= Solution
When $|k|h\gg1$, the two shear layers are separated by many disturbance decay lengths. Since $\coth(|k|h)\to1$, the roots reduce to
$$
\boxed{\sigma\sim-\frac{ikU}{2}\mathbin{\pm}\frac{|k|U}{2}}.
$$
This 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>.
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