Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 39D (Waves)

Mass and momentum conservation areFor homentropic ideal-gas flow, $c^2=dp/d\rho$ andCombining the two conservation equations then givesSubtracting the harmless constant $\pm 2c_0/(\gamma -1)$ proves that each Riemann invariantis constant on $C_{\pm }$, where $dx/dt=u\pm c$.

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.

Put $A=(\gamma +1)/2$. A shock formation by characteristic intersection first occurs when the map from emission time to position loses monotonicity:ThusFor $X=c_0{t}^3/T^2$, write $s=\tau /T$ to obtainMinimizing over $s>0$ gives $s^2=[3(3A-1){]}^{-1}$ and hence