Geography
Earthquakes: P and S Waves, Magnitude & 8 Solved Problems
An earthquake is a sudden release of energy along a fault: the focus is where it starts, magnitude is how big it is. Tables, scales and 8 solved problems.
Recommended for: Grade 6 · Grade 7 · Grade 8
An earthquake is a sudden shaking of the ground caused by rock breaking deep underground. The rocks of the crust are pushed slowly by moving tectonic plates and store energy like a spring being wound up. When the stress is bigger than the rock can take, the fault snaps, the two blocks slip past each other, and the stored energy is released within seconds as seismic waves.
The question that organizes the whole topic is: how do you describe an earthquake with numbers? Where it began (focus), how big it is (magnitude), how strongly a place felt it (intensity) and how far away it happened (the delay between waves).
Key terms
| Term | What it means |
|---|---|
| fault | a crack in the crust along which two blocks of rock slide |
| focus (hypocenter) | the point underground where the rock breaks: the earthquake starts here |
| epicenter | the point on the surface directly above the focus |
| seismic waves | vibrations that spread out from the focus through the Earth |
| seismograph | an instrument that records ground vibrations |
| seismogram | the graph drawn by the seismograph |
| foreshock | a smaller shake that comes before the main earthquake |
| aftershock | a weaker shake that follows the main earthquake, for days or months |
Seismic waves: P, S and surface waves
| Wave | How the ground moves | Typical speed | Arrives | Effect |
|---|---|---|---|---|
| P (primary) | pushes and pulls back and forth, like a spring | 5-7 km/s in the crust | first | a sharp jolt, less destructive |
| S (secondary) | shakes side to side or up and down | 3-4 km/s | after P | strong shaking, very damaging |
| Love | horizontal shaking at the surface | under 4 km/s | last | very destructive |
| Rayleigh | rolling motion, like ocean waves | under 4 km/s | last | very destructive |
P waves travel through solids, liquids and gases; S waves only through solids. That is how scientists learned that the Earth’s outer core is liquid: S waves cannot pass through it, as you can see in the structure of the Earth. The delay between P and S is also the tool for locating an earthquake, as problem 4 shows.
Why they happen: tectonic plates
Earthquakes are not scattered randomly: about 90% happen along plate boundaries, especially around the Pacific “Ring of Fire”, the same belt that holds most of the world’s volcanoes.
| Boundary | What happens | Example |
|---|---|---|
| convergent | one plate collides with or slides under another: the strongest and deepest earthquakes | Japan, Chile, Alaska |
| divergent | plates move apart: weak, shallow earthquakes | Mid-Atlantic Ridge, East African Rift |
| transform | plates slide sideways past each other | San Andreas Fault (California) |
Almost all earthquakes are tectonic. There are other kinds, much less common: volcanic (linked to moving magma), collapse (underground caves giving way) and induced by people (fluid injection, large reservoirs).
Depth of the focus
| Type | Depth | Notes |
|---|---|---|
| shallow | 0-70 km | about 3 out of 4 earthquakes, the most damaging |
| intermediate | 70-300 km | common at convergent boundaries |
| deep | 300-700 km | felt over wide areas but with modest damage |
Depth explains why two earthquakes of the same magnitude can have completely different effects. Problem 7 puts numbers on it.
Magnitude and intensity: two different scales
| Magnitude | Intensity | |
|---|---|---|
| what it measures | energy released at the source | effects at a place |
| values per earthquake | just one | many, one for each location |
| how it is found | seismograph | observing damage and what people felt |
| scales | Richter (ML), moment magnitude (Mw) | Modified Mercalli (MMI), EMS-98 |
| values | decimal numbers (6.0; 7.9) | Roman numerals, I-XII |
The magnitude scale is logarithmic: +1 means 10 times the amplitude and 31.6 times the energy. If you are comfortable with the laws of exponents, you see it at once: energy grows like 10^(1.5 M), and problems 5 and 6 do the arithmetic.
