Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 38B / Solution

For smooth isentropic flow with $p=K{\rho }^{\gamma }$, $\gamma >1$, the continuity and momentum equations givewhere $c^2=dp/d\rho$. Adding and subtracting these equations showsThese are the Riemann invariants on the two characteristic families.

Before any shock, the incoming $C_{-}$ characteristics originate in the initially uniform gas, so their invariant is zero. Thus $c=c_0+(\gamma -1)u/2$ throughout the disturbed simple wave. A $C_{+}$ characteristic leaving the piston at time $\tau$ has constant $u=\dot{X}(\tau )$ and speed $u+c=c_0+(\gamma +1)\dot{X}(\tau )/2$. ThereforeThis parametrization applies for $0\le \tau \le t$ in $X(t)\le x\le c_{0}t$ as long as the characteristics remain ordered. The gas for $x\ge c_{0}t$ is still at rest. The values $\tau =t$ and $\tau =0$ give the piston and leading disturbance respectively.

Characteristic crossing first occurs when the derivative of the displayed $x(\tau ,t)$ with respect to $\tau$ vanishes. With $U(\tau )=\dot{X}(\tau )$ this givesFor the printed $X=2c_0{t}^3/(3T^2)$, $U=2c_0{\tau }^2/T^2$, soIts minimum over $\tau >0$ occurs at ${\tau }_{\ast }=T/\sqrt{3\gamma +1}$ and satisfies $t_s>{\tau }_{\ast }$. Hence the first shock forms atThe characteristic formula is restricted to earlier times; crossing afterwards must be replaced by a shock satisfying the conservation laws.