Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A / a

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- Solution a

Solution

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^{'} .

For potential flow,

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,

so

ϕ_{tt} - c_{0}^{2} \nabla^{2} ϕ = 0.

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