Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 2 / b

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- Solution b

Solution

Let

m = ρ_{1} u_{1} = ρ_{2} u_{2}

be the signed mass flux. Momentum conservation gives

m^{2} (\frac{1}{ρ _{1}} - \frac{1}{ρ _{2}}) = p_{2} - p_{1} .

Dividing the energy-flux equation by

m

and eliminating

m^{2}

yields the pressure-density Hugoniot relation for a perfect gas:

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

With

P = p_{2} / p_{1}

and

D = ρ_{2} / ρ_{1}

, this becomes

\frac{2 γ}{γ - 1} (P - D) = (P - 1) (D + 1) .

Solving for the mass density ratio gives

D = \frac{( γ + 1 ) P + γ - 1}{( γ - 1 ) P + γ + 1} .

For a compressive normal shock wave,

P > 1

and

D > 1

P \to \infty

, the finite shock compression ratio is

(γ + 1) / (γ - 1)

.

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