Seismic surface-wave dispersion

Seismic surface-wave dispersion is the frequency dependence of Rayleigh wave or Love wave propagation caused by depth-dependent elastic properties, mode structure and possibly intrinsic anelasticity. Phase velocity is $\omega /k$; group velocity is $d\omega /dk$. Different periods and branches have different depth sensitivities, enabling surface-wave tomography.

For surface phase speed $c$ on a spherical Earth, the conserved angular ray parameter is $p=R_E/c$. The local horizontal slowness at radius $r$ is $p/r$, giving squared radial shear wavenumber $q_r^2={\omega }^2[\beta (r{)}^{-2}-p^2/r^2]$ in ray theory. The displayed inequality makes it negative. A jump in shear speed can turn a propagating wave above the interface into an evanescent wave below it. This is a local condition, not proof of evanescence at every greater depth.