A one-dimensional shock wave propagates at a constant speed along a tube aligned with the x-axis and containing a perfect gas. In the reference frame where the shock is at rest at x=0, the gas has speed U0, density ρ0 and pressure p0 in the region x<0 and speed U1, density ρ1 and pressure p1 in the region x>0.
Write down equations of conservation of mass, momentum and energy across the shock. Show that
γ−1γ(ρ1p1−ρ0p0)=2p1−p0(ρ11+ρ01)
where γ is the ratio of specific heats.
From now on, assume γ=2 and let P=p1/p0. Show that 31<ρ1/ρ0<3.
The increase in entropy from x<0 to x>0 is given by ΔS=CVlog(p1ρ02/p0ρ12), where CV is a positive constant. Show that ΔS is a monotonic function of P.
If ΔS>0, deduce that P>1,ρ1/ρ0>1,(U0/c0)2>1 and (U1/c1)2<1, where c0 and c1 are the sound speeds in x<0 and x>0, respectively. Given that ΔS must have the same sign as U0 and U1, interpret these inequalities physically in terms of the properties of the flow upstream and downstream of the shock.