Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 10 / b / Solution

Fixed endpoints select sine normal modes with $k_n=n\pi /L$ and ${\omega }_n=v{k}_n$. The product identity $\sin z\cos z=\frac{1}{2}\sin 2z$ decomposes the held initial profile into $n=2$ and $n=4$, with amplitudes $1$ and $1/2$. Zero initial velocity selects cosine time dependence. ThereforeBoth endpoints vanish at every time. At $t=0$ the product-form displacement is recovered and $y_t=0$. The implicit leading displacement amplitude is one in the source's chosen length units; including a physical amplitude $A$ multiplies the whole solution by $A$.