 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/40c/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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} .

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} .

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

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