 Solution

Before characteristics intersect, the right-running compression is a simple wave. The left-running invariant retains its undisturbed value $R_{-}=0$, soA $C_{+}$ characteristic emitted by the piston at time $\tau$ carries the constant value $u=\dot{X}(\tau )$ and has speedIt therefore has equationwhich gives the required parametrization. It applies between the piston and the leading undisturbed characteristic,for $0\le \tau \le t$, until the first characteristic intersection; for $x>c_{0}t$ the gas remains at rest.