Bounded piecewise-linear shear layer

Let $\zeta =z/L\in [-1,1]$, $h=L_s/L\in (0,1)$, and normalize velocity by its maximum magnitude. The profile is $U=\zeta /h$ for $|\zeta |<h$ and $U=\mathrm{sgn}\zeta$ outside. Each region has exponential vertical-velocity modes; the two walls and vorticity-jump matching for an inviscid shear flow determine their coupling. With $\alpha =kL$, $q=\alpha h$, $X=\tanh q$, $Y=\tanh [\alpha (1-h)]$, $H=q(1+XY)$ and $J=q(X+Y)$, elimination gives $c^2=(H-Y)(J-XY)/(HJ)$, with $c$ normalized by the velocity scale.

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.