= Solution
Put $q=|k|$. Decay at infinity and the symmetry of the disturbance suggest the mode potentials
$$
\varphi_+=Ae^{-q(y-h)}e^{ikx+\sigma t},\qquad
\varphi_-=Ae^{q(y+h)}e^{ikx+\sigma t},
$$
$$
\varphi_0=B\cosh(qy)e^{ikx+\sigma t}.
$$
The middle potential is an <even function> of $y$ because its vertical derivative must take opposite values at the two interfaces. The upper kinematic conditions give
$$
-qA=(\sigma+ikU)\eta_0,\qquad
qB\sinh(qh)=\sigma\eta_0.
$$
The lower conditions give the same two relations. The upper dynamic condition is
$$
(\sigma+ikU)A=\sigma B\cosh(qh).
$$
Eliminating $A$, $B$ and $\eta_0$ yields the <dispersion relation>
$$
\boxed{(\sigma+ikU)^2+\sigma^2\coth(qh)=0}.
$$
Writing $C=\coth(qh)$, its two roots are
$$
\boxed{\sigma=\frac{-ikU\mathbin{\pm}qU\sqrt C}{1+C}}.
$$
For every real $k\ne0$, $C>0$ and one root has the positive <growth rate>
$$
\operatorname{Re}\sigma=\frac{|k|U\sqrt{\coth(|k|h)}}{1+\coth(|k|h)}>0.
$$
\b[Thus every nonzero wavenumber is linearly unstable in this inviscid <vortex sheet> model; $k=0$ is neutral.]
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