The shudders that let us x-ray the Earth. Two body waves race outward from every quake: P (compressional) runs ahead, S (shear) lags — and their arrival gap gives the distance to the source. Because shear cannot cross a liquid, S carves a shadow past ~103° — the evidence that the outer core is molten. Down the center, data flows: elastic constants go in, the wave engine runs, the shadow comes out. Blue builds it; red tries to break it.
source R. D. Oldham, "The Constitution of the Interior of the Earth, as Revealed by Earthquakes," Quart. J. Geol. Soc. London 62 (1906) 456–475 — DOI 10.1144/gsl.jgs.1906.062.01-04.21 (paywalled; scan at archive.org); Moho after A. Mohorovičić (1909). Rendered, not quoted.
A wave's speed is set by the medium's stiffness and density. Two kinds of stiffness give two waves:
P — compression, resisted by bulk K and shear G:
vP = √((K + 4G/3)/ρ).
S — shear only, resisted by G alone:
vS = √(G/ρ).
Since K > 0, the P term is always larger, so vP > vS — P arrives first. For a Poisson solid (K = 5G/3) the ratio is exactly √3 ≈ 1.732.
| term | restored by | in liquid |
|---|---|---|
| vP | K + shear | still runs |
| vS | shear G only | = 0 |
Oldham (1906) read a discontinuity in S arrivals near 120° and inferred a dense central core; the modern S-shadow edge sits near 103°. G = 0 in a liquid → vS = 0 → S cannot cross the outer core. That single fact weighed the core molten.
Every wave that arrives is then sized: log10 of its amplitude is the magnitude, measured on the-richter-scale. This sphere's output — an amplitude on a trace — is that sphere's input.
The blue team's live check: re-run the wave engine over every material and confirm vP > vS for all, that shear vanishes in a liquid, and that the shadow begins past 103°. If red tampers, this badge catches it.
Feed the engine three numbers about the rock (or fluid) at the path: the bulk modulus K (resistance to squeeze), the shear modulus G (resistance to twist), and the density ρ. A liquid is simply a medium with G = 0.
Pick a material below and a source distance. That is all the wave equations need — the speeds and the shadow follow.
The seismogram below is computed live from vP, vS and the distance — P lands, then S, then the gap is read back as a range.
Change the material or the distance — every speed, the arrival gap, and the shadow verdict are computed on the spot from the elastic moduli, never looked up.
What the machine proves: vP > vS always (P first); the S–P gap grows with distance and inverts to give the range; and a liquid (G=0) kills S entirely, so the outer core throws an S-wave shadow past ~103° — Oldham's molten core. The P-wave core shadow (103°–142°) is refraction, marked AMBER.
The blue witness (left) confirms these live; the red team (right) tries to make S beat P.
And vP/vS = √3 holds only for an ideal Poisson solid (ν = ¼). Real rocks scatter around it; fluids, partial melt, and cracks push vS down further. The engine reports the real ratio per material — √3 is the anchor, not a law of every rock.
"S-waves are slower, so they lose energy and stop in the core." Cut. S stops because a liquid has no shear stiffness (G=0 → vS=0), not because it is slow. Speed and existence are different claims.
"Oldham found the shadow at 103°." Cut. He inferred the core from a delay near ~120°; the crisp 103° S-shadow edge is later work. The engine uses the modern edge, labelled — not pretended to be his figure.
"P and S mean primary and secondary rock." Kept, corrected. They mean primary/secondary arrival — the same rock, two wave modes, P simply first.
The red team's move: rewrite the shear-wave law so vS uses the compressional stiffness and comes out faster than vP — S would arrive first and the distance method inverts. The blue witness (window 7) is watching.
Flip it and every material reports vS > vP; the witness recomputes, finds P is no longer first, and turns red. Nothing is faked — the attack is real and it is caught.