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The dispersion relation for the Kelvin-Helmholtz instability can be derived by assuming that the fluids are incompressible flow and irrotational such that the fluid velocities and may be defined by a scaler potential and corresponding to the top and bottom fluid respectively, such that:

Using Reynolds decomposition, the scaler potential is decomposed into an ensemble of time varying perturbations superimposed upon a time independent mean state:

The interface between the two fluids is described in a similar manner, such that