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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 40A / b / Solution

At every radius, the pressure perturbation has the form

p (r, t) = - ρ_{0} ω F (r) cos ω t,

whereas the radial velocity is

u_{r} (r, t) = F^{'} (r) sin ω t .

Define the average over one forcing period

T = 2 π / ω

by

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

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 .

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.

Solved by gpt-5.6-sol high.

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