Snell's law for seismic rays

Remarks

The common value in parts 2 and 3 is the ray parameter of the ray. Part 1 follows from the boundary conditions: displacement and traction must be continuous at every point of the interface and at every time. This forces all the waves to share the same frequency and the same apparent speed $v_k / \sin i_k$ along the interface. Parts 2 and 3 follow from part 1 applied to thin layers or shells. They also follow from the ray equation, since the travel time is invariant under horizontal translations, respectively rotations. Example: a P-wave in a crust with $\alpha_1 = 6$ km/s hits a mantle with $\alpha_2 = 8$ km/s. The transmitted P-wave grazes the interface when $\sin i_0 = 6/8$, that is $i_0 \approx 48.6°$; beyond this critical angle there is no transmitted P-ray, only a head wave along the interface. An incident P-wave also produces converted S-waves, which leave at a smaller angle because $\beta < \alpha$. Isotropy is needed: in an anisotropic medium the law holds for the slowness vectors, but the rays need not be parallel to them. It is the elastic form of the optical Snell's law, with $1/v$ in place of the refractive index.

Going deeper

For a student

In a layered or spherical Earth the conserved number $p$ labels each ray from source to receiver. The ray turns where $\sin i = 1$, that is where $v(z) = 1/p$, or $v(r) = r/p$ in a sphere. This gives the distance and travel time of each ray as integrals over depth, which is the basis of travel-time curves. The amplitudes of the reflected and converted waves are given by the Zoeppritz equations, which use the angles found here.