At , the vertical velocities match, . Using the stream function representation, this gives Expanding about gives where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition. The free-surface condition: At the free surface , the kinematic condition holds: Linearizing, this is simply where the velocity is linearized on to the surface . Using the normal-mode and streamfunction representations, this condition is , the second interfacial condition. Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at is given by the Young–Laplace equation: where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is Thus, However, this condition refers to the total pressure (base+perturbed), thus (As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form this becomes The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations, with to yield Putting this last equation and the jump condition on together, Substituting the second interfacial condition and using the normal-mode representation, this relation becomes where there is no need to label (only its derivatives) because at Solution Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation with the boundary conditions has the solution The first interfacial condition states that at , which forces The third interfacial condition states that Plugging the solution into this equation gives the relation The A cancels from both sides and we are left with To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then, and clearly If , and c is real. This happens when the lighter fluid sits on top; If , and c is purely imaginary. This happens when the heavier fluid sits on top. Now, when the heavier fluid sits on top, , and where is the Atwood number. By taking the positive solution, we see that the solution has the form and this is associated to the interface position η by: Now define The time evolution of the free interface elevation initially at is given by: which grows exponentially in time. Here B is the amplitude of the initial perturbation, and denotes the real part of the complex valued expression between brackets.