Equal-density top-hat planar-jet dispersion relation

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 a top-hat planar jet in stationary fluid of the same mass density, odd and even velocity potentials have the same temporal growth rate. Put $s=\tanh (|k|L)$. Their dispersion relations use $s$ and $1/s$, but $\sqrt{s}/(1+s)=\sqrt{1/s}/(1+1/s)$. Their real phase velocities are respectively $Vs/(1+s)$ and $V/(1+s)$. The common growth rate is strictly below $|kV|/2$ for finite $|k|L>0$, approaching the isolated vortex sheet value when the sheets decouple.