Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 39C / a

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- Solution a

Solution

For one-dimensional homentropic flow,

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

and

d p = c^{2} d ρ

. For a perfect gas,

d c / c = (γ - 1) d ρ / (2 ρ)

. Combining the two equations therefore shows that the Riemann invariants

R_{\pm} = u \pm \frac{2 c}{γ - 1}

satisfy

(\partial_{t} + (u \pm c) \partial_{x}) R_{\pm} = 0 .

Their characteristic curves are parametrized by

\frac{d x _{\pm}}{d t} = u (x_{\pm} (t), t) \pm c (x_{\pm} (t), t),

and

R_{\pm}

is constant along the corresponding curve until characteristic intersection creates a shock.

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