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An inviscid uniform-density self-gravitating cylinder in vacuum, with no rotation or surface tension, has the displayed axisymmetric dispersion relation, where $\kappa =|k_z|R$. To derive it, let $W=\psi +\pi /\rho =ϵ{\Pi }_0{I}_0(kr)e^{i(kz-\omega t)}$. The linear kinematic boundary condition gives ${\Pi }_0={\omega }^2/[k{I}_0^{\prime }(\kappa )]$. The Lagrangian pressure perturbation vanishes on the moving free surface, giving $\pi (R)=2\pi G{\rho }^{2}R\eta$. Substitute the axisymmetric gravitational potential of a displaced cylindrical interface and use $I_0^{\prime }=I_1$ to obtain the relation. Its nonzero neutral root is ${\kappa }_c\simeq 1.0667$: wavelengths longer than $2\pi R/{\kappa }_c\simeq 5.89R$ have ${\omega }^2<0$ and grow. The fastest-growing mode has $\kappa \simeq 0.58$, so its wavelength is about $11R$. Negative ${\omega }^2$ must be interpreted through an imaginary frequency, not through a real oscillatory ansatz.

Write total pressure and gravitational potential asHere $\pi$ is the Eulerian perturbation of a fluid variable applied to pressure, while $\psi$ is the corresponding gravitational potential perturbation. They have units $\mathrm{Pa}$ and ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, respectively. The constant interior mass density is $\rho$.

The Euler equations areSubtract the hydrostatic equilibrium relation $\nabla p_0/\rho +\nabla {\Phi }_0=0$ and discard terms quadratic in the perturbation. Mass conservation gives incompressible flow, and the gravitational Poisson equation has no bulk density perturbation. Thus, strictly inside the fluid,The last equation is Laplace equation, not a statement that self-gravity has disappeared. The displaced surface carries a first-order surface mass density $\rho \eta$, where $\eta$ is its radial displacement, and this supplies the source and matching conditions for $\psi$ at the moving interface. The axisymmetric gravitational potential of a displaced cylindrical interface, expressed using modified Bessel functions, incorporates that boundary source.

It is useful to define $W=\psi +\pi /\rho$. Taking the divergence of the momentum equation gives ${\nabla }^{2}W=0$. For an axisymmetric Fourier mode with axial wavenumber $k>0$, regularity at $r=0$ selects radial dependence $I_0(kr)$, the Modified Bessel function of the first kind, since $K_0(kr)$ is singular there. In what follows $k=|k_z|$ can always be chosen nonnegative.