= Solution
Take $k>0$ without loss of generality, so $\alpha=kh/2>0$, and require decay as $Z\to\pm\infty$. The base velocity is $\tilde U=Z$ everywhere; there is no jump in its derivative. Let $C=\tilde c$ and $E=e^{-2\alpha}$. A decaying representation that already solves the layer equations is
$$
\tilde\psi=Ae^{-\alpha|Z+1|}+Be^{-\alpha|Z-1|}.
$$
It is continuous at both interfaces. At $Z=-1$ the value is $A+BE$ and the derivative jump is $-2\alpha A$; at $Z=1$ they are $B+AE$ and $-2\alpha B$. The density anomaly drops by one across each interface. The <jump conditions for stratified inviscid shear flow> therefore reduce to
$$
\boxed{\begin{pmatrix}
2\alpha(1+C)^2-J&-JE\\
-JE&2\alpha(1-C)^2-J
\end{pmatrix}\begin{pmatrix}A\\B\end{pmatrix}=0.}
$$
For nonreal $C$ the displacement denominators do not vanish. Neutral limiting values are obtained by continuation, rather than by dividing by zero at an interface.
Set $q=J/(2\alpha)$. A nonzero mode requires the <determinant> to vanish:
$$
[(1+C)^2-q][(1-C)^2-q]-q^2E^2=0.
$$
Expansion gives the <dispersion relation for two density interfaces in uniform shear>,
$$
\boxed{C^4-\left(2+\frac J\alpha\right)C^2+
\frac{(2\alpha-J)^2-J^2e^{-4\alpha}}{4\alpha^2}=0.}
$$
Uniform shear throughout the exterior is important: replacing it with constant exterior velocities would introduce <vorticity> jumps and give a different <dispersion relation>.
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