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

Insert

h = γ p / [ρ (γ - 1)]

, eliminate

p_{2}

and the downstream normal speed using mass conservation, and solve for the density ratio. This gives

\frac{ρ _{1}}{ρ _{2}} = \frac{2 c _{1}^{2} + ( γ - 1 ) U ^{2}}{( γ + 1 ) U ^{2}}, c_{1}^{2} = \frac{γ p _{1}}{ρ _{1}} .

Solved by gpt-5.6-sol high.

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