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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-1/40b/b/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 40B / b / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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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