 Solution

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

Substitute the three nonzero initial-velocity coefficients into the general Fourier sine series, with

ω_{1} = π c / L

y (x, t) = \frac{V _{0}}{ω _{1}} [sin \frac{π x}{L} sin ω_{1} t + \frac{1}{4} sin \frac{2 π x}{L} sin 2 ω_{1} t + \frac{2}{9} sin \frac{3 π x}{L} sin 3 ω_{1} t] .

Every term satisfies the wave equation and the endpoint conditions. Differentiating in time at zero recovers the three coefficients

V_{0}, V_{0} /2, 2 V_{0} /3

, providing a direct initial-data check.

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