= Discrete Neumann mode selection
{c}
{title2=$q_m=(m\pi/\ell)^2$}
On $0<x<\ell$, homogeneous <Neumann boundary conditions> give spatial <Fourier modes> $\cos(m\pi x/\ell)$, $m=0,1,\ldots$, with squared <wavenumbers> $q_m$. Differentiating twice proves that their <Laplacian> <eigenvalues> are $-q_m$, and their endpoint <derivatives> vanish. A <dispersion relation> $\sigma(q)$ therefore predicts the fastest permitted growing mode by maximizing $\operatorname{Re}\sigma(q_m)$ over integers $m$, rather than over all real <wavenumbers>. A continuous unstable band is insufficient if it contains no $q_m$.
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