Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 39C / d / Solution

Apply the normal-shock relations to $M_{1n}=M_1\sin {\theta }_1$. Tangential velocity continuity and mass conservation giveAs $M_1\to \infty$, maximize ${\theta }_1-{\theta }_2$; with $k=(\gamma -1)/(\gamma +1)$ the optimum has ${\tan }^2{\theta }_1=1/k$, and the maximum deflection is $\arcsin (1/\gamma )$.