 Unstable band of a three-layer stratified shear flow

For the three-layer stratified piecewise-linear shear flow, set $\alpha =kh/2$. The dispersion relation is quadratic in squared nondimensional phase velocity. Its constant term is $[2\alpha -(1+J)(1+e^{-2\alpha })][2\alpha -(1+J)(1-e^{-2\alpha })]/(4{\alpha }^2)$. Throughout the displayed band this is negative, so one squared speed is negative and gives an exponentially growing normal mode.