= Energy identity for free-surface disk waves
For radial and vertical <Lagrangian fluid displacements> of a local axisymmetric mode, integrate the momentum equations and use <incompressible flow> plus the <free-surface pressure condition of an incompressible disk>. The result is
$$
\omega^2\int_{-H}^H(|\xi_r|^2+|\xi_z|^2)dz
=\kappa^2\int_{-H}^H|\xi_r|^2dz+\Omega_z^2H\bigl(|\xi_z(H)|^2+|\xi_z(-H)|^2\bigr).
$$
For nonnegative $\kappa^2$ and positive $\Omega_z^2$, every nontrivial oscillatory mode has real nonnegative $\omega^2$. The surface term is the gravitational restoring energy of the displaced vacuum interfaces.
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