Offshore Earthquake: Reading the Waves and Making the Call
A coastal town of about 4 000 people sits on a low-lying spit roughly 2 m above sea level, with a single narrow road to high ground 3 km inland. At 14:32 a town seismograph records the first P-wave arrival from an offshore earthquake; the S-wave arrival follows 12.0 s later. The duty officer assumes P waves travel at 6.0 km/s and S waves at 3.5 km/s through the local crust. The epicentre is offshore, and the earthquake depth is later estimated at 25 km. Civil Defence issues a national advisory stating that there is no tsunami threat to New Zealand, based on the preliminary magnitude, while the district council posts 'long or strong, get gone'. On the school field, students felt rolling shaking for about 45 seconds and several had to steady themselves against a fence. The school has a roll of 600 students and a tsunami evacuation route that takes about 25 minutes on foot.
1. Calculate the distance from the seismograph to the epicentre using the P and S arrival times. Show your working and explain why the time gap can be used this way.
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The two waves leave the epicentre together but travel at different speeds, so the faster P wave pulls ahead. If d is the epicentre distance:
- Time for P: t_P = d / 6.0
- Time for S: t_S = d / 3.5
- Gap: t_S - t_P = 12.0 s
d/3.5 - d/6.0 = 12.0
d(0.2857 - 0.1667) = 12.0
0.1190 d = 12.0
d = 12.0 / 0.1190 = 100.8 km, about 101 km
The gap grows in proportion to distance, because each kilometre adds a fixed extra travel time to the slower wave. That is the basis of the standard 'lag' method: measure the gap on one seismograph, read off the distance, then draw a circle of that radius around the station. One station gives distance, not direction; three stations give the epicentre by where the circles cross.
A useful check: the P wave took about 101/6.0 = 17 s to arrive, so the shaking started at about 14:31:43 and the S wave arrived around 14:32:00.
2. The shaking lasted about 45 seconds and was strong enough that students had to steady themselves. Use this, the epicentre distance and the 25 km depth to assess how much weight the 'no tsunami threat' advisory should carry.
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Three pieces of evidence matter.
Duration. Strong shaking that continues for 45 s at a site 100 km away implies a long rupture, because a bigger fault takes longer for the rupture to sweep past and for its energy to arrive. Long duration is therefore a sign of a substantial earthquake, not a minor one. It also means a large area of seafloor may have been deformed.
Depth. At 25 km the rupture is moderately deep. A deep rupture displaces the seafloor less vertically than a shallow one, and vertical displacement is what lifts the water column and generates a tsunami. This is the strongest reason the advisory might be right, and it is consistent with a preliminary magnitude that looks moderate.
Uncertainty and source. The advisory is national, based on a preliminary magnitude, and automatic solutions are often revised upward within tens of minutes as more stations report. It also does not rule out a smaller local wave or a submarine landslide triggered by the shaking, which can generate a wave even from a modest earthquake.
Judgement: the advisory is useful information but not a reason to stay put. The felt evidence is a natural warning that overrides a generic national statement.
3. Tsunami warnings can be wrong. Decide what the school should do in the minutes after the shaking stops and justify your decision as a risk judgement, including an estimate of how quickly a wave could arrive.
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Travel-time estimate: a tsunami in deep water travels at roughly 200 m/s (about 700 km/h). Over 100 km:
t = d / v = 100 000 / 200 = 500 s, about 8 minutes
That is faster than the 25-minute evacuation walk, before allowing for the time taken to decide, take a roll and move 600 students. Waiting even ten minutes for confirmation could mean being caught on the spit.
Decision: evacuate to high ground immediately, on foot, without waiting for a siren or social media confirmation.
Justification as a risk judgement:
- The expected cost of evacuating if the advisory is right is a disrupted afternoon and some lost lesson time; the cost of not evacuating if it is wrong is fatalities. Asymmetric consequences mean the threshold for acting should be low.
- The school is on a 2 m spit with one narrow road, so the route itself is a bottleneck; starting early keeps the road clear of cars.
- Long, strong shaking is a recognised natural warning, and 45 s of shaking that made people grab a fence qualifies as strong.
- Tsunamis arrive as a series, so the first wave is not necessarily the biggest; students stay on high ground until an official all-clear, not until the first wave passes.
Practical points: take the roll on the field if it takes under a minute, walk rather than drive, and pre-assign staff to the route in advance.