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

For a single equal-density vortex sheet with stream speeds $0$ and $V$, the gravitational contribution vanishes. The Kelvin-Helmholtz dispersion relation with gravity reduces to $c^2+(V-c{)}^2=0$, with roots $c=V/2\pm i|V|/2$. Its positive temporal growth rate is ${\gamma }_{\mathrm{sheet}}=k|V|/2$.

Both jet parity branches have the common growth rate derived above, henceEquivalently, $2\sqrt{s}\le 1+s$ follows from $(\sqrt{s}-1{)}^2\ge 0$. Equality is approached when the two interfaces are separated by many disturbance decay lengths, $kL\to \infty$. The zero-shear case has both growth rates zero. For small $kL$, the growth rate ratio is approximately $2\sqrt{kL}$: interaction between the sheets suppresses amplification relative to an isolated sheet.