Use P-wave and S-wave velocities, reflection and refraction to probe the internal Earth and locate an earthquake

Learning goal: Describe P-waves, S-waves and surface waves and how each is recognized on a seismogram; explain how wave velocity, reflection and refraction depend on the material a seismic wave crosses; use travel-time curves to find the distance to an epicenter, locate it by triangulation and calculate the origin time; and explain what all of this reveals about the internal Earth.

Why seismic waves are the geologist's probe. The internal Earth cannot be sampled directly: the deepest borehole reaches only about 12 km, while the center lies about 6,371 km down. Almost everything known about the interior comes from seismic waves, the vibrations released when rock breaks along a fault or when a large explosion is set off. These waves spread through the whole planet, and thousands of seismometers record their arrival times and ground motions. Because the speed of a wave depends on the stiffness and density of the material it travels through, and because waves bend and bounce at boundaries between different materials, the record of arrivals at the surface is a record of what the waves passed through on the way. Physical geology reads that record quantitatively, which is the step beyond grade 8.

Body waves: P and S. Two kinds of wave travel through the body of Earth. A P-wave (primary, or pressure wave) is a compressional wave: particles move back and forth along the direction of travel, squeezing and stretching the rock. Every kind of matter resists compression, so P-waves travel through solids, liquids and gases. An S-wave (secondary, or shear wave) moves particles sideways, perpendicular to the direction of travel. Only a solid resists being sheared, so S-waves travel through solids only and stop at a liquid. In any given rock the P-wave is faster: in the continental crust P-waves travel at roughly 6 km/s and S-waves at roughly 3.5 km/s; in the upper mantle P-waves reach about 8 km/s. The ratio of P speed to S speed in ordinary rock is about 1.7. Because P arrives first and S second, the delay between them grows with distance, and that delay is the key to locating earthquakes.

Surface waves. When body waves reach the surface they generate surface waves that travel along the outside of Earth, more slowly than either body wave, with larger amplitudes that die away with depth. Rayleigh waves roll the ground in an ellipse; Love waves shake it side to side. They arrive last and, at most distances from a large shallow earthquake, cause most of the damage. On a typical three-component seismogram (one vertical channel and two horizontal channels) from a distant earthquake, the P-wave usually shows most strongly on the vertical channel as a small, sharp first arrival, the S-wave appears as a larger onset on the horizontal channels, and the surface waves are the long, high-amplitude train that follows.

Velocity, reflection and refraction. Wave speed rises with the stiffness of rock and falls with its density; in practice deeper rock is both stiffer and denser, and stiffness wins, so speed generally increases with depth. Two things happen when a wave meets a boundary where speed changes. Part of the wave reflects, bouncing back toward the surface; a reflected arrival on a seismogram shows that a boundary exists, and its travel time gives the boundary's depth once the wave speeds along the path are known. In the simplest case, a wave going straight down and back at a steady speed v, depth = v × time ÷ 2; real paths are slanted and speeds change with depth, so seismologists work the depth out with a velocity model. The rest of the wave refracts: crossing the boundary at an angle, it bends because one side of the wavefront speeds up or slows down before the other. Refraction toward the surface as speed increases with depth is why seismic ray paths through the mantle curve back upward, and why a wave that started downward can emerge thousands of kilometers away. Sharp refraction at the core boundary, where P-wave speed drops abruptly on entering the liquid outer core, bends P-waves inward so strongly that a ring of the surface between roughly 104 and 140 degrees from the earthquake receives no direct P-wave. That ring is the P-wave shadow zone, and the absence of direct S-waves beyond about 103 to 104 degrees, where they would have to cross the core, is the evidence that the outer core is liquid. (Some converted waves do arrive there: they cross the liquid core as P-waves and travel the mantle as S-waves, which is itself consistent with a liquid core.)

Reading distance from the S minus P interval. Since P and S start together but travel at different speeds, the time between their arrivals increases with distance from the source. A travel-time curve plots arrival time against distance for each wave type. A fully worked model: suppose (model values) P-waves travel at a uniform 6.0 km/s and S-waves at 3.5 km/s. Assume a shallow focus, so the wave path is about the same length as the distance along the surface. For a station 300 km from the epicenter, P takes 300 ÷ 6.0 = 50.0 s and S takes 300 ÷ 3.5 = 85.7 s, so S arrives 35.7 s after P. In general the interval is d × (1/3.5 − 1/6.0) = d × 0.119 s/km, so in this model every 11.9 s of S minus P interval means 100 km of distance. Reading the interval at a station therefore gives the distance, not the direction. A circle of that radius around the station contains the epicenter somewhere on its edge. Three stations give three circles that cross at one point: this is what seismologists call triangulation (because it uses distances rather than angles, a surveyor would call it trilateration). Two circles cross at two points, which is why a third station is needed. Real travel-time curves are not straight lines, because velocity increases with depth and the waves curve, but the method is identical: read the S minus P interval, find the distance on the curve.

Origin time. Once the distance is known, the P-wave travel time to that station follows, and subtracting it from the recorded P arrival gives the moment the earthquake happened. If a station 300 km away recorded P at 10:15:50, the earthquake began 50.0 s earlier, at 10:15:00.

Focus, epicenter, magnitude and intensity. The focus (hypocenter) is the point inside Earth where the rupture began; the epicenter is the point on the surface directly above it. Depth of focus matters: where earthquake foci get deeper in a band that slopes down under a continent, they trace a subducting plate, which is why foci depth data can be used to build a three-dimensional model of subduction. Magnitude measures the size of the earthquake at its source. The original Richter scale was calculated from the amplitude on a seismogram corrected for distance, and each whole step corresponds to ten times that amplitude; seismologists now usually report moment magnitude, calculated from the fault area, the amount of slip and the stiffness of the rock, and each whole step corresponds to roughly 32 times the energy released. One earthquake can be given slightly different magnitudes by different methods. Intensity (the Mercalli scale, rated from I to XII) describes the effects at a place, from barely felt to total destruction, and is assigned from damage reports and eyewitness accounts; it falls off with distance from the epicenter and depends on the ground, so one earthquake has one magnitude but many intensities.

