= Solution
Seek a separated <normal mode> $y(x,t)=X(x)Q(t)$. Substitution in the <wave equation> gives $XQ''=c^2X''Q$. For an oscillatory mode choose the separation constant $-\omega^2$:
$$
Q''+\omega^2Q=0,\qquad\boxed{X''+k^2X=0,\qquad k=\omega/c}.
$$
The fixed endpoints impose $X(0)=X(l)=0$. In $X=A\sin kx+B\cos kx$, the first condition gives $B=0$, and the second requires $\sin kl=0$. Nonzero displacement therefore requires
$$
\boxed{k_n=\frac{n\pi}{l},\quad\omega_n=\frac{n\pi c}{l},\quad f_n=\frac{nc}{2l},\qquad n=1,2,3,\ldots}.
$$
Each spatial shape $\sin(n\pi x/l)$ has fixed nodes and an oscillating overall amplitude: a <standing wave>. A general lossless motion is a superposition of these standing <normal modes>, not necessarily one single-frequency standing wave. The $n=0$ solution is trivial under both fixed-end conditions and does not supply an additional vibrational mode.
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