 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/40c/a/solution

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

Write

ρ = ρ_{0} + ρ

p = p_{0} + p

, and let

u

be small. Linearized mass and momentum conservation are

\partial_{t} ρ + ρ_{0} \nabla \cdot u = 0, ρ_{0} \partial_{t} u = - \nabla p .

The linearized equation of state is

p = c_{0}^{2} ρ

. Differentiate the first equation in time and eliminate the velocity with the second to obtain the wave equation

\partial_{t}^{2} p - c_{0}^{2} \nabla^{2} p = 0 .

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