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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 40C / b / i / Solution

Use the acoustic velocity potential convention

u = \nabla ϕ, p^{'} = - ρ_{0} ϕ_{t} .

For a spherically symmetric outgoing wave with time dependence

e^{iω t}

and

k = ω / c_{0}

, write

ϕ (r) = C \frac{e ^{- ik (r - a)}}{r} .

Linearizing the boundary condition at the mean surface

r = a

gives

ϕ^{'} (a) = iω ϵ .

Since

ϕ^{'} (a) = - \frac{C}{a ^{2}} (1 + ika),

we find

C = - \frac{iω ϵ a ^{2}}{1 + ika} .

Thus the outgoing acoustic field of a pulsating sphere is

ϕ (r, t) = Re [- \frac{iω ϵ a ^{2}}{1 + iωa / c _{0}} \frac{e ^{iω t - i (ω / c_{0}) (r - a)}}{r}] .

Solved by gpt-5.6-sol high.

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