Instability threshold for a bounded piecewise-linear shear layer ($L_s/L<1/2$)

The bounded piecewise-linear shear layer has exponentially growing modes exactly when $h<1/2$. Its squared phase speed tends to $2h-1$ as $kL\to 0$ and to $1$ as $kL\to \infty$. For $h<1/2$, sufficiently long nonzero waves are therefore unstable. Conversely $J-XY>0$, and for $h\ge 1/2$, $q\ge \alpha (1-h)>Y$, so $H-Y>0$ for every positive wavenumber and $c^2>0$. This proves the threshold without assuming monotonicity of the dispersion curve.