# User:Loren36/kelvin

## Theory

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 ${\displaystyle u_{1}}$ and ${\displaystyle u_{2}}$ may be defined by a scaler potential ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$ corresponding to the top and bottom fluid respectively, such that:

${\displaystyle u_{i}=\nabla \phi _{i}}$

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

${\displaystyle \phi _{i}(x,y,z,t)=\Phi _{i}+\phi _{i}'(x,y,z,t)}$
${\displaystyle \phi _{i}'(x,y,z,t)={\hat {\phi _{i}}}(z)e^{i(kx+ly)+st}}$

The interface between the two fluids ${\displaystyle \zeta }$ is described in a similar manner, such that

${\displaystyle \zeta (x,y,z,t)={\hat {\zeta }}(z)e^{i(kx+ly)+st}}$