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

$$u \pm \frac{2}{\gamma-1} c=\text { constant }$$

on characteristics

$$C_{\pm}: \frac{d x}{d t}=u \pm c$$

A piston moves smoothly down a long tube, with position $x=X(t)$. Gas occupies the tube ahead of the piston, $x>X(t)$. Initially the gas and the piston are at rest, and the speed of sound in the gas is $c_{0}$. For $t>0$, show that the $C_{+}$characteristics are straight lines, provided that a shock-wave has not formed. Hence find a parametric representation of the solution for the velocity $u(x, t)$ of the gas.