= Solution
Use <separation of variables>, $y(x,t)=X(x)Q(t)$. The <wave equation> implies $X''/X=\ddot Q/(c^2Q)=-k^2$, giving $X''+k^2X=0$ and $\ddot Q+c^2k^2Q=0$. The fixed-end <boundary conditions> $X(0)=X(l)=0$ select
$$
X_n(x)=\sin\frac{n\pi x}l,\qquad k_n=\frac{n\pi}l,\qquad \boxed{\omega_n=\frac{n\pi c}l,\quad f_n=\frac{nc}{2l},\quad n=1,2,\ldots.}
$$
Each such product is a <standing wave>. A general released string is a superposition of these <normal modes>, not necessarily a single separated standing wave. Absence of damping keeps their amplitudes constant.
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