Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 39C / Solution

Work in the shock frame. The undisturbed upstream gas enters the stationary discontinuity at speedwhile the downstream gas, which moves with the piston in the laboratory frame, leaves atThe Rankine-Hugoniot conditions for a perfect gas areandEliminating the velocities between these equations gives the perfect-gas density jump

The mass and momentum equations also implysoSubstitution of the density ratio into mass conservation, followed by this momentum relation, givesUsing the adiabatic sound speed $c_1^2=\gamma p_1/{\rho }_1$ therefore yieldsFinally, eliminating $U$ from $V=\beta p_1/({\rho }_{1}U)$ gives the piston-driven normal shock relationFor a compressive shock $\beta >0$, the first boxed equation has $U>c_1$: the shock propagates supersonically into the undisturbed gas, approaching the sound speed in the weak shock limit $\beta \to 0$.