THEORY We have all noticed how the surface of a lake gets ruffled by a gentle breeze. To understand such phenomena, we have to study perturbations at an interface between two fluids. Let us consider a plane horizontal interface separating two fluids. If this interface is disturbed, we expect that, under certain circumstances, the interface may oscillate in the form of waves and, under other circumstances, the disturbance may grow to give rise to an instability.

We shall first develop a general mathematical analysis to show how a perturbation at such an interface evolves.

To simplify the analysis, we assume that the fluids on both sides of the interface are incompressible and ideal. We suppose that the viscosity is insignificant if the boundary layer thickness is weak face to the disturbance.

Hence, if there is no vorticity inside one of these fluids at the beginning, Kelvin's vorticity theorem asserts that the velocities induced inside the fluid as a result of the perturbation remain irrotational. So velocity potentials can be introduced.
We neglect the surface tension at the interface between the two fluids and the gravity effect so that we consider the special case of vortex sheet in a homogeneous fluid ( ).
So we consider fluids in two horizontal parallel infinite streams of different velocity, one stream above the other in a two-dimensional problem.
The study is made in the system of reference moving at the average two fluid velocity which gives us the next figure : Then consider an initial disturbance which slighty displaces the sheet so that its elevation is sinusoidal and equal to z=e(x,t).
So we assume the existence of a velocity potential on each side of the interface between the two streams. So we obtain : where V.x represent the velocity potential of the undisturbed flow and the disturbed one's.
The calculation is now made with the lineary hypothesis, ie The fluid particuls at the interface must move with the interface without the two fluids occupying the same point at the same time without a cavity forming between the fluids. The vertical velocity at the interface is given by : (1)

We could now write the pressure continuity at the interface : pressure of the two streams are equal at the interface z=e.

Moreover, the Bernoulli's theorem gives us the previous relation: which gives at the interface, considering the equality of pressure: (2), by considering y=e like y=0 in first order.

We are seeking for solution of the form : where k represent the wavenumber and the growth rate.

By introducing the last relation in the equations (1) and (2), we obtain the previous relations between constants : We obtain the compatibility between these equation by writting that his determinant is null, which gives us: This is the dispersion relation which shows that unstable mode  always exists.

The next figures show us the formes of different Kelvin-Helmholtz instability steps:  In the next part, the numerical simulations represent some of those steps.