Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 38B / iii

If the lower density ${\rho }_{-}$ exceeds the upper density ${\rho }_{+}$, stable density stratification adds a gravitational restoring term. Including surface tension $\gamma$, the same calculation has the schematic dispersion relationThe gravity term dominates the shear term as $K\to 0$ and therefore suppresses sufficiently long waves. The capillary term dominates as $K\to \infty$ and suppresses sufficiently short waves. Together they reduce the Kelvin-Helmholtz instability growth rate and can leave only a finite unstable band, or eliminate instability if the restoring effects are strong enough.