  A numerical model is proposed to compute the eigenmodes and the forced
response of multilayered elastic spheres. The main idea is to describe
analytically the problem along the angular coordinates with spherical harmonics
and to discretize the radial direction with one-dimensional finite elements.
The proper test function must be carefully chosen so that both vector and
tensor spherical harmonics orthogonality relationships can be used. The
proposed approach yields a general one-dimensional formulation with a fully
analytical description of the angular behaviour, suitable for any interpolating
technique. A linear eigenvalue problem, simple and fast to solve, is then
obtained. The eigensolutions are the spheroidal and torsional modes. They are
favourably compared with literature results for a homogeneous sphere. The
eigensolutions are superposed to compute explicitly the forced response. The
latter is used to reconstruct the propagation of surfaces waves. In particular,
the collimation of a Rayleigh wave (non-diffracted surface wave propagating
with a quasi-constant width) excited by a line source in a homogeneous sphere
is recovered with the model. Based on the vibration eigenmodes, a modal
analysis shows that such a wave is a superposition of fundamental spheroidal
modes with a displacement confined at the equator of the sphere. These modes
are the so-called Rayleigh modes, of sectoral type and high polar wavenumbers.
When a thin viscoelastic coating is added to the sphere, the Rayleigh mode
behaviour is recovered in a limited frequency range, allowing the generation of
a collimating wave at the interface between the sphere and the coating.
