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 Incoming links: Reactive acoustic energy flux

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 40B / b / iii / Solution Created 2026-09-24 Updated 2026-10-03

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The normal acoustic energy flux is

I_{y} = p^{'} u_{y} = p^{'} ϕ_{y} .

(1)

In either half-space, the pressure fields above are proportional to

cos (k x - ω t)

, whereas the normal velocities are proportional to

sin (k x - ω t)

. They are therefore in temporal quadrature. Averaging over one period gives

⟨ I_{y} ⟩ \propto ⟨ cos (k x - ω t) sin (k x - ω t)⟩ = 0.

(2)

Thus the field is evanescent and stores reactive acoustic energy flux near the membrane without transporting net acoustic energy away from it in the normal direction; the wave's mean energy transport is along the membrane.

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 40A / b / Solution Created 2026-09-24 Updated 2026-09-25

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At every radius, the pressure perturbation has the form

p (r, t) = - ρ_{0} ω F (r) cos ω t,

(1)

whereas the radial velocity is

u_{r} (r, t) = F^{'} (r) sin ω t .

(2)

Define the average over one forcing period

T = 2 π / ω

⟨ I_{r} ⟩ = \frac{1}{T} \int_{0}^{T} p (r, t) u_{r} (r, t) d t .

(3)

Then

⟨ I_{r} ⟩ = - ρ_{0} ω F (r) F^{'} (r) \frac{1}{T} \int_{0}^{T} cos ω t sin ω t d t = 0 .

(4)

The same is true for the net flux through every concentric sphere.

Pressure and velocity are in temporal quadrature, so this is a reactive acoustic energy flux. The forcing establishes a standing wave in the closed, lossless annulus: compressional and kinetic energy exchange during each cycle, but no energy is transported outward on average. This is the pressure-forced standing acoustic wave in a spherical annulus.

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