 Quartic dispersion relation for a three-layer stratified shear flow

For the bounded-shear three-layer profile with density contrasts minus one, zero and plus one, the localized-mode matching determinant is $[a(1-c{)}^2-(1-c)-J][a(1+c{)}^2-(1+c)-J]-b^2(1-c^2{)}^2=0$, where $a=\alpha [1+\coth (2\alpha )]$ and $b=\alpha \mathrm{csch}(2\alpha )$. Dividing by $a^2-b^2=2\alpha a$ gives a quartic in $c$ and a quadratic in $c^2$. Its constant term is $\{[2\alpha -(1+J){]}^2-e^{-4\alpha }(1+J{)}^2\}/(4{\alpha }^2)$, whose negative sign proves the unstable band of a three-layer stratified shear flow.