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 Incoming links: Mean acoustic power of a pulsating sphere

Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 38C / Solution Created 2026-09-24 Updated 2026-09-24

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / ii / Solution Created 2026-09-24 Updated 2026-09-25

For real harmonic quantities

f (t) = Re (f e^{iω t}), g (t) = Re (g e^{iω t}),

define the period average by

⟨ f g ⟩ = \frac{ω}{2 π} \int_{0}^{2 π / ω} f (t) g (t) d t = \frac{1}{2} Re (f g^{*}) .

This is the time average of harmonic power.

At

r = a

, put

x = ωa / c_{0}

. The velocity and pressure amplitudes are

u_{r} (a) = iω ϵ, p (a) = - i ρ_{0} ω ϕ (a) = - \frac{ρ _{0} ω ^{2} ϵ a}{1 + i x} .

Hence the mean outward work rate per unit area is

⟨ p^{'} u_{r} ⟩ = \frac{1}{2} Re (p u_{r}^{*}) = \frac{ρ _{0} ω ^{4} ϵ ^{2} a ^{2}}{2 c _{0} ( 1 + x ^{2} )} .

Multiplication by the area

4 π a^{2}

yields

⟨ P ⟩ = \frac{2 π ρ _{0} ω ^{4} ϵ ^{2} a ^{4} / c _{0}}{1 + ω ^{2} a ^{2} / c _{0}^{2}} = 2 π a^{2} ρ_{0} ω^{2} ϵ^{2} c_{0} \frac{ω ^{2} a ^{2}}{c _{0}^{2} + ω ^{2} a ^{2}} .

This is the mean acoustic power of a pulsating sphere.

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