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. Derive the Rankine-Hugoniot relations across the shock. 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. Now consider a change of frame such that the shock is stationary and the gas has a component of velocity $V$ parallel to the shock. Deduce that the angle of deflection $\delta$ of the flow which is produced by a stationary shock inclined at an angle $\alpha=\tan ^{-1}(U / V)$ to an oncoming stream of Mach number $M=\left(U^{2}+V^{2}\right)^{\frac{1}{2}} / c_{1}$ is given by

$$\tan \delta=\frac{2 \cot \alpha\left(M^{2} \sin \alpha^{2}-1\right)}{2+M^{2}(\gamma+\cos 2 \alpha)}$$

[Note that

$$\left.\tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi} . \quad\right]$$