 Solution

For either normal mode, let $k_n=n\pi /L$ and write its contribution as $B_n\sin (k_{n}x)\sin ({\omega }_{n}t)$. The product-to-sum identity givesThis standing wave is a superposition of two travelling waves of equal amplitude $|B_n|/2$, moving oppositely with phase velocity ${\omega }_n/k_n=c$. Their cancellation at $x=0,L$ enforces the fixed-end boundary conditions at every time.