 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014/11/c/solution

Use the product identity

sin z cos z = \frac{1}{2} sin 2 z

to decompose the initial shape:

y (x, 0) = A sin \frac{π x}{l} + \frac{A}{2} sin \frac{2 π x}{l} .

The Fourier sine series is already explicit. Zero initial velocity gives cosine time factors, so

A_{1} = A, A_{2} = A /2, A_{n} = 0 (n \geq 3),

and

y (x, t) = A sin (π x / l) cos (π c t / l) + (A /2) sin (2 π x / l) cos (2 π c t / l)

. Orthogonality of the sine modes makes this decomposition unique in the usual square-integrable fixed-end setting.

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