Equal-density top-hat planar-jet dispersion relation
= 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 $e^{ikx+\sigma t}$ satisfy
$$
\sigma^2+\coth(|k|h)(\sigma+ikU)^2=0
$$
for a <varicose mode of a planar jet>, and
$$
\sigma^2+\tanh(|k|h)(\sigma+ikU)^2=0
$$
for a <sinuous mode of a planar jet>. Both have unstable growth rate
$$
\operatorname{Re}\sigma
=\frac{|kU|}{2}\sqrt{1-e^{-4|k|h}}.
$$