A plane shock is moving with speed $U$ into a perfect gas. Ahead of the shock the gas is at rest with pressure $p_{1}$ and density $\rho_{1}$, while behind the shock the velocity, pressure and density of the gas are $u_{2}, p_{2}$ and $\rho_{2}$ respectively.

(a) Write down the Rankine-Hugoniot relations across the shock, briefly explaining how they arise.

(b) Show that

$$\frac{\rho_{1}}{\rho_{2}}=\frac{2 c_{1}^{2}+(\gamma-1) U^{2}}{(\gamma+1) U^{2}}$$

where $c_{1}^{2}=\gamma p_{1} / \rho_{1}$ and $\gamma$ is the ratio of the specific heats of the gas.

(c) Now consider a change of frame such that the shock is stationary and the gas has a component of velocity $U$ parallel to the shock on both sides. Deduce that a stationary shock inclined at a 45 degree angle to an incoming stream of Mach number $M=\sqrt{2} U / c_{1}$ deflects the flow by an angle $\delta$ given by

$$\tan \delta=\frac{M^{2}-2}{\gamma M^{2}+2}$$

$\left[\right.$ Note that $\left.\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta} .\right]$