Seismic Waves — How Earthquakes See Inside the Earth

Every earthquake sends mechanical waves radiating outward through the planet. These seismic waves refract, reflect, and convert at every compositional boundary — and by recording their arrival times at seismograph stations worldwide, Earth scientists have mapped the planet's interior in extraordinary detail, from the Mohorovičić discontinuity to the inner-core boundary.

1. P-waves, S-waves and Surface Waves

Seismic waves are elastic waves: they temporarily deform rock and restore it without permanent damage. They fall into two broad families — body waves that travel through the interior, and surface waves confined near the crust.

P-waves (Primary / Compressional)

Particle motion is parallel to the direction of propagation — the rock compresses and dilates like a sound wave. P-waves travel through solids and liquids (including Earth's outer core), and arrive first at seismograph stations.

v_P = \sqrt((K + 4/3 μ) / ρ) K = bulk modulus (resistance to compression) μ = shear modulus (resistance to shear, = 0 in fluids) ρ = density Upper mantle: v_P \approx 8.0 km/s Outer core: v_P \approx 8.0-10.3 km/s (liquid iron, μ = 0) Inner core: v_P \approx 11.2 km/s

S-waves (Secondary / Shear)

Particle motion is perpendicular to propagation. S-waves require a solid medium (shear modulus μ > 0) — they cannot propagate through the liquid outer core. This observation was the key evidence that the outer core is liquid.

v_S = \sqrt(μ / ρ) Upper mantle: v_S \approx 4.5 km/s Outer core: v_S = 0 (no shear transmission — liquid) Inner core: v_S \approx 3.5 km/s (solid iron)

Surface Waves

Love waves (L): horizontal shear motion transverse to propagation direction; fastest surface wave; destructive in earthquakes.
Rayleigh waves (R): elliptical retrograde motion in the vertical plane; responsible for the rolling sensation felt during earthquakes — travel at ~90% of S-wave speed.

Seismic arrival order: P-waves arrive first, then S-waves, then surface waves. The time difference (S–P delay) × 8 km/s gives rough epicenter distance. Three stations with different delays allow triangulation of the earthquake source.

2. Wave Speeds and Earth's Layers

Earth's interior is divided into the crust (5–70 km thick), mantle (70–2 890 km), outer core (2 890–5 150 km) and inner core (5 150–6 370 km). Wave speeds increase with depth in the mantle due to rising pressure (which stiffens rock) and show jumps at major compositional boundaries.

The PREM (Preliminary Reference Earth Model) is the standard 1-D velocity profile against which all seismological data is compared. Modern tomographic models perturb PREM to create 3-D images of thermal and compositional anomalies — essentially CT-scanning the planet.

Layer Depth (km) v_P (km/s) v_S (km/s) ρ (g/cm³) ────────────────────────────────────────────────────────────── Upper crust 0-20 5.8-6.5 3.2-3.7 2.7 Lower crust 20-35 6.5-7.1 3.7-3.9 2.9 Upper mantle 35-400 8.0-9.0 4.5-4.9 3.3 Transition zone 400-660 9.0-10.6 4.9-5.6 3.7-4.4 Lower mantle 660-2890 11.0-13.7 6.2-7.3 4.4-5.6 Outer core 2890-5150 8.0-10.3 — 9.9-12.2 Inner core 5150-6370 11.0-11.3 3.5 12.8-13.1

3. Snell's Law and Wave Refraction

At any boundary between materials of different wave speeds, seismic waves obey Snell's law — exactly as light does at an air-glass interface:

sin(i₁) / v₁ = sin(i₂) / v₂ = p (ray parameter, constant along ray) Where i₁, i₂ are angles of incidence and refraction from the normal. If v₂ > v₁: the ray bends away from normal → i₂ > i₁ If v₂ < v₁: the ray bends toward normal → i₂ < i₁

Because wave speed generally increases with depth in the mantle, rays continuously bend upward — following curved paths through the interior and emerging at the surface far from the earthquake source. This is the basis of the ray parameter method for locating earthquakes.

At boundaries, some energy is also reflected (following the law of reflection) and some energy converts between P and S modes (P–S conversion). This mode conversion creates a rich set of seismic phases (pP, sS, ScS, SS, PKIKP…) that provide independent information about layer depths and properties.

4. The Seismic Shadow Zone

Of all the discoveries seismology has produced, the shadow zone stands out. Between about 103° and 143° angular distance from any earthquake epicenter, P-wave arrivals are significantly weakened. Beyond 143°, P-waves resume, but S-waves are absent.

