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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2022/12/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2022 / 12 / b / Solution by

Codex  0 2026-10-25

Expand the initial velocity in the Fourier sine basis,

V (x) = \sum_{n \geq 1} V_{n} sin (nπ x / L)

, where

V_{n} = (2/ L) \int_{0}^{L} V (x) sin (nπ x / L) d x

. Zero initial displacement removes the cosine-in-time coefficients. Thus

y (x, t) = n \geq 1 \sum \frac{V _{n}}{ω _{n}} sin \frac{nπ x}{L} sin (ω_{n} t) .

The trigonometric identity

2 sin a sin b = cos (a - b) - cos (a + b)

rewrites each summand as

\frac{V _{n}}{2 ω _{n}} [cos (\frac{nπ x}{L} - ω_{n} t) - cos (\frac{nπ x}{L} + ω_{n} t)] .

It is therefore a superposition of travelling waves of speeds

+ c

and

- c

, whose interference enforces the fixed endpoints. For general square-integrable initial velocity, the series is interpreted in the standard finite-energy sense.

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