= 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 $U=\operatorname{sgn}\zeta$ 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.
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