 Solution

For one sine mode, $y_{xx}=-k_n^2{A}_n\sin (k_{n}x)\cos ({\omega }_{n}t)$ and $y_{tt}=-{\omega }_n^2{A}_n\sin (k_{n}x)\cos ({\omega }_{n}t)$. The wave equation on a string therefore holds whenThe superposition principle gives the same equation for the sum, assuming enough regularity to differentiate termwise, or interpreting finite-energy data in the usual modal sense. Fixed endpoints require $\sin (k_{n}L)=0$, so $k_n=n\pi /L$. Every cosine time factor has zero time derivative at $t=0$, giving the held-at-rest initial condition.

The coefficients must represent the actual held profile $f(x)$ through its Fourier sine series, $A_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$ for $n\ge 1$. Thus the series accommodates the displacement initial condition once these coefficients are chosen; it does not select an unspecified initial shape by itself. The printed $n=0$ term is identically zero because $\sin 0=0$, so the nontrivial fixed-end modes start at $n=1$.