= Semidiscrete fixed-string normal mode
{title2=$v_j^{(k)}=\sqrt{2/(n+1)}\sin(jk\pi/(n+1))$}
The <second-order central difference> approximation to the fixed-end <wave equation> on $n$ interior points has <normal modes>
$$
v_j^{(k)}=\sqrt{\frac2{n+1}}\sin\frac{jk\pi}{n+1},\qquad
\lambda_k=-\frac{4c^2}{h^2}\sin^2\frac{k\pi}{2(n+1)},\qquad
\omega_k=\sqrt{-\lambda_k},\qquad 1\le j,k\le n.
$$
Here $h=L/(n+1)$ and the endpoint <displacements> are zero. The <eigenvectors> form an <orthonormal basis> in the ordinary <Euclidean norm>, without an extra grid-spacing weight. They sample the continuous <standing wave> shapes exactly, while their <angular frequencies> differ from $k\pi c/L$ because of spatial discretization. For fixed $k$, the difference tends to zero as $h\to0$.
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