| Magnitude | Typical effects | How many per year worldwide |
|---|---|---|
| below 3 | almost never felt | hundreds of thousands |
| 3-3.9 | felt by few, no damage | about 130,000 |
| 4-4.9 | objects shake, damage rare | about 13,000 |
| 5-5.9 | damage to weak buildings | about 1,300 |
| 6-6.9 | destructive within about 100 km | about 130 |
| 7-7.9 | major, widespread damage | about 15 |
| 8 and above | catastrophic, hundreds of km | about 1 |
The largest earthquakes on record
| Earthquake | Moment magnitude | Notes |
|---|---|---|
| Chile, 1960 | 9.5 | the strongest ever measured |
| Alaska, 1964 | 9.2 | strongest in North America |
| Sumatra, 2004 | about 9.1 | Indian Ocean tsunami, around 230,000 deaths |
| Tōhoku, Japan, 2011 | 9.1 | tsunami and nuclear accident, about 20,000 dead or missing |
| Haiti, 2010 | 7.0 | shallow (about 13 km), more than 100,000 deaths by most estimates |
| San Francisco, 1906 | about 7.9 | fires after the quake; about 3,000 deaths |
Haiti is the key lesson: magnitude 7.0 is far below the giants above, yet it killed more people than most of them. A shallow focus, poorly built houses and a crowded capital explain why.
What to do during an earthquake
- During the shaking: Drop, Cover and Hold On. Get under a sturdy table or desk and hold on; stay away from windows, mirrors and tall furniture.
- If outdoors: move away from buildings, trees, power lines and anything that could fall.
- Afterward: expect aftershocks. Use stairs, not elevators, and follow local emergency instructions.
Common mistakes
- Mixing up focus and epicenter. The focus is underground; the epicenter is above it, on the surface.
- Mixing up magnitude and intensity. Magnitude is one per earthquake, intensity is one per place.
- Treating the scale as linear. Magnitude 8 is not twice magnitude 4: it is 10,000 times the amplitude and 1,000,000 times the energy.
- Calling every measurement ‘Richter’. Seismologists now mostly use moment magnitude (Mw), which is more reliable for large earthquakes.
- Thinking the earthquake starts at the epicenter. It starts at the focus; the epicenter is just its projection.
- Expecting predictions. Scientists calculate probabilities, not exact days.
- Believing deep earthquakes are the most dangerous. For the same magnitude, shallow ones shake the surface hardest.
The eight problems below go from easiest to hardest, and no two use the same method. The first two fix vocabulary and P and S wave arithmetic, the third uses the Pythagorean theorem to go from epicenter to focus, the fourth locates an epicenter from wave delay, the fifth and sixth work with the logarithmic scale and energy, the seventh explains why equal earthquakes cause different damage, and the eighth finds a tsunami’s speed using the same idea as uniform motion.
Solved exercises
1. Match each description to the correct term: (a) the point deep underground where the rock breaks; (b) the point on the surface directly above it; (c) the crack along which two blocks of rock slide past each other; (d) the instrument that records ground vibrations; (e) the waves that reach a station first. base
Show solution
- (a) The point deep underground where the rock breaks is the focus (also called the hypocenter): this is where the earthquake really starts.
- (b) The point on the surface directly above the focus is the epicenter, where the shaking is usually strongest.
- (c) The crack along which two blocks of rock slide is a fault.
- (d) The instrument that records ground vibrations is a seismograph, and the trace it draws is a seismogram.
- (e) The first waves to arrive are the P waves (primary waves), the fastest ones.
- Common mistake to avoid: the epicenter is not where the earthquake 'starts'. It is only the point on the surface above where it starts.
Answer: (a) focus (hypocenter); (b) epicenter; (c) fault; (d) seismograph; (e) P waves.
2. A seismic station is 120 km from the focus. Assume P waves travel at 6 km/s and S waves at 3 km/s. How long does each wave take to arrive, and how many seconds pass between the two? base
Show solution
- Use the uniform motion formula: time = distance ÷ speed.
