Fourier mode ($e^{i\omega x}$)

A Fourier mode is a single complex oscillatory component $e^{i\omega x}$. Its real and imaginary parts are a cosine and a sine; $\omega$ is the spatial angular frequency. Every constant-coefficient differential operator acts on it by multiplication: $P({\partial }_x)e^{i\omega x}=P(i\omega )e^{i\omega x}$. In particular, ${\partial }_x^2{e}^{i\omega x}=-{\omega }^2{e}^{i\omega x}$. These eigenfunctions reduce a partial differential equation such as the heat equation to separate equations for each frequency. On the real line a pure Fourier mode is not in the L2 space; the Fourier transform uses such modes in a continuous superposition.

The spatially constant Fourier mode. On a periodic domain its coefficient represents the spatial mean, up to the chosen Fourier transform normalization. Derivatives annihilate this mode, so homogeneous periodic diffusion preserves it.

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 $q_m$.