 Unbounded piecewise-linear shear layer

A uniform-density inviscid shear flow with $U=U_{0}z/d$ for $|z|<d$ and $U=\pm U_0$ outside has two vorticity jumps but no walls. Decaying normal modes satisfy $w^{\prime \prime }-k^{2}w=0$ 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 $1-2kd\pm 2kdc/U_0$ 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 $c=\pm i{U}_0$.