- P waves: 120 km ÷ 6 km/s = 20 s.
- S waves: 120 km ÷ 3 km/s = 40 s.
- Gap between the two arrivals: 40 s - 20 s = 20 s.
- Check: the gap must be positive, because S waves are slower and arrive second.
Answer: P waves arrive after 20 s, S waves after 40 s, so there is a 20 s gap. That gap is exactly what lets us calculate how far away the earthquake was.
3. A seismograph sits 15 km from the epicenter, measured along the ground, and the focus is 8 km deep. How far is the station from the focus in a straight line? How long do P waves (6 km/s) take to reach it, and how much later do S waves (3 km/s) arrive? intermedio
Show solution
- The horizontal distance and the depth are the two legs of a right triangle. The straight-line distance is the hypotenuse.
- Pythagorean theorem: d² = 15² + 8² = 225 + 64 = 289.
- d = √289 = 17 km.
- P waves: 17 km ÷ 6 km/s = 2.83 s.
- S waves: 17 km ÷ 3 km/s = 5.67 s.
- S waves arrive 5.67 s - 2.83 s = 2.83 s after the P waves.
Answer: The station is 17 km from the focus (not 15 km). P waves arrive after about 2.8 s and S waves about 2.8 s later, at 5.7 s. Depth counts even when the epicenter looks close.
4. A seismogram shows that the S waves arrive 15 seconds after the P waves. With P waves at 6 km/s and S waves at 3 km/s, how far away is the focus? Why can't one station alone pinpoint the epicenter? intermedio
Show solution
- Call the distance d. P waves take d/6 seconds and S waves take d/3 seconds.
- The delay is d/3 - d/6 = d/6.
- So d/6 = 15 s, which gives d = 6 × 15 = 90 km.
- Check: 90 ÷ 3 = 30 s for S, 90 ÷ 6 = 15 s for P, a difference of 15 s. It works.
- One station only gives a distance, not a direction. The epicenter could be anywhere on a circle of radius 90 km around the station.
- Two stations give two circles that cross at two points; three stations give one point. This is called triangulation.
Answer: 90 km. You need at least three stations so that the three circles cross at a single point, the epicenter.
5. The 2011 Tōhoku earthquake had a moment magnitude of 9.1; the 2010 Haiti earthquake had 7.0. How many times larger was the wave amplitude in Japan, and how many times more energy was released? (Amplitude grows 10 times per magnitude unit; energy grows about 31.6 times.) intermedio
Show solution
- Difference in magnitude: 9.1 - 7.0 = 2.1.
- Amplitude: 10^2.1 = 125.9 times larger.
- Energy: 10^(1.5 × 2.1) = 10^3.15 = 1,413 times larger.
- Check with whole numbers: 2 units would give 31.6² = 1,000 times the energy; 2.1 units gives a bit more, and 1,413 fits.
Answer: Wave amplitude about 126 times larger, energy about 1,400 times larger. A gap of just two magnitude units means more than a thousand times the energy.
6. Use the formula log E = 4.8 + 1.5 M (E in joules) to find the energy released by the Haiti earthquake (M = 7.0). Then convert it to tons of TNT, knowing that 1 ton of TNT releases 4.184 × 10^9 J. intermedio
Show solution
- Put in M = 7.0: log E = 4.8 + 1.5 × 7.0 = 4.8 + 10.5 = 15.3.
- So E = 10^15.3 = 2.0 × 10^15 J (because 10^0.3 = 2.0).
- Convert: 2.0 × 10^15 ÷ 4.184 × 10^9 = about 477,000 tons of TNT.
- In kilotons: 477 kt, or about 0.48 megatons.
- For scale: that is roughly 30 times the energy of the Hiroshima bomb (about 15 kt), but spread over a huge volume of rock instead of a single point.
Answer: About 2.0 × 10^15 J, equal to roughly 477,000 tons (0.48 megatons) of TNT.
