 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 isFor 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.