Minimum shear for capillary Kelvin-Helmholtz instability ($U_{\min }^2=\frac{2({\rho }_1+{\rho }_2)}{{\rho }_1{\rho }_2}\sqrt{({\rho }_2-{\rho }_1)g\gamma }$)

For two infinitely deep inviscid layers with stable density ordering ${\rho }_2>{\rho }_1$ and positive surface tension $\gamma$, the velocity jump destabilizes wavenumber $k$ precisely whenThe threshold diverges at both small and large wavenumber. Minimizing the bracket gives $|k|=\sqrt{({\rho }_2-{\rho }_1)g/\gamma }$ and the stated minimum velocity jump. A larger jump opens a bounded band of unstable wavenumbers, obtained from the two positive roots of $\gamma q^2-[{\rho }_1{\rho }_2{U}^2/({\rho }_1+{\rho }_2)]q+({\rho }_2-{\rho }_1)g=0$. Equality at threshold is marginal rather than exponential growth.