The propagation of short elastic surface waves on a slowly rotating earth

The effects of slow rotation with angular velocity Ω on Love and Rayleigh waves with given horizontal wave vector k on a plane-layered, transversely isotropic half-space have been calculated to first order in Ω. The frequency of Love waves is unaffected by rotation, while the frequency of Rayleigh waves is increased by k⁻¹R(k)Ω · (z^ × k) where z^ is the unit outward normal to the boundary of the half-space and R(k) is a dimensionless function of k, the length of k. R(k) lies between −1 and 1, and vanishes identically for a homogeneous, isotropic halfspace with Poisson ratio $14$⁠. The vertical and longitudinal particle motions in a Love wave do not vanish and are neither in phase nor in quadrature with the transverse motion. The transverse particle motion in a Rayleigh wave does not vanish and is neither in phase nor in quadrature with the vertical and longitudinal motions. It has been shown that a group of short, multicomponent waves on an anisotropic curved surface behaves like a classical particle, whose Hamiltonian is generated by the local dispersion relation and the curvilinear coordinates used to describe the surface. This extension of Hamilton's eikonal theory, together with the plane dispersion relations just derived, has been applied to Love and Rayleigh waves of given wave number k on a radially stratified, transversely isotropic sphere of radius a rotating slowly with angular velocity Ω. Correct to first order in Ω and (ak)⁻¹, the trajectory of a Love wave group is unaffected by rotation, while the plane of the great circle trajectory of a Rayleigh wave group maintains its inclination to the axis of rotation and precesses about that axis with angular velocity (ak)⁻)R(k)Ω. Because of this precession, at a fixed seismograph the direction of arrival of 333 second mantle Rayleigh waves from a point impulsive source changes systematically by about 2R degrees after each circuit of the earth.