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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 39B / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

The one-dimensional Isentropic Euler equations are

ρ_{t} + u ρ_{x} + ρ u_{x} = 0, u_{t} + u u_{x} + \frac{c ^{2}}{ρ} ρ_{x} = 0, c^{2} = \frac{d p}{d ρ} .

Define the Riemann invariants for one-dimensional isentropic flow

R_{\pm} = u \pm \int_{ρ_{0}}^{ρ} \frac{c ( s )}{s} d s .

Combining the two equations directly gives

[\partial_{t} + (u \pm c) \partial_{x}] R_{\pm} = 0,

so

R_{\pm}

is constant on the corresponding characteristic curve

d x / d t = u \pm c

.

Initially both invariant disturbances are supported in

0 \leq x \leq L

. Once the two characteristic families have separated, the left-moving packet is a simple wave with

R_{+}

at its equilibrium value, while the right-moving packet is a simple wave with

R_{-}

at equilibrium. Between and outside them, both invariants retain their equilibrium values. The schematic spacetime diagram below uses the equilibrium slopes

\pm c_{0}

and shows separation after

t = L / (2 c_{0})

.

Figure 1.
Characteristic curves and simple-wave separation
.

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