Axisymmetric inviscid pipe stability equation ($(U-c)[{\varphi }^{\prime \prime }-{\varphi }^{\prime }/r-k^2\varphi ]-(U^{\prime \prime }-U^{\prime }/r)\varphi =0$)

An axisymmetric Stokes streamfunction disturbance $\varphi (r)e^{ik(z-ct)}$ of an axial basic flow obeys the displayed ordinary differential equation. Regularity requires $\varphi =O(r^2)$ at the axis, and an impermeable wall requires $\varphi (a)=0$ for nonzero $k$. Multiplication by ${\varphi }^{\ast }/r$ and integration by parts shows that a growing real-wavenumber mode requires $(U^{\prime }/r{)}^{\prime }$ to change sign; if it vanishes identically, the positive energy integral rules out such a mode. The condition concerns this cylindrical vorticity gradient rather than $U^{\prime \prime }$ alone.

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 $c$.