{0}$, by

$$x(t)=\frac{\gamma+1}{\gamma-1}\left(\frac{c_{0} t}{x_{0}}\right)^{2 /(\gamma+1)} x_{0}-\frac{2 c_{0} t}{\gamma-1}$$

Deduce that the velocity of any given particle tends to $-2 c_{0} /(\gamma-1)$ as $t \rightarrow \infty$.