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 Incoming links: Isentropic Euler equations

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 39A / a / Solution Created 2026-09-24 Updated 2026-10-03

The one-dimensional Isentropic Euler equations are

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

For the polytropic equation of state,

c^{2} = \frac{d p}{d ρ} = \frac{γ p}{ρ}, \frac{c}{c _{0}} = (\frac{ρ}{ρ _{0}})^{(γ - 1) /2} .

The continuity equation therefore implies

c_{t} + u c_{x} + \frac{γ - 1}{2} c u_{x} = 0,

while the momentum equation becomes

u_{t} + u u_{x} + \frac{2 c}{γ - 1} c_{x} = 0.

Adding and subtracting these equations gives

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

Thus the Riemann invariants

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

are constant along the characteristic curves

C_{\pm} : \frac{d x}{d t} = u \pm c .

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 39B / a / Solution Created 2026-09-24 Updated 2026-10-14

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
. Details.

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