Seismic Rays Through the Earth

Rays in a sphere. Seismic waves travel on rays that bend wherever the wave speed changes. In a spherically symmetric Earth Snell's law takes the form p = r sin(i) / v(r) = constant along a ray (the ray parameter p). A ray turns where r / v(r) falls to p. Since speed generally increases with depth, rays curve back up and surface at an epicentral distance Delta(p) = 2 integral of p dr / (r sqrt((r/v)^2 - p^2)), after a time T(p) = 2 integral of (r/v)^2 dr / (r sqrt((r/v)^2 - p^2)). The sim evaluates these integrals numerically for each family of rays and finds every ray that reaches the seismometer.

The core. At the core-mantle boundary (3480 km) P waves slow down from about 13.7 to 8.1 km/s and bend sharply towards the vertical, which leaves a P shadow zone and a bright re-emergence beyond it (PKP). S waves cannot travel in the liquid outer core at all, so there is no direct S beyond the shadow edge; S can still cross the core as a P wave (SKS). That is how the liquid core was discovered, and the faint inner-core arrivals revealed the solid inner core.

What is simplified. The velocity profiles are piecewise linear and PREM-like, with no crust, no 410 and 670 km jumps, no inner-core S wave, no anisotropy, no attenuation and no lateral variation; the source is at the surface and amplitudes are not computed (the seismogram shows one equal wavelet per arrival). Travel times therefore differ from real reference tables, by tens of seconds at long range. The Earth is drawn cut open so the ray plane is visible.

Numerical checks (run on load): the ray integrals against the straight chord in a homogeneous sphere; the Herglotz relation dT/dDelta = p on the computed curves of P, S, PKP and SKS; the shadow edge of a core in a constant-velocity mantle against 2 acos(r_core/R); and the radii of the Blender model against the radii used in the physics, with the drawn ray ending at the seismometer.

• Press Shadow zone (120 deg): no direct P or S arrives, only waves that went through the core; the red and purple arcs on the rim mark the shadows.
• Press PKP region (150 deg): the liquid core bends P rays so strongly that a new arrival appears beyond the shadow.
• Look at the travel-time graph: the slope of each curve at the station distance equals the ray parameter p (Herglotz relation).