Discrete Neumann mode selection ($q_m=(m\pi /\ell {)}^2$)

On $0<x<\ell$, homogeneous Neumann boundary conditions give spatial Fourier modes $\cos (m\pi x/\ell )$, $m=0,1,\dots$, 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 $\mathrm{Re}\sigma (q_m)$ over integers $m$, rather than over all real wavenumbers. A continuous unstable band is insufficient if it contains no $q_m$.