Inviscid instability of a piecewise-linear mixing layer ($4k^2{c}^2=(2k-1{)}^2-e^{-4k}$)

For $U=-1$ below $z=-1$, $U=z$ between $z=-1$ and $z=1$, and $U=1$ above $z=1$, decaying normal modes solve ${\varphi }^{\prime \prime }-k^2\varphi =0$ in each region. Pressure matching at a piecewise-linear shear interface gives continuity of $\varphi$ and $(U-c){\varphi }^{\prime }-U^{\prime }\varphi$. The two corner conditions yield the displayed dispersion relation. At $k=1/2$ the wave speeds are $c=\pm i/e$, so one mode grows exponentially with rate $1/(2e)$.