 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/39a/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Tangential velocity is unchanged, while the downstream normal velocity is

U ρ_{1} / ρ_{2}

. Thus the incident and outgoing angles to the shock have tangents

1

and

ρ_{1} / ρ_{2}

. Using the tangent-difference identity, the ratio from part (b), and

M^{2} = 2 U^{2} / c_{1}^{2}

gives

tan δ = \frac{M ^{2} - 2}{γ M ^{2} + 2} .

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