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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 $2\pi$.

The higher-order Euler-Lagrange equation for a functional depending on $h_x$ and $h_{xx}$ givesWith the membrane elastic lengththe general free-membrane profile isTwo integrations by parts, followed by use of the field equation, turn the energy of any free interval $[a,b]$ into the boundary term

Take the undeformed membrane to have $h,h_x\to 0$ at infinity. In the small-contact approximation, the cylindrical profile isContinuity of height and slope at the right contact point and exponential decay then givewith its reflected copy on the left. The two free tails carryWithin the contact, $h_{xx}=-1/R$. Since ${\delta }_o\ll \xi$, the tension contribution there is smaller than the bending contribution by $O({\delta }_o^2/{\xi }^2)$, soWriting the dimensionless adhesion strength as $U=R^2\mathcal{U}/k_c$, the total energy isIts stationary point isIt is an admissible bound state only for $U>1/2$; otherwise the constrained minimum is the unbound state ${\delta }_o=0$. Substitution gives

Write the membrane height above its mean plane as a sum of Fourier modes, $h(\mathbf{r})={\sum }_{\mathbf{q}}{h}_{\mathbf{q}}{e}^{i\mathbf{q}\cdot \mathbf{r}}$. The supplied average is a thermal mean-squared modal height; for complex Fourier coefficients it is understood as $⟨|h_{\mathbf{q}}{|}^2⟩$, with $h_{-\mathbf{q}}=h_{\mathbf{q}}^{\ast }$ for a real height field. The terms are:

In the small-slope approximation, Helfrich membrane energy containsThe equipartition theorem, with the independent real modes counted consistently, gives the stated membrane fluctuation spectrum under tension. Its units are length squared: $\sigma q^2$ and $\kappa q^4$ both have units $\mathrm{J}{\mathrm{m}}^{-4}$, so $k_{B}T/[A(\sigma q^2+\kappa q^4)]$ has units ${\mathrm{m}}^2$.

At small length scales, $q$ is large and the bending term dominates:The fluctuation amplitude is controlled mainly by the membrane bending modulus and thermal energy, rather than tension. The crossover occurs when $\sigma q_c^2=\kappa q_c^4$, henceThe bending-dominated regime extends up to the membrane elastic length when scale means $1/q$, or up to ${\lambda }_c$ when it means wavelength. At still larger scales tension gives a $q^{-2}$ spectrum. At the opposite end, the continuum formula requires wavelengths larger than the microscopic membrane thickness or molecular cutoff. Without numerical $\kappa$ and $\sigma$, no numerical crossover scale is determined.