Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 39A (Waves)

The one-dimensional Isentropic Euler equations areFor the polytropic equation of state,The continuity equation therefore implieswhile the momentum equation becomesAdding and subtracting these equations givesThus the Riemann invariantsare constant along the characteristic curves

The undisturbed gas has $u=0$, $c=c_0$, and hence $R_{-}=0$. The disturbance from the receding piston is a right-moving simple wave, so this invariant remains zero throughout it:

Let $s$ be the time at which the relevant $C_{+}$ characteristic leaves the piston. The boundary condition is $u={\dot{X}}_p(s)$ there. Both $u$ and $c$ are constant along this characteristic, whose speed isHence $s=s(x,t)$ is determined implicitly by the receding-piston rarefaction wave equationWithin $X_p(t)<x<c_{0}t$, the solution is thereforeFor $x\ge c_{0}t$, the gas remains undisturbed: $u=0$ and $\rho ={\rho }_0$. The speed bound keeps $c$ and $\rho$ positive, and monotonic increase of $|{\dot{X}}_p|$ makes this an expanding rarefaction wave rather than a characteristic-crossing compression.

At $t=t_c$, the piston velocity reachesThe simple-wave relation then gives $c=0$ and $\rho =0$. For later times the formal continuation would require negative sound speed, so the gas can no longer follow the piston. By vacuum formation behind a receding piston, a vacuum region opens between $X_p(t)$ and the gas edgeThe earlier rarefaction solution remains to the right of $x_v(t)$, while $X_p(t)<x<x_v(t)$ is vacuum.

HereWrite $A=(\gamma +1)/2$. Substituting $x=(c_0-a)t$ into the $C_{+}$ characteristic equation givesFor $0<a<A$ and $0<\alpha \ll 1$, the relevant emission time is exponentially small. The terms containing an explicit factor $s$ are therefore beyond all algebraic orders, leavingThus the singular small-power piston emission time isAlthough $s\to 0$, the limiting state is nontrivial:which is the centered-rarefaction state on the ray $x/t=c_0-a$ when $\alpha =0$.