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Linear homentropic acoustic equations Created 2026-09-29 Updated 2026-10-06

For perturbations

(ρ^{'}, p^{'}, u)

about a uniform fluid of density

ρ_{0}

, the linearized continuity equation, momentum equation, and homentropic pressure perturbation relation are

ρ_{t}^{'} + ρ_{0} \nabla \cdot u = 0, ρ_{0} u_{t} = - \nabla p^{'}, p^{'} = c_{0}^{2} ρ^{'} .

For irrotational flow

u = \nabla ϕ

, they imply

p^{'} = - ρ_{0} ϕ_{t}

and

ϕ_{tt} = c_{0}^{2} \nabla^{2} ϕ

.

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / a / Solution Created 2026-09-24 Updated 2026-09-25

Let

ρ^{'}

and

p^{'}

be the density and pressure perturbations and let

u

be the velocity perturbation. The linear acoustics equations about a uniform quiescent state are

ρ_{t}^{'} + ρ_{0} \nabla \cdot u = 0, ρ_{0} u_{t} = - \nabla p^{'}, p^{'} = c_{0}^{2} ρ^{'} .

The last relation is the homentropic pressure perturbation law.

Taking the scalar product of the momentum equation with

u

gives

\frac{\partial}{\partial t} (\frac{1}{2} ρ_{0} ∣ u ∣^{2}) = - u \cdot \nabla p^{'} = - \nabla \cdot (p^{'} u) + p^{'} \nabla \cdot u .

The continuity and pressure relations imply

p^{'} \nabla \cdot u = - \frac{p ^{'} p _{t}^{'}}{ρ _{0} c _{0}^{2}} = - \frac{\partial}{\partial t} (\frac{p ^{'2}}{2 ρ _{0} c _{0}^{2}}) .

Therefore the acoustic energy conservation law is

\frac{\partial}{\partial t} (K + W) + \nabla \cdot I = 0,

with

K = \frac{1}{2} ρ_{0} ∣ u ∣^{2}, W = \frac{p ^{'2}}{2 ρ _{0} c _{0}^{2}}, I = p^{'} u .

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