Free Oscillations of the Laterally Inhomogeneous Earth: Quasi-degenerate Multiplet Coupling

The scheme leading to the computation of the normal mode eigenfrequencies of the laterally inhomogeneous Earth is derived. Rayleigh's variational principle is used to calculate the first-order corrections to the eigenfrequencies of a spherically symmetric, non-rotating and elastic Earth model due to the lateral variations in the real Earth. The Wigner-Eckart theorem is used, whenever applicable, to simplify many complicated angular integrals involving the strain deviator tensor to more basic and manageable integrals involving three spherical harmonics. In particular, near-resonance multiplet coupling for spheroidal-spheroidal, spheroidal-toroidal and toroidal-toroidal multiplets is presented along with the uncoupled case of a single arbitrary spheroidal or toroidal multiplet.