Bending-membrane Kelvin-Helmholtz dispersion relation ($(\sigma +ikU/2{)}^2=U^2{k}^2/4-\beta |k{|}^5/(2\rho )$)

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 $e^{ikx+\sigma t}$ satisfiesThe 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 $|U||k|\sqrt{3/20}$.