 Solution

A material interface obeys ${\eta }_t+{\varphi }_{j,x}{\eta }_x={\varphi }_{j,y}$ at its actual height. Keeping only first-order terms and evaluating at the undisturbed interface gives ${\varphi }_{1,y}={\varphi }_{2,y}={\eta }_t$ at $y=0$. Linearized unsteady Bernoulli gives the pressure perturbation $-{\rho }_j({\varphi }_{j,t}+g\eta )$, after absorbing time-only constants into the potentials. Pressure continuity therefore givesFor $\vartheta =kx-\omega t$, choose potentials decaying away from the interface:They are harmonic and their normal derivatives equal $A\omega \sin \vartheta ={\eta }_t$ at zero. The dynamic condition reduces toIf the denser fluid is above, the squared frequency is negative. The corresponding perturbations grow or decay exponentially instead of oscillating: this is Rayleigh-Taylor instability, with growing rate $\sqrt{gk({\rho }_2-{\rho }_1)/({\rho }_1+{\rho }_2)}$ in the ideal model without surface tension.