Wave speeds from elastic moduli
For an isotropic solid with bulk modulus K, shear modulus mu and density rho: Vp = sqrt((K + 4mu/3)/rho) and Vs = sqrt(mu/rho). P waves compress along the direction of travel, S waves shear perpendicular to it. Vp/Vs = sqrt(K/mu + 4/3) is always above 1.15, and for a liquid (mu = 0) there is no S wave at all.
Rays in a layered medium
Each layer is homogeneous and flat. Snell's law keeps the ray parameter p = sin(i)/v constant. A ray leaving the source at depth z_s reaches a station at distance x after T = sum t_j / (v_j cos i_j) and x = sum t_j tan i_j. Three families are traced: the direct ray, reflected rays, and head (refracted) waves which travel along the top of a faster layer at its speed. The first arrival is the smallest time over all families. For two layers a head wave arrives at T = x/v2 + 2h sqrt(v2^2 - v1^2)/(v1 v2) and overtakes the direct wave beyond the crossover distance 2h sqrt((v2+v1)/(v2-v1)); beyond it the Moho head wave (Pn) is the first P arrival.
Wavefronts in the cut-away
The wavefronts on the cut face come from a grid solution of the eikonal equation |grad T| = 1/v(z) in (distance, depth) by fast sweeping. It is a first-order scheme, so it differs from the ray tracer by about a percent, and it is shown only as the picture; all numbers come from the ray tracer. The surface ripples use the ray-traced arrival times.
Surface (Rayleigh) waves
Rayleigh waves travel just below the free surface at vR = Vs (0.862 + 1.14 nu)/(1 + nu) (Viktorov approximation, nu = Poisson's ratio). Here Vs and nu are thickness-averaged over the top two layers; real surface waves are dispersive (long periods sample deeper, faster rock) and this model is not.
Locating an epicentre
S travels slower than P, so the gap t_S - t_P grows with distance. Inverting the ray-traced S - P curve gives a radius around each station; three circles meet at the epicentre (a least-squares fit with Gauss-Newton is used when picking noise makes them miss each other). In a uniform crust the rule is d = (t_S - t_P) / (1/Vs - 1/Vp); using a single speed pair in a layered crust biases the answer, which the "Uniform rule" option shows. Depth is not solved for here: the focus depth is taken as known, which in reality needs more stations.
Magnitude and energy
Radiated energy log10 E = 1.5 M + 4.8 (E in joules; Gutenberg-Richter) and moment M0 = 10^(1.5 M + 9.1) N m (Hanks-Kanamori). Amplitudes use a Wood-Anderson local-magnitude relation with a Southern-California distance term (Hutton and Boore); it is a regional calibration, not universal.
Shadow zones
In a spherical Earth with Vp(r) and Vs(r) the ray parameter p = r sin(i)/v is conserved. Rays that reach the core-mantle boundary at r = 3480 km are bent by the sudden velocity drop (P) or cannot enter at all (S, because the outer core is liquid, mu = 0). Between the last mantle-bounded ray and the first core-refracted ray no direct wave arrives: the shadow zone. This page uses a simplified radial velocity profile (linear pieces following the PREM trend) so its edges are approximate: real S shadow begins near 103 degrees and the P shadow is about 104 to 140 degrees. Reflected phases (PcP, ScS) are drawn dashed and are not counted as direct arrivals.
What is simplified
Flat layers, no attenuation, no anisotropy, no dispersion, a point source, and no amplitude physics beyond the ML relation. Layer values are typical textbook figures, not measurements of a specific region. The page contains no hazard or damage estimates.