Past exam of the mathematical methods II course of the University of Oxford / 2014 / 11 / c / Solution

Use the product identity $\sin z\cos z=\frac{1}{2}\sin 2z$ to decompose the initial shape:The Fourier sine series is already explicit. Zero initial velocity gives cosine time factors, soand $y(x,t)=A\sin (\pi x/l)\cos (\pi ct/l)+(A/2)\sin (2\pi x/l)\cos (2\pi ct/l)$. Orthogonality of the sine modes makes this decomposition unique in the usual square-integrable fixed-end setting.