= Solution
Use complex <normal modes> and take their real parts at the end. Write $\eta=Ae^{i(kx-\omega t)}$, with $k>0$. The <Laplace equation> and the wall <Neumann boundary conditions> give the vertical forms
$$
\phi_+=C_+\cosh(k(y-h))e^{i(kx-\omega t)},\qquad
\phi_-=C_-\cosh(k(y+h))e^{i(kx-\omega t)}.
$$
At $y=0$, the two <kinematic boundary conditions> become
$$
-k\sinh(kh)C_+=-i(\omega-kU)A,\qquad
k\sinh(kh)C_-=-i\omega A.
$$
Thus
$$
C_+=\frac{i(\omega-kU)A}{k\sinh(kh)},\qquad
C_-=-\frac{i\omega A}{k\sinh(kh)}.
$$
The dynamic condition is
$$
\omega C_-\cosh(kh)=(\omega-kU)C_+\cosh(kh).
$$
Substituting the two amplitudes and cancelling the common nonzero depth factor gives the <dispersion relation> for a <vortex sheet in a symmetric rigid channel>:
$$
\omega^2+(\omega-kU)^2=0.
$$
Equivalently,
$$
\left(\omega-\frac{kU}{2}\right)^2=-\frac{k^2U^2}{4},
\qquad
\boxed{\omega=\frac{kU}{2}\pm i\frac{kU}{2}.}
$$
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 therefore
$$
\boxed{\eta(x,t)=A_0e^{kUt/2}\cos\left(kx-\frac{kU}{2}t\right).}
$$
The other branch decays at the same rate. Consequently \b[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 $U\ne0$; 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.
\Image[/past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2014-channel-instability.png]
{title=A symmetric-channel vortex sheet develops an exponentially growing interface wave}
{description=Small disturbances are advected at half the upper-fluid speed and grow at rate $kU/2$. The left panel keeps all three snapshots small compared with the wall spacing; the right panel shows the real and growing imaginary parts of the frequency.}
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