Free modes of layered oblate planets

The equations governing the motion of an elastic gravitating nonhomogeneous isotropic field are considered, and a general solution for the Laplace or Fourier transform of the displacement vector is given in spherical coordinates. The perturbation method is used to set the boundary conditions for an oblate layered spheroid and it is seen that, except when there is rotational symmetry in the boundary conditions, in the geometry of the field, and in the displacements, the two different sets of usually uncoupled modes do not separate, and generally there is coupling in the displacements associated with these two sets of modes. If the displacements have axial symmetry these modes degenerate into the usual torsional and normal modes. The theoretical study refers to a model consisting of an indefinite number of homogeneous layers having different density and different elastic parameters, separated by spheroids of different flattening; but the computations of the first-order perturbation of the torsional periods caused by the oblateness have been made for two planet models, one consisting of a homogeneous shell limited inside by a sphere and outside by a spheroid and one consisting of a solid homogeneous spheroid. Since the influence of the flattening on the lower periods of oscillation in these two cases is of the order of only a few parts in a thousand no further computations have been made for the higher-order periods or harmonics or for more sophisticated models. Finally, it is shown that the broadening of the spectral lines caused by the flattening of the earth could not seriously interfere with the measurement of the dissipation function from the width of the same lines.