Free oscillations of a slightly anisotropic earth

Summary The effect of anisotropy on the free oscillations of the Earth is examined under the assumptions that anisotropy is general, i.e. there are 21 independent components, and a small perturbation from an isotropic media. It is shown that the deviation of a multiplet frequency δω from an average spherically symmetric earth depends upon 15 functions of θ (co‐latitude) and φ (longitude) as opposed to three in the isotropic case shown previously by Woodhouse & Girnius. In the asymptotic limit, δω is an average of a local functional δωbar; local (θ, φ, ψ) along a great circle path which contains the source and the receiver, where δωbar; local (θ, φ, ψ) is a function of θ, φ and azimuth ψ. Also δωbar; local (θ, φ, ψ)is a function of the elastic constants beneath a location (θ, φ). The azimuth dependence of δωbar; local (θ, φ, ψ)is 1, cos 2ψ, sin2ψ, cos 4ψ and sin4ψ, which is the same as a flat‐layered case. It is shown that formulae equivalent with those of Smith & Dahlen are recovered by an earth flattening transformation. In the non‐asymptotic case, in which the angular order l is low, δω shows odd‐order azimuthal dependence, i.e. ψ and 3ψ dependence. There exist seven radial kernels in the spherical case as opposed to four in the flat‐layered case, although radial resolution remains similar because of similarity of the kernels.