7. Two earthquakes have the same magnitude and the same epicenter. The first has its focus 12 km deep, the second 120 km deep. A town lies 5 km from the epicenter. Compare the distances to the two foci and say which shakes the town harder, assuming shaking is inversely proportional to distance. avanzato
Show solution
- First earthquake: d1 = √(5² + 12²) = √(25 + 144) = √169 = 13 km.
- Second earthquake: d2 = √(5² + 120²) = √(25 + 14,400) = √14,425 = 120.1 km.
- Ratio: 120.1 ÷ 13 = 9.2.
- If shaking is proportional to 1/d, the shallow earthquake shakes the town about 9 times harder.
- This is a simplification (waves also lose strength by absorption), but the point is clear: the same magnitude does not mean the same damage.
- That is why there are two scales: magnitude measures the earthquake (one number), intensity measures the effects at each place (it changes from town to town). The 2010 Haiti earthquake, at only about 13 km deep, was devastating for exactly this reason.
Answer: The shallow one, about 9 times stronger in the town. Magnitude is one number per earthquake; intensity depends on distance and depth, which is why very deep earthquakes rarely cause damage.
8. An earthquake on the floor of the Pacific, where the ocean is 4,000 m deep, causes a tsunami. Its speed is v = √(g h) with g = 9.8 m/s². Find v in m/s and km/h, the time to reach a coast 1,500 km away, and the speed over a continental shelf 40 m deep. avanzato
Show solution
- In the open ocean: v = √(9.8 × 4,000) = √39,200 = 198 m/s.
- In km/h: 198 × 3.6 = 713 km/h.
- Time for 1,500 km = 1,500,000 m: 1,500,000 ÷ 198 = 7,576 s, which is 126 minutes, or 2 hours and 6 minutes.
- Over 40 m of water: v = √(9.8 × 40) = √392 = 19.8 m/s = 71 km/h.
- Ratio of the speeds: 198 ÷ 19.8 = 10. The depth dropped by 100 times, but the speed only by 10 (a square root).
- The wave slows but its energy does not disappear: the water piles up and the wave grows taller. That is why a tsunami is almost invisible at sea and destructive at the coast.
Answer: 198 m/s (713 km/h) in the open ocean, reaching the coast in about 2 h 6 min; over the shelf 19.8 m/s (71 km/h). It moves like a jet plane in deep water, then slows and rises near the shore.
FAQ
What is the difference between the focus and the epicenter of an earthquake?
The focus (or hypocenter) is the point underground where the rock breaks and the earthquake begins. The epicenter is the point on the surface directly above it. The epicenter is usually where the damage is greatest, but the earthquake starts at the focus. The shallower the focus, the more destructive the shaking.
What is the difference between magnitude and intensity?
Magnitude measures the energy released at the source and is a single number for each earthquake (the Richter scale or, today, moment magnitude Mw). Intensity measures the effects felt and seen at a particular place, using scales like the Modified Mercalli Scale (I to XII) or EMS-98. One earthquake has one magnitude but many intensities, depending on distance, depth and the type of buildings.
How do you find the epicenter of an earthquake?
Measure the delay between the arrival of P waves and S waves on a seismogram: the farther the station, the longer the delay. With typical speeds of 6 and 3 km/s, distance is about 6 times the delay in seconds. One station gives a circle of possible epicenters; three stations give three circles that cross at a single point (triangulation).
How much stronger is a magnitude 7 earthquake than a magnitude 6?
The wave amplitude is 10 times larger and the energy released is about 31.6 times larger. The scale is logarithmic, so a magnitude 8 releases about 1,000 times the energy of a magnitude 6, because 31.6 × 31.6 = 1,000.
Can earthquakes be predicted?
No. Scientists cannot say exactly when and where the next earthquake will strike. They can map the most dangerous regions and estimate the probability of a strong earthquake over a number of years, which is the basis of building codes. Early warning systems exist that can send alerts seconds before the damaging S waves arrive.