 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/39b/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

For

p (ρ) = A - \frac{B ^{2}}{ρ},

c^{2} = \frac{d p}{d ρ} = \frac{B ^{2}}{ρ ^{2}}, c = \frac{B}{ρ} .

Since

\int (c / ρ) d ρ = - B / ρ = - c

, choose invariants that vanish in the equilibrium state

(u, c) = (0, c_{0})

R_{+} = u - c + c_{0}, R_{-} = u + c - c_{0} .

Consequently

u + c = c_{0} + R_{-}, u - c = - c_{0} + R_{+} .

In a right-moving simple wave with

R_{-} = 0

, the

R_{+}

u + c = c_{0}

, independent of amplitude. The transport equation becomes

(\partial_{t} + c_{0} \partial_{x}) R_{+} = 0

, so the whole velocity profile translates rigidly at the equilibrium sound speed

c_{0}

without nonlinear distortion.

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