= Solution
Before characteristics intersect, the right-running compression is a simple wave. The left-running invariant retains its undisturbed value $R_-=0$, so
$$
c=c_0+\frac{\gamma-1}{2}u.
$$
A $C_+$ characteristic emitted by the piston at time $\tau$ carries the constant value $u=\dot X(\tau)$ and has speed
$$
u+c=c_0+\frac{\gamma+1}{2}\dot X(\tau).
$$
It therefore has equation
$$
x=X(\tau)+\left[c_0+\frac{\gamma+1}{2}\dot X(\tau)\right](t-\tau),
$$
which gives the required parametrization. It applies between the piston and the leading undisturbed characteristic,
$$
X(t)\leq x\leq c_0t,
$$
for $0\leq\tau\leq t$, until the first characteristic intersection; for $x>c_0t$ the gas remains at rest.
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