= Quartic dispersion relation for a three-layer stratified shear flow
{title2=$c^4+Pc^2+Q_0=0$}
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 $b=\alpha\operatorname{csch}(2\alpha)$. 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>.
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