= Unbounded piecewise-linear shear layer
{title2=$\frac{c^2}{U_0^2}=\frac{(2kd-1)^2-e^{-4kd}}{4k^2d^2}$}
A uniform-density inviscid <shear flow> with $U=U_0z/d$ for $|z|<d$ and $U=\pm U_0$ outside has two <vorticity> jumps but no walls. Decaying <normal modes> satisfy $w''-k^2w=0$ in each region. Continuity of $w$ and <pressure matching at a piecewise-linear shear interface> gives a two-by-two determinant whose diagonal entries are $1-2kd\pm2kd\,c/U_0$ and off-diagonal entries $e^{-2kd}$. Its vanishing gives the displayed <dispersion relation>. The <Kelvin-Helmholtz instability> band is $0<kd<\alpha_s$, where $2\alpha_s-1=e^{-2\alpha_s}$ has its unique positive root above $1/2$, approximately $0.63923227$. Short waves decouple the two edges and remain neutral; long waves approach the <vortex sheet> instability with $c=\pm iU_0$.
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