When an earthquake strikes, seismographs around the world record it within minutes. The first question is where it was. In Earthquakes and Seismic Waves you saw P waves outrun S waves. That race is the key, and solving a formula for one letter turns it into a distance.
Why it works
Both waves leave the focus at the same instant and travel the same distance $d$, but at different speeds. The S wave arrives later by $\Delta t = \dfrac{d}{v_S} - \dfrac{d}{v_P}$. Factor out $d$ and solve: $d = \Delta t \cdot \dfrac{v_P v_S}{v_P - v_S}$. With typical crustal speeds $v_P \approx 6\,\text{km/s}$ and $v_S \approx 3.5\,\text{km/s}$, the fraction is $\dfrac{6 \times 3.5}{2.5} = 8.4$: each second of S−P gap means about 8.4 km of distance. Seismologists use travel-time curves built from thousands of earthquakes, because real speeds change with depth, but the idea is the same.
Key idea
The gap between P and S arrivals grows with distance: $d = \Delta t \cdot \dfrac{v_P v_S}{v_P - v_S}$, about 8.4 km per second of gap for crustal speeds. Once you know $d$, you can find when the quake began.
Worked examples
Worked example: Distance from the gap
At a station, S waves arrive 20 s after P waves. How far away was the earthquake ($v_P = 6$, $v_S = 3.5\,\text{km/s}$)?
- $d = 20 \times 8.4 = 168\,\text{km}$.
- Check: P takes $168 \div 6 = 28\,\text{s}$; S takes $168 \div 3.5 = 48\,\text{s}$; the gap is $20\,\text{s}$.
Answer: $168\,\text{km}$.
Worked example: When did it start?
For that station, the P wave arrived at 10:42:30. When did the earthquake begin?
- P travel time: $168 \div 6 = 28\,\text{s}$.
- Start: $10{:}42{:}30 - 28\,\text{s} = 10{:}42{:}02$.
Answer: 10:42:02.
The same race works with a thunderstorm: the flash arrives almost at once and the sound follows at about 343 m/s, so each second between flash and thunder means about a third of a kilometer.
Common mistake
It seems as if one arrival time should give the distance: distance = speed × time. But the time in that formula is travel time, and nobody knew when the earthquake began. Check by asking what you know: the arrival clock time, not the travel time. The gap between two arrivals needs no start time.