Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 1 / b / iv / Solution

The odd-potential branch uses $s=\tanh (kL)$, whereas the even-potential branch uses $1/s$. The imaginary parts are identical becauseConsequently the equality of top-hat jet parity growth rates givesHere $\gamma =k{c}_i$ is the positive temporal growth rate. The real phase velocities differ: $c_{r,\mathrm{odd}}=\frac{V}{2}(1-e^{-2kL})$ and $c_{r,\mathrm{even}}=\frac{V}{2}(1+e^{-2kL})$. Thus equal amplification does not imply equal propagation. As $kL\to \infty$, both approach the isolated-vortex sheet phase velocity $V/2$ and growth rate $k|V|/2$.