= Inviscid instability of a triangular jet
{title2=$U(y)=\max(1-|y|,0)$}
Decaying <normal modes> of this continuous piecewise-linear profile satisfy $\phi''-k^2\phi=0$ between the corners. <Pressure matching at a piecewise-linear shear interface> yields an odd <varicose mode of a planar jet> with real speed $c=(1-e^{-2k})/(2k)$ and an even <sinuous mode of a planar jet> whose speeds satisfy
$$
2k^2c^2+k(1-2k-e^{-2k})c-[1-k-(1+k)e^{-2k}]=0.
$$
Its reduced <discriminant> factors as $(2k-3-e^{-2k}-4e^{-k})(2k-3-e^{-2k}+4e^{-k})$. The second factor is positive for $k>0$, while the first is strictly increasing and crosses zero once. Thus exactly the long-wave even branch is exponentially unstable, for $0<k<k_c$, where $2k_c-3-e^{-2k_c}-4e^{-k_c}=0$.
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