Paper 4, Section II, C

Waves
Part II, 2014

A one-dimensional shock wave propagates at a constant speed along a tube aligned with the xx-axis and containing a perfect gas. In the reference frame where the shock is at rest at x=0x=0, the gas has speed U0U_{0}, density ρ0\rho_{0} and pressure p0p_{0} in the region x<0x<0 and speed U1U_{1}, density ρ1\rho_{1} and pressure p1p_{1} in the region x>0x>0.

Write down equations of conservation of mass, momentum and energy across the shock. Show that

γγ1(p1ρ1p0ρ0)=p1p02(1ρ1+1ρ0)\frac{\gamma}{\gamma-1}\left(\frac{p_{1}}{\rho_{1}}-\frac{p_{0}}{\rho_{0}}\right)=\frac{p_{1}-p_{0}}{2}\left(\frac{1}{\rho_{1}}+\frac{1}{\rho_{0}}\right)

where γ\gamma is the ratio of specific heats.

From now on, assume γ=2\gamma=2 and let P=p1/p0P=p_{1} / p_{0}. Show that 13<ρ1/ρ0<3\frac{1}{3}<\rho_{1} / \rho_{0}<3.

The increase in entropy from x<0x<0 to x>0x>0 is given by ΔS=CVlog(p1ρ02/p0ρ12)\Delta S=C_{V} \log \left(p_{1} \rho_{0}^{2} / p_{0} \rho_{1}^{2}\right), where CVC_{V} is a positive constant. Show that ΔS\Delta S is a monotonic function of PP.

If ΔS>0\Delta S>0, deduce that P>1,ρ1/ρ0>1,(U0/c0)2>1P>1, \rho_{1} / \rho_{0}>1,\left(U_{0} / c_{0}\right)^{2}>1 and (U1/c1)2<1\left(U_{1} / c_{1}\right)^{2}<1, where c0c_{0} and c1c_{1} are the sound speeds in x<0x<0 and x>0x>0, respectively. Given that ΔS\Delta S must have the same sign as U0U_{0} and U1U_{1}, interpret these inequalities physically in terms of the properties of the flow upstream and downstream of the shock.