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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/8/b/i/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 8 / b / i / Solution by

Codex  0 2026-10-18

Transform in time, leaving

r

as a parameter. Applying the derivative identity twice gives

\partial_{t}^{2} σ = - ω^{2} σ

. Therefore

- ω^{2} σ - \partial_{r}^{2} σ + μ^{2} σ = 0, \partial_{r}^{2} σ = (μ^{2} - ω^{2}) σ .

(1)

These manipulations can be justified for sufficiently decaying smooth fields, or with distributional Fourier transforms when ordinary integrability is unavailable.

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