= Inviscid instability of a piecewise-linear mixing layer
{title2=$4k^2c^2=(2k-1)^2-e^{-4k}$}
For $U=-1$ below $z=-1$, $U=z$ between $z=-1$ and $z=1$, and $U=1$ above $z=1$, decaying <normal modes> solve $\phi''-k^2\phi=0$ in each region. <Pressure matching at a piecewise-linear shear interface> gives continuity of $\phi$ and $(U-c)\phi'-U'\phi$. 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 $1/(2e)$.
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