= Axisymmetric inviscid pipe stability equation
{title2=$(U-c)[\phi''-\phi'/r-k^2\phi]-(U''-U'/r)\phi=0$}
An axisymmetric <Stokes streamfunction> disturbance $\phi(r)e^{ik(z-ct)}$ of an axial basic flow obeys the displayed <ordinary differential equation>. Regularity requires $\phi=O(r^2)$ at the axis, and an impermeable wall requires $\phi(a)=0$ for nonzero $k$. Multiplication by $\phi^*/r$ and <integration by parts> shows that a growing real-wavenumber mode requires $(U'/r)'$ 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 $U''$ alone.
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