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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 39A / a / Solution

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}

. The Rankine-Hugoniot conditions for a perfect gas are therefore

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.

Solved by gpt-5.6-sol high.

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