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For a jet of velocity $U$ occupying $|y|<h$ in stationary fluid of the same density, disturbances proportional to $e^{ikx+\sigma t}$ satisfyfor a varicose mode of a planar jet, andfor a sinuous mode of a planar jet. Both have unstable growth rate

For an even velocity potential, take ${\Phi }_I=C\cosh (kz)$ and the decaying exterior amplitudes equal at the upper and lower interfaces. At $z=L$, kinematics gives $A=ic{B}_{+}$ and $C\sinh (kL)=i(V-c)B_{+}$. The dynamic condition becomes $-cA=(V-c)C\cosh (kL)$. ThereforeWriting $s=\tanh (kL)$, its two roots areThe interior vertical velocity is odd, so $B_{-}=-B_{+}$; this is a varicose mode of a planar jet. As with the odd potential, $k>0$, finite $L>0$ and $V\ne 0$ give a growing root. The parity of the potential and the parity of the interface displacements are opposite because the vertical velocity is its derivative.

Take $k>0$; reflection in $x$ supplies the result for negative wavenumber. Since each region has uniform base velocity, the perturbation is an incompressible, irrotational flow with a velocity potential satisfying the Laplace equation. Its decaying outer forms areFor a varicose mode of a planar jet, reflection symmetry makes the inner potential proportional to $\cosh (ky)$; for a sinuous mode of a planar jet, it is proportional to $\sinh (ky)$.

At the upper interface, the linearized kinematic boundary condition isThe first equation gives ${\varphi }_{+}(h)=-\sigma \eta /k$. The second givesThe linearized Unsteady Bernoulli equation gives $p^{\prime }=-\rho ({\partial }_t+U{\partial }_x)\varphi$. Applying pressure continuity therefore produces the equal-density top-hat planar-jet dispersion relationSolving the quadratic equation givesEquivalently,The plus sign is the unstable Kelvin-Helmholtz instability branch. In particular, the two modes have exactly the same growth rate,Its limiting forms areThus neither type grows faster in this equal-density idealization, although their phase speeds differ. The unbounded large-$k$ growth is the familiar short-wave pathology of an inviscid vortex sheet.