= Bending-membrane Kelvin-Helmholtz dispersion relation
{title2=$(\sigma+ikU/2)^2=U^2k^2/4-\beta|k|^5/(2\rho)$}
For equal fluid density $\rho$ on opposite sides of a massless flat membrane, a velocity jump $U$ and bending pressure jump $\beta\eta_{xxxx}$ with $\beta>0$, a mode $e^{ikx+\sigma t}$ satisfies
$$
(\sigma+ikU/2)^2=U^2k^2/4-\beta|k|^5/(2\rho).
$$
The decaying upper and lower perturbation potentials are $A_+e^{-|k|y}$ and $A_-e^{|k|y}$. Their kinematic conditions are $-|k|A_+=(\sigma+ikU)\eta$ and $|k|A_-=\sigma\eta$. The linearized <Bernoulli equation> gives $p_- -p_+=-\rho[\sigma^2+(\sigma+ikU)^2]\eta/|k|$. Equating it to the bending pressure yields the relation. Bending stabilizes $|k|^3\ge\rho U^2/(2\beta)$. The maximum positive growth is at $|k|^3=\rho U^2/(5\beta)$ and is $|U|\,|k|\sqrt{3/20}$.
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