Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 38C / f / Solution

When $|k|h\gg 1$, the two shear layers are separated by many disturbance decay lengths. Since $\coth (|k|h)\to 1$, the roots reduce toThis is exactly the Kelvin-Helmholtz instability of one isolated vortex sheet separating equal-density streams of speeds $U$ and $0$: the pattern travels at their mean speed $U/2$, while its unstable amplitude grows at rate $|k|U/2$. The exponentially weak interaction between the two layers disappears in this short-wavelength limit.