 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/40a/a/solution

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

Let

ρ^{'}

and

p^{'}

be the density and pressure perturbations. Homentropy gives

p^{'} = c_{0}^{2} ρ^{'}

. The linearized mass and momentum equations are

ρ_{t}^{'} + ρ_{0} \nabla \cdot u = 0, ρ_{0} u_{t} = - \nabla p^{'} .

u = \nabla ϕ

, and the momentum equation integrates to

p^{'} = - ρ_{0} ϕ_{t},

where a function of time has been absorbed into

ϕ

. Substitution into mass conservation yields

- \frac{ρ _{0}}{c _{0}^{2}} ϕ_{tt} + ρ_{0} \nabla^{2} ϕ = 0,

ϕ_{tt} - c_{0}^{2} \nabla^{2} ϕ = 0.

 New to topics? Read the docs here!