Two regions of inviscid fluid with the same density are separated by a thin membrane at $y=0$. The fluid in $y>0$ has the uniform velocity $(U, 0,0)$ in Cartesian coordinates, while the fluid in $y<0$ is at rest.

The membrane is now slightly perturbed to $y=\eta(x, t)$. The dynamical effect of the membrane is to induce a pressure difference across it equal to $\beta \partial^{4} \eta / \partial x^{4}$, where $\beta$ is a constant and the sign is such that the pressure is higher below the interface when $\partial^{4} \eta / \partial x^{4}>0$.

On the assumption that the flow remains irrotational and all perturbations are small, derive the relation between $\sigma$ and $k$ for disturbances of the form $\eta(x, t)=\operatorname{Re}\left(C e^{i k x+\sigma t}\right)$, where $k$ is real but $\sigma$ may be complex. Show that there is instability only for $|k|<k_{\max }$, where $k_{\max }$ is to be determined. Find the maximum growth rate and the value of $|k|$ for which this is obtained.