 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/39a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

In the frame in which the shock is stationary, the upstream gas approaches it with speed

V

, while the downstream gas recedes from it with speed

V - u_{1}

ρ_{0} V = ρ_{1} (V - u_{1}),

p_{0} + ρ_{0} V^{2} = p_{1} + ρ_{1} (V - u_{1})^{2},

and

\frac{γ p _{0}}{( γ - 1 ) ρ _{0}} + \frac{1}{2} V^{2} = \frac{γ p _{1}}{( γ - 1 ) ρ _{1}} + \frac{1}{2} (V - u_{1})^{2} .

These equations express conservation of mass flux, momentum flux, and total specific enthalpy respectively.

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