Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 40A / c / ii

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- Solution ii

Solution

For real physical fields of period

T = 2 π / (k c)

⟨ I_{z} ⟩ (x, z) = \frac{1}{T} \int_{t_{0}}^{t_{0} + T} p^{'} (x, z, t) u_{z} (x, z, t) d t .

Equivalently, for complex amplitudes,

⟨ I_{z} ⟩ = \frac{1}{2} Re (p u_{z}^{*}) .

Here

p = i ρ_{0} k c Φ e^{- α z}, u_{z} = - α Φ e^{- α z} .

Their product is purely imaginary, so

⟨ I_{z} ⟩ = 0.

Pressure and normal velocity are in temporal quadrature: energy is stored and returned by the evanescent field, but none is transported away from the membrane on average.

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