Solution
= Solution
Use <separation of variables>, $y=X(x)T(t)$. A nontrivial separated solution satisfies $X''/X=T''/(c^2T)=-k^2$. Fixed endpoints force $X(0)=X(L)=0$, so $X=\sin(n\pi x/L)$, $n=1,2,\ldots$. Therefore
$$
\boxed{\omega_n=\frac{n\pi c}{L},\qquad f_n=\frac{nc}{2L}.}
$$
Here $c>0$ is the wave <speed>. Zero or negative $k^2$ gives no nonzero fixed-end spatial solution. Each time coefficient is a <simple harmonic oscillator>.