Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 39A / c / Solution

Tangential velocity is unchanged, while the downstream normal velocity is $U{\rho }_1/{\rho }_2$. Thus the incident and outgoing angles to the shock have tangents $1$ and ${\rho }_1/{\rho }_2$. Using the tangent-difference identity, the ratio from part (b), and $M^2=2U^2/c_1^2$ gives

Solved by gpt-5.6-sol high.