This pattern is explained by the liquid outer core:

P-waves entering the core refract strongly (the speed drops from ~13.7 km/s in the lower mantle to ~8 km/s at the core-mantle boundary, bending rays toward the vertical).
This strong inward bending creates a zone where no direct P-waves emerge between 103°–143°.
S-waves cannot pass through the liquid outer core at all → no S-waves beyond 103°.

Critical angle for P at CMB: sin(i_c) = v_mantle / v_core = 13.7 / 8.0 ≈ 0.584 i_c ≈ 35.7° Rays entering at shallower angles undergo total reflection, rays entering more steeply refract into the core — this geometry produces the shadow zone angular range.

PKIKP phase: P-waves that traverse the inner core (P–outer core–inner core–outer core–P) arrive in the shadow zone because the inner core is solid and its high wave speed focuses energy there. The inner core's solidity was proven by identifying finite v_S in inner core — the first observation of seismic waves converting to S inside a metallic core.

5. Major Discontinuities

Mohorovičić discontinuity (Moho): 5–70 km depth; marks the base of the crust. P-wave speed jumps from ~7 km/s (lower crust) to ~8 km/s (upper mantle). Discovered by Andrija Mohorovičić in 1909 when he noticed two P-wave arrivals from a Croatian earthquake — direct waves and waves refracted along the Moho.
410 km discontinuity: olivine transforms to wadsleyite (a denser polymorph), increasing density and wave speed. A global seismic reflector visible in deep earthquakes.
660 km discontinuity: wadsleyite to bridgmanite + ferropericlase. The deepest limit of the upper mantle transition zone. Debated as a partial barrier to whole-mantle convection.
Core-Mantle Boundary (CMB), 2 890 km: dramatic compositional jump from silicate mantle to liquid iron-nickel core. Produces strong reflections (ScS, PcP phases). Above the CMB, the ultra-low velocity zone (ULVZ) possibly contains partial melt.
Inner Core Boundary (ICB), 5 150 km: liquid to solid iron transition. The inner core grows by ~1 mm/yr as the planet slowly cools.

6. Seismographs and Ground Motion

A classic seismograph uses an inertial mass suspended so it remains stationary while the ground moves around it. Modern instruments are broadband velocity seismometers or MEMS accelerometers:

Broadband seismometer: records ground velocity from 0.001 Hz (normal modes) to 30 Hz (local seismicity). Sensitivity: nanometre displacement. Used in the Global Seismographic Network (GSN).
Strong-motion accelerometer: records ground acceleration unclipped up to 2g. Used in earthquake engineering to determine shaking experienced by buildings.

Ground motion is recorded in three components: two horizontal (N-S and E-W) and one vertical. P-waves produce mainly vertical motion; S-waves and Love waves produce mainly horizontal.

7. Measuring Earthquake Size

Richter Local Magnitude (M_L)

Defined by Charles Richter in 1935 as the logarithm of the maximum amplitude recorded on a standard Wood-Anderson seismometer at 100 km distance. Each integer step represents a 10× amplitude increase and ~31.6× energy increase. Only valid for small-to-moderate earthquakes within ~600 km.

Moment Magnitude (M_w)

The modern standard, defined in terms of seismic moment M₀ — proportional to the fault area × slip × shear modulus:

M₀ = μ \cdot A \cdot D (N\cdotm) μ = shear modulus of rock (~3 \times 10¹⁰ Pa) A = rupture area (m²) D = average slip (m) M_w = (2/3) log₁₀(M₀) - 6.07 2011 Tōhoku earthquake: M₀ \approx 3.9 \times 10²² N\cdotm \to M_w = 9.1 Fault area \approx 450 km \times 150 km, mean slip \approx 20 m

Energy Release

E_s \approx M₀ / 20 000 (seismic energy \approx 5 \times 10⁻⁵ \times M₀) M_w 9.0 \to E_s \approx 2 \times 10¹⁸ J — equivalent to ~500 gigatons TNT M_w 6.0 \to E_s \approx 2 \times 10¹³ J — equivalent to Hiroshima atomic bomb

Magnitude–frequency relation (Gutenberg-Richter law): log₁₀(N) = a − b·M, where N is the number of earthquakes ≥ M per year and b ≈ 1. For every M 8 earthquake, there are roughly 10 M 7s, 100 M 6s, and 1000 M 5s — globally, about 1.3 million earthquakes per year are detectable.

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