Because the tidal potential (14) superposes many different spherical harmonics, the separation of variables (7) applied to the oscillation equations must be replaced by the more-general expressions
(the notation for the sums has been abbreviated). Substituting these solution forms into the linearized equations, and taking advantage of the orthonormality of the spherical harmonics, leads to a set of differential equations for each combination of \(\ell\), \(m\) and \(k\). A given set resembles the corresponding oscillation equations, with just a couple changes:
Rather than being an eigenvalue parameter, the oscillation frequency is set by \(\sigma = k \Oorb\), representing the forcing frequency of the partial tidal potential in an inertial frame.
The perturbation \(\tPhi'\) is replaced by \(\tPsi' \equiv \tPhi' + \tPhiT\), representing the radial part of the total (self + tidal) gravitational potential perturbation.