= Solution
For one sine mode, $y_{xx}=-k_n^2A_n\sin(k_nx)\cos(\omega_nt)$ and $y_{tt}=-\omega_n^2A_n\sin(k_nx)\cos(\omega_nt)$. The <wave equation on a string> therefore holds when
$$
\boxed{\omega_n^2=\frac T\mu k_n^2,\qquad \omega_n=vk_n,\quad v=\sqrt{T/\mu}.}
$$
The <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 $\sin(k_nL)=0$, 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>, $A_n=(2/L)\int_0^Lf(x)\sin(n\pi x/L)dx$ for $n\geq1$. 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 $\sin0=0$, so the nontrivial fixed-end modes start at $n=1$.
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