Earthquakes at the surface. For earthquakes that happened before seismometers, magnitude has to be estimated from the size of the area over which each Mercalli intensity was reported, which is why historic magnitudes are quoted as estimates. The New Madrid earthquakes of 1811 and 1812 in the central Mississippi valley are the classic case: eyewitnesses on the river reported that the Mississippi flowed backward for a time where the ground was suddenly uplifted across its channel, and Reelfoot Lake in Tennessee formed as land subsided. Engineers design for shaking rather than trying to stop it: steel or reinforced-concrete frames that bend without breaking, diagonal cross-bracing and shear walls, base isolators that let a building ride on flexible bearings, and avoidance of loose, water-saturated ground that can liquefy.

The misconception. Students often say the S-wave is stopped by the core because the core is dense. Density alone does not stop a wave; P-waves cross the same dense core. What stops an S-wave is the absence of shear strength, which is the property of a liquid. The shadow zones are evidence about the state of the outer core, not simply its density.

What seismic data cannot tell directly. Wave speed constrains stiffness and density together, not composition on its own; chemistry has to be supplied by other evidence such as meteorites, high-pressure experiments and the rocks brought up by volcanoes. Nor do seismic waves predict earthquakes: they record ruptures that have already happened, and long-term hazard comes from the pattern of past events at plate boundaries and faults.

Worked velocity calculation: A P-wave and an S-wave from the same earthquake reach a station 240 km away. Model velocities: P at 6.0 km/s, S at 3.5 km/s. Travel times: P = 240 ÷ 6.0 = 40.0 s; S = 240 ÷ 3.5 = 68.6 s. S minus P = 28.6 s. Check with the model rule of 11.9 s per 100 km: 2.4 × 11.9 = 28.6 s. The step students skip is the unit sense: seconds are what the seismogram shows, so the calculation must end in seconds, and the interval, not either travel time alone, is what an observer can read, because the origin time is unknown until the distance is found.

Worked distance and origin time from a seismogram: A station records the P arrival at 14:02:36 and the S arrival at 14:03:24. The interval is 48 s. Distance = 48 ÷ 0.119 ≈ 403 km. P travel time = 403 ÷ 6.0 ≈ 67.2 s, so the origin time is 14:02:36 minus 67.2 s ≈ 14:01:29 (to the nearest second). Reasoning: the earthquake happened before the first wave arrived, so the origin time is earlier than the P arrival by exactly the P travel time.

Worked triangulation: Three stations report S minus P intervals of 23.8 s (station A), 35.7 s (station B) and 59.5 s (station C). Model distances: A = 23.8 ÷ 0.119 = 200 km; B = 300 km; C = 500 km. On a map at 1 cm = 100 km, draw circles of radius 2 cm, 3 cm and 5 cm around A, B and C. The three circles meet at a single point, the epicenter. If the circles nearly but not exactly meet, the small triangle between them reflects the fact that real velocities are not uniform and the focus has depth; the epicenter is taken inside that triangle.

Worked refraction reasoning: A P-wave leaves the focus traveling downward at an angle into rock where speed increases smoothly with depth. Each deeper part of the wavefront moves faster than the shallower part, so the front swings around, and the path curves back toward the surface. The wave surfaces far from the focus without ever having been reflected. Now contrast the core boundary: P-wave speed drops sharply, from about 13 km/s in the lower mantle to about 8 to 9 km/s in the liquid outer core, so the ray bends sharply toward the vertical on entering and away again on leaving; the net effect is that rays are gathered inward, leaving the shadow ring without direct P arrivals.

Representation task: Sketch a three-component seismogram (vertical, north-south, east-west) for a station 300 km from an earthquake using the model velocities. Mark the P arrival at 50.0 s, the S arrival at 85.7 s and the start of the surface waves after that; draw the typical pattern, with P sharpest on the vertical trace, S largest on the horizontal traces, and the surface-wave train the largest and longest. Then draw a cross-section of Earth showing one P-wave ray that refracts gently in the mantle and one that enters the core and is bent inward, and shade the shadow zone with its angular limits.

Safe investigation, wave types and triangulation: (a) With a long spring toy on the floor, send a push-pull pulse and sketch the coil motion (a P-wave model); send a side-to-side pulse and sketch it (an S-wave model). The spring shows the two directions of particle motion only; it cannot model a liquid, so the liquid outer core is argued from the seismic records, not from the spring. (b) On a printed map at 1 cm = 100 km, place three stations and, for a hidden epicenter chosen by a partner, have the partner compute each station's S minus P interval with the model rule. Convert each interval to a distance, draw the circles with a compass and find the epicenter. Record: the intervals, the distances, the radius drawn and the error between your point and the partner's. What the model cannot show: the spring shows particle motion, not the speeds of real P- and S-waves or why S-waves stop in a liquid, and the uniform-velocity map exercise cannot show the curved travel-time relation of a real Earth.

Independent performance: Write an explanation of at least 300 words titled "Reading the internal Earth from a seismogram." Success criteria: it defines P-waves, S-waves and surface waves by particle motion and speed; it explains what refraction and reflection are and what each tells a seismologist; it shows one full calculation of distance from an S minus P interval and one origin-time calculation using the model velocities; it explains why three stations are needed for triangulation; and it states one thing that seismic wave data cannot tell us on its own.