Representation of the Elastic - Gravitational Excitation of a Spherical Earth Model by Generalized Spherical Harmonics

Abstract

The generalized spherical harmonics, which arise as representations of the rotation group, provide a natural basis for the expansion of tensors of any order in spherical co-ordinates. By using the covariant differentiation rules of Burridge, it is possible to obtain economically the separated differential equations of elastic vibration in a radially symmetric sphere. Derivation of the excitation of an Earth model by a point force or point dislocation is also carried out with the aid of these functions. While all the results obtained are known, the methods used have a substantially greater simplicity than conventional methods.

Seismic Excitation by Single Force and Asymmetric Moment Tensor.

“…(B4) where the strain and the rotation are evaluated at the source. The first term due to the symmetric moment tensor has been studied (Phinney and Burridge, 1973;Gilbert and Dziewonski, 1975). The second term is discussed in this appendix.…”

“…Following Phinney and Burridge (1973), the canonical components are defined by (19) The canonical components are expanded in terms of the generalized spherical harmonics of Phinney and Burridge (1973) (20) We may obtain the displacement outside the source radius on which the traction is given in Eq. (20) as a boundary condition.…”

Generalized spherical harmonic expansion of a solenoidal vector.

“…Formulae of the differentiation for higher order tensors are found in BURRIDGE (1969) and PHINNEY and BURRIDGE (1973).…”

Seismic wave propagation in fully anisotropic axisymmetric media

“…To evaluate the elements of the coupling matrix, it is useful to express the strain E and the perturbed elastic stiffness tensor c ijkl (r, θ, ϕ) in terms of generalized spherical harmonics Y N lm (Phinney & Burridge 1973;Dahlen & Tromp 1998, (C.164)):…”