 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/39c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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.

 New to topics? Read the docs here!