= Membrane fluctuation spectrum under tension
{title2=$\langle|h_{\mathbf q}|^2\rangle=k_BT/[A(\sigma q^2+\kappa q^4)]$}
For small-slope height fluctuations of a nearly planar membrane, $A$ is projected area, $\sigma$ is <membrane tension>, $\kappa$ is <membrane bending modulus>, and $q$ is a <wavenumber>. With a Fourier convention in which the mode amplitude $h_{\mathbf q}$ has units of length, the quadratic <Helfrich membrane energy> and the <equipartition theorem> give the displayed thermal variance. Bending dominates for $q\gg\sqrt{\sigma/\kappa}$, giving a $q^{-4}$ spectrum. Tension dominates at smaller $q$, giving $q^{-2}$. The corresponding <membrane elastic length> is $\sqrt{\kappa/\sigma}$ when length is measured as $1/q$; a wavelength convention multiplies it by $2\pi$.
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