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 Incoming links: Stratified acoustic pressure equation

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 75 / 1 / b / Solution Created 2026-10-03 Updated 2026-10-06

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At rest, all time derivatives and advective terms vanish. The continuity equation and the stated energy equation are then identities, while the momentum equation reduces to

\nabla (p_{0} + χ) = 0

. The reference state need not have uniform mass density or entropy. Assume smooth positive reference pressure and mass density, and retain only first-order perturbations.

The linearized continuity equation, momentum balance and adiabatic energy relation are

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

(1)

p_{t}^{'} + u^{'} \cdot \nabla p_{0} = c_{0}^{2} (ρ_{t}^{'} + u^{'} \cdot \nabla ρ_{0}) = - ρ_{0} c_{0}^{2} \nabla \cdot u^{'} .

(2)

The perturbation of

c^{2}

multiplies a vanishing reference material derivative, so it does not enter at first order. Differentiate the last equation in time, then substitute the linearized momentum equation:

p_{tt}^{'} = \frac{\nabla p _{0}}{ρ _{0}} \cdot \nabla p^{'} + ρ_{0} c_{0}^{2} \nabla \cdot (\frac{\nabla p ^{'}}{ρ _{0}}) .

(3)

Expanding the divergence yields the stratified acoustic pressure equation

\frac{p _{tt}^{'}}{c _{0}^{2}} - \nabla^{2} p^{'} = (\frac{\nabla p _{0}}{c _{0}^{2} ρ _{0}} - \frac{\nabla ρ _{0}}{ρ _{0}}) \cdot \nabla p^{'} .

(4)

For a perfect gas,

c_{0}^{2} ρ_{0} = γ p_{0}

. Put

a (x) = p_{0}^{1/ γ} / ρ_{0}

. Then

\nabla lo g a = \frac{\nabla p _{0}}{γ p _{0}} - \frac{\nabla ρ _{0}}{ρ _{0}}, \frac{1}{a} \nabla \cdot (a \nabla p^{'}) = \nabla^{2} p^{'} + \nabla lo g a \cdot \nabla p^{'} .

(5)

Thus the divergence-form stratified acoustic pressure equation is

c_{0}^{- 2} p_{tt}^{'} - a^{- 1} \nabla \cdot (a \nabla p^{'}) = 0.

(6)

This is homogeneous linear propagation through the static medium, including its inhomogeneity. In the acoustic analogy of part (a),

W

has no first-order contribution when viscosity is neglected and the reference velocity is zero. Nevertheless the chosen left-hand operator lacks the gradient terms in the genuine propagation equation. Those effects must consequently appear in

Q_{tt}

. They describe propagation of an existing disturbance, rather than an independent source. Since reference-field choices also change the division between the operator and forcing, $Q_{tt}$ cannot be identified unambiguously with newly generated noise. Its presence in the right-hand side is a consequence of the chosen analogy, not a source-classification theorem.

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Propagation-source ambiguity in an acoustic analogy 2026-10-06

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The distinction between generation and propagation depends partly on the reference operator chosen in an acoustic analogy. For example, a homogeneous-Laplacian operator with spatially varying reference sound speed omits the first-gradient terms of the stratified acoustic pressure equation. These terms then reappear as effective forcing, even for a freely propagating small disturbance in a static medium. A right-hand-side term is therefore not automatically an independent physical noise source.

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