 Solution

For stable density jumps, $J>0$ and $q=J/(2\alpha )>0$. Let $y=C^2$. The two roots of the dispersion relation for two density interfaces in uniform shear areThey are real, and $y_{+}>0$. An exponentially growing normal mode exists precisely when $y_{-}<0$, giving a pair of imaginary phase speeds; the member with positive imaginary part grows in $e^{ik(x-ct)}$ for $k>0$.

Since $y_{+}{y}_{-}=(1-q{)}^2-q^2{E}^2$, this condition is $|1-q|<qE$. Solving both inequalities yieldsAt either endpoint $y_{-}=0$, so the exponential growth rate vanishes. Outside the open band both squared phase speeds are nonnegative and the interfacial modes have real frequencies. This is a modal instability test for the stably stratified configuration; unstable density inversions would require a separate analysis. Within the band the dimensional growth rate is $k\Delta U\sqrt{-y_{-}}/2$.