= Evanescent potential modes of inviscid Poiseuille flow
{title2=$\phi=rJ_1(j_{1,l}r/a),\quad k=\pm i j_{1,l}/a$}
When $U'/r$ is constant and $U\ne c$, the <axisymmetric inviscid pipe stability equation> reduces to $\phi''-\phi'/r-k^2\phi=0$. Axis regularity selects $rI_1(kr)$ and impermeability gives $I_1(ka)=0$. These imaginary-wavenumber disturbances are spatially evanescent potential fields, not bounded whole-pipe temporal normal modes; their radial equation supplies no eigenvalue condition on $c$.
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