Seismic Wave Propagation - 2D Crust Cross-Section and Travel-Time Curves

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Speed

Live readouts

Graphs

Numerical checks

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How to use

Press a preset, then watch the fronts leave the red focus. Try: (1) Continental crust: move the station until the green head wave overtakes the gold direct wave - that is the crossover distance; (2) Weak contrast: the critical angle approaches 90 degrees and the head wave barely helps; (3) Surface blast: focal depth 0 gives the textbook formula for the crossover; (4) turn S waves on and read the distance from the S-P time. Space pauses, R restarts.

The science

Waves in an elastic solid. P waves are compressional and S waves are shear waves:

Vp = sqrt((K + 4/3 mu)/rho), Vs = sqrt(mu/rho)

so S is always slower (Vp/Vs about 1.73 for a Poisson solid) and cannot travel in a liquid.

Layer over half-space. With crust speed V1 below mantle speed V2 the critical angle is sin(theta_c) = V1/V2 (Snell's law). For focal depth hs and layer thickness h:

t_direct = sqrt(x^2+hs^2)/V1, t_refl = sqrt(x^2+(2h-hs)^2)/V1t_head = x/V2 + (2h-hs) cos(theta_c)/V1, valid for x > (2h-hs) tan(theta_c)

For hs = 0 the head wave overtakes the direct wave at the crossover distance xc = 2h sqrt((V2+V1)/(V2-V1)). The slope of the head-wave branch gives 1/V2 and its intercept gives the crustal thickness - this is how Mohorovicic found the Moho.

S-P method. The lag S-P = r (1/Vs - 1/Vp) is proportional to the distance r, so one station gives r and three stations give the epicentre.

Wavefield view. The coloured picture is a finite-difference solution of the scalar wave equation u_tt = c^2 (u_xx + u_zz) on a 200 x 112 grid (dx = 0.5 km) with a free surface on top, a mirror wall at x = 0 and absorbing sponge layers on the other sides. Stability needs the Courant number V_max dt/dx to stay below 1/sqrt(2); the page picks dt at 0.9 of that limit and prints it. The overlay rays and the ray-theory pulses in the seismogram come from the formulas above, so the cyan finite-difference trace can be compared with them directly.

Reflection coefficient. At the Moho a plane P wave arriving at angle theta1 is reflected with R = (Z2 cos theta1 - Z1 cos theta2)/(Z2 cos theta1 + Z1 cos theta2), where Z = rho V is the acoustic impedance and Snell's law sin theta2 / V2 = sin theta1 / V1 gives theta2. Densities come from Gardner's relation rho = 310 V^0.25 (V in m/s). Beyond the critical angle the transmitted wave becomes evanescent and |R| = 1. This fluid-fluid formula ignores the converted S waves of the full Zoeppritz equations.

Checks. (1) bisection of t_head = t_direct against the closed-form crossover; (2) reflection and head wave meet exactly at the critical distance; (3) the Courant number of the finite-difference grid is below 0.707; (4) the finite-difference move-out between two stations matches the straight ray in uniform rock on a coarse and a fine grid; (5) R^2 plus the transmitted energy equals 1, R(0) = (Z2-Z1)/(Z2+Z1) and |R| = 1 past the critical angle; (6) a Poisson solid gives Vp/Vs = sqrt(3) and granite Vs = 3.33 km/s; (7) distance recovered from S-P.

Simplifications. Two homogeneous layers and a flat Moho, constant density in the finite-difference model (scalar waves, no S waves or surface waves), acoustic reflection coefficients instead of the full Zoeppritz equations, schematic Ricker pulse amplitudes in the ray seismogram, no attenuation, and a 2D cross-section of what is a 3D spreading wavefield.