= Solution
Take $k>0$, without loss for this symmetric real base state, and write $s=1+J$, $e=e^{-2\alpha}$, and $y=\widetilde c^{\,2}$. The given quartic becomes a real quadratic $y^2+Ay+B=0$. Its constant term factors as
$$
B=\frac{[2\alpha-s(1+e)][2\alpha-s(1-e)]}{4\alpha^2}.
$$
Therefore $B<0$ exactly when
$$
\frac{2\alpha}{1+e^{-2\alpha}}<1+J<\frac{2\alpha}{1-e^{-2\alpha}}.
$$
A negative product means the two quadratic roots are real and of opposite sign: the <discriminant> is $A^2-4B>0$. The negative root gives a pair $\widetilde c=\pm i\sqrt{-y_-}$. One sign has positive <temporal growth rate> $kc_i$ and is unstable. Using the definitions of the hyperbolic functions gives \b[the required unstable band]:
$$
\boxed{\frac{\alpha e^\alpha}{\cosh\alpha}<1+J<\frac{\alpha e^\alpha}{\sinh\alpha}.}
$$
This proves the <unstable band of a three-layer stratified shear flow> directly, without needing to solve all four quartic roots.
For the wave interpretation, consider either interface in isolation with decaying <normal modes> $\widehat w=w_0e^{-k|z-z_i|}$. Put $S=\Delta U/h$, $U_i=\overline U(z_i)$, and $v=c-U_i$, the intrinsic <phase velocity>. At either interface $[\overline\rho]=-\Delta\rho/2$, while $[\overline U']=-S$ at the upper interface and $+S$ at the lower one. The <jump conditions for stratified inviscid shear flow> give the <gravity-vorticity interface wave> relation
$$
2kv^2-[\overline U']v-\frac{g\Delta\rho}{2\rho_0}=0.
$$
For stable density jumps, $J\geq0$, both intrinsic branches are real. In units $C=2c/\Delta U$, the isolated wave speeds are
$$
C_u=1+\frac{-1\pm\sqrt{1+8\alpha J}}{4\alpha},\qquad
C_l=-1+\frac{1\pm\sqrt{1+8\alpha J}}{4\alpha}.
$$
The upper wave travelling against its positive background current has $C_u=1-[1+\sqrt{1+8\alpha J}]/(4\alpha)$. The lower wave travelling against its negative background current has the opposite speed, $C_l=-C_u$. Their <counterpropagating wave resonance in a three-layer shear flow> occurs at $C_u=C_l=0$, which gives
$$
\boxed{1+J=2\alpha\quad\text{for the isolated-wave resonance}.}
$$
At large <wavenumber>, coupling across the separation $h$ is exponentially weak, of order $e^{-kh}=e^{-2\alpha}$. The unstable band becomes
$$
1+J=2\alpha+O(\alpha e^{-2\alpha}),
$$
with full width $4\alpha e^{-2\alpha}+O(\alpha e^{-6\alpha})$. It is a <counterpropagating wave instability>: the two waves propagate oppositely relative to their local currents but have almost equal laboratory <phase velocities>, allowing weak coupling to lock their phases and extract mean-flow energy. The <unstable band of a three-layer stratified shear flow> thus becomes narrowly concentrated around this resonance. For fixed $J$, arbitrarily large $k$ lies outside that band; large-$k$ resonance requires $J$ to grow with $\alpha$.
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