Evanescent potential modes of inviscid Poiseuille flow ($\varphi =r{J}_1(j_{1,l}r/a),k=\pm i{j}_{1,l}/a$)

When $U^{\prime }/r$ is constant and $U\ne c$, the axisymmetric inviscid pipe stability equation reduces to ${\varphi }^{\prime \prime }-{\varphi }^{\prime }/r-k^2\varphi =0$. Axis regularity selects $r{I}_1(kr)$ 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 $c$.