 Solution

Use complex normal modes and take their real parts at the end. Write $\eta =A{e}^{i(kx-\omega t)}$, with $k>0$. The Laplace equation and the wall Neumann boundary conditions give the vertical formsAt $y=0$, the two kinematic boundary conditions becomeThusThe dynamic condition isSubstituting the two amplitudes and cancelling the common nonzero depth factor gives the dispersion relation for a vortex sheet in a symmetric rigid channel:Equivalently,For the specified $U>0$, the plus sign gives positive temporal growth rate $\gamma =kU/2$ in the convention $e^{-i\omega t}$. A real growing interface displacement is thereforeThe other branch decays at the same rate. Consequently the interface is linearly unstable, exhibiting Kelvin-Helmholtz instability. There is no nonzero-amplitude mode with a purely real temporal frequency for $k>0$ and $U\ne 0$; the paper's sinusoidal notation must allow a complex frequency, with the physical displacement obtained by a real part.

The wave pattern is advected at speed $U/2$, the mean of the two base velocities. The growth rate is independent of $h$ here because both layers have the same mass density and the same depth, making their factors $\coth (kh)$ cancel. In an asymmetric channel those inertia factors would differ. In the zero-shear limit $U=0$, the two roots merge at zero and there is no exponential growth.