Seismic Waves: P-Waves, S-Waves and the Shadow Zone

CT-scanning a planet with sound

Nobody has ever drilled more than about 12 kilometres into Earth, barely scratching the 6,371 km radius, yet we can describe the composition, density and physical state of every layer down to the very centre. The tool is seismology: earthquakes release elastic waves that travel through the whole planet, refracting and reflecting at every compositional boundary along the way, and a global network of seismographs times their arrivals precisely enough to reconstruct the interior — essentially CT-scanning the Earth using its own tremors as the imaging pulse.

P-waves, S-waves and the race to arrive first

P-waves (primary, compressional) push particles back and forth parallel to their direction of travel, exactly like sound in air, and they can pass through solids and liquids alike. S-waves (secondary, shear) move particles perpendicular to the direction of travel — a motion that only a solid, with nonzero shear modulus, can support. Because P-waves travel faster, they always arrive at a seismograph first, followed some seconds or minutes later by the S-wave; that lag, multiplied by roughly 8 km/s, gives a rough distance estimate, and three or more stations let you triangulate an epicenter exactly.

v_P = sqrt((K + 4/3*mu) / rho)     // K: bulk modulus, mu: shear modulus, rho: density
v_S = sqrt(mu / rho)               // mu = 0 in any fluid -> v_S = 0

// Upper mantle:  v_P ~ 8.0 km/s     v_S ~ 4.5 km/s
// Outer core:    v_P ~ 8-10.3 km/s  v_S = 0   (liquid iron, mu = 0)
// Inner core:    v_P ~ 11.2 km/s    v_S ~ 3.5 km/s  (solid again)

A third family, surface waves, is confined to the crust and does the actual shaking damage in an earthquake: Love waves whip side to side and travel fastest, while Rayleigh waves move the ground in a rolling, elliptical path at roughly 90% of S-wave speed — the sensation most people describe as "the ground rolling like water."

Snell's law, underground

Wherever wave speed changes — at a compositional boundary, or gradually with the rising pressure of depth — seismic rays bend according to exactly the same law that bends light at a glass-air interface:

sin(i1) / v1  =  sin(i2) / v2  =  p     // p: ray parameter, constant along the ray

// v2 > v1  ->  ray bends away from the normal  (i2 > i1)
// v2 < v1  ->  ray bends toward the normal      (i2 < i1)

Because wave speed generally rises with depth through the mantle, rays continuously curve back upward rather than travelling in straight lines, tracing broad arcs that surface again far from the source — the geometric basis for locating distant earthquakes purely from travel times. At sharp boundaries some energy also reflects and some converts between P and S modes entirely, generating a rich alphabet soup of seismic phases (pP, sS, ScS, PKIKP…) that each independently constrain a different layer's depth and properties.

The shadow zone: a hole shaped like a liquid core

Between roughly 103° and 142° of angular distance from any earthquake, direct P-waves essentially vanish, and beyond 103° S-waves disappear entirely and never come back. This double signature is exactly what a liquid outer core produces: P-wave speed collapses from about 13.7 km/s in the lowermost mantle to about 8 km/s crossing into the core, bending rays so sharply toward the vertical that no direct path threads through the 103°–142° window, while S-waves — unable to shear a fluid at all — simply stop propagating the instant they reach the core-mantle boundary.

// Critical angle for P at the core-mantle boundary:
sin(i_c) = v_mantle / v_core = 13.7 / 8.0  ≈  0.584
i_c ≈ 35.7°
// Shallower incidence -> total reflection; steeper -> refraction into the core

Richard Oldham first identified the missing S-waves in 1906 and correctly inferred a liquid core; Inge Lehmann later found faint arrivals inside the shadow zone (the PKIKP phase) that could only be explained by a small, denser, solid inner core nested within the liquid outer core — its high wave speed focuses energy right back into the region the outer core alone would leave silent. Detecting a finite S-wave speed inside that inner core was the confirmation that it, unlike its surroundings, is solid iron.

From wiggly lines to magnitude numbers

Modern seismographs are broadband velocity sensors or MEMS accelerometers recording three components of ground motion — two horizontal and one vertical — with nanometre sensitivity. The old Richter scale measured the raw amplitude on a specific instrument at a fixed distance and breaks down for large or distant events; the modern moment magnitude scale instead derives directly from the physical size of the rupture:

M0 = mu * A * D             // shear modulus x rupture area x average slip
Mw = (2/3) * log10(M0) - 6.07

// 2011 Tohoku earthquake: M0 ~ 3.9e22 N*m  ->  Mw = 9.1
// each whole step in Mw ~ 32x more energy released

The Gutenberg-Richter law, log₁₀(N) = a − b·M with b ≈ 1, describes how earthquake frequency falls off with size: for roughly every magnitude-8 event, there are about 10 magnitude-7s, 100 magnitude-6s and 1,000 magnitude-5s, adding up to over a million detectable earthquakes worldwide every year.

Frequently asked questions

Try it live

Every wave front, refraction and shadow-zone geometry above runs live in Seismic Wave Propagation — P-waves, S-waves & Shadow Zone. Trigger an earthquake, watch the rays bend through the layers, and see the shadow zone appear in real time.