Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 39D / a

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- Solution a

Solution

Mass and momentum conservation are

ρ_{t} + (ρ u)_{x} = 0, u_{t} + u u_{x} + \frac{1}{ρ} p_{x} = 0.

For homentropic ideal-gas flow,

c^{2} = d p / d ρ

and

\frac{d c}{c} = \frac{γ - 1}{2} \frac{d ρ}{ρ} .

Combining the two conservation equations then gives

(\partial_{t} + (u \pm c) \partial_{x}) (u \pm \frac{2 c}{γ - 1}) = 0.

Subtracting the harmless constant

\pm 2 c_{0} / (γ - 1)

proves that each Riemann invariant

R_{\pm} = u \pm \frac{2 ( c - c _{0} )}{γ - 1}

is constant on

C_{\pm}

, where

d x / d t = u \pm c

.

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