  A method is presented for solving elastodynamic problems in radially
inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such
that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system
${r,\theta,\phi}$. The time harmonic displacement field $\mathbf{u}(r,\theta
,\phi)$ is expanded in a separation of variables form with dependence on
$\theta,\phi$ described by vector spherical harmonics with $r$-dependent
amplitudes. It is proved that such separation of variables solution is
generally possible only if the spherical anisotropy is restricted to transverse
isotropy with the principal axis in the radial direction, in which case the
amplitudes are determined by a first-order ordinary differential system.
Restricted forms of the displacement field, such as $\mathbf{u}(r,\theta)$,
admit this type of separation of variables solutions for certain lower material
symmetries. These results extend the Stroh formalism of elastodynamics in
rectangular and cylindrical systems to spherical coordinates.
