Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 2 / b / Solution

Integrating the three conservation laws across a stationary normal shock wave gives the Rankine-Hugoniot conditions for a perfect gas

ρ_{1} u_{x 1} = ρ_{2} u_{x 2},

p_{1} + ρ_{1} u_{x 1}^{2} = p_{2} + ρ_{2} u_{x 2}^{2},

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

Eliminate

u_{x 2}

with the conserved mass flux and write

v_{s}^{2} = γ p / ρ

and

M_{x} = u_{x} / v_{s}

. Solving the remaining two algebraic equations gives

\frac{u _{x 1}}{u _{x 2}} = \frac{ρ _{2}}{ρ _{1}} = \frac{( γ + 1 ) M _{x 1}^{2}}{( γ - 1 ) M _{x 1}^{2} + 2},

\frac{p _{2}}{p _{1}} = \frac{2 γ M _{x 1}^{2} - ( γ - 1 )}{γ + 1} .

For a compressive shock,

M_{x 1} > 1

, the density and pressure increase, and the downstream normal flow is subsonic in the shock frame.

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