Mathsquake – the Maths of Earthquakes

Justin Leung

February 27, 2010 – the same as any other at work in El Centro Sismológico Nacional (National Seismological Center Chile) in Santiago, Chile. That is, until the ground starts to shake. Things start falling around you, and instinctively you dive underneath a table to protect yourself. After what seems like forever, you come back out and immediately get to work. You need to figure out the epicenter of the earthquake so you can send out a warning to anyone at risk of further earthquakes in the area…

The epicenter is the point of the surface above the focus of the earthquake, where the focus is the source of the earthquake below ground.

An earthquake might seem simple in the sense that it just creates a lot of intense shaking that lasts for around a minute. But in that minute, many different waves have travelled past you. These different types of waves can tell us a lot about the earthquake, such as the type and its location.

As its name suggests, the primary wave, or P wave, is the first type of wave that hits you. It pushes you back and forth because it is a longitudinal wave: the motion you feel is in the same direction in which the wave is moving – think of pushing a ‘slinky’ as shown below.

After a short while, the secondary wave, or S wave, arrives. Unlike the P wave, the S wave is a transverse wave: the motion you feel is at a right angle or perpendicular to the wave direction. This means you feel an up and down motion when this wave passes through.

As you emerge from under the table and regain your composure, the first thing you do is look at the data from your seismometers – devices used to record shaking in the earth. The oldest seismometers work by hanging a weight that is attached to a pen, which records wiggles on a piece of paper attached to a rotating drum. A stronger vibration is reflected by a larger amplitude (height of the wave) in the wiggle. Current seismometers largely work in the same way, but record the wiggles digitally.

The P and S waves are different waves, and therefore have different frequencies and amplitudes. However, the differences are quite subtle so it’s unlikely that you would be able to feel the difference as they pass through you. We can work out the arrival time of P and S waves from the data, and the difference in arrival times helps us to calculate the distance between the seismometer and the epicenter. We can rearrange the speed = distance/time formula to get:

∆T = d*(1/vs – 1/vp)

• ∆T = Ts-Tp = Difference in arrival times where Ts is the time of arrival of the S wave, and Tp the arrival time of the P wave
• d = distance
• vs and vp = S and P wave speeds

Using the formula above, can you help to solve the following?

1. The red and blue lines below mark the time of arrival of the P and S waves respectively. What is the difference in the arrival time of the P and S waves?
1. The difference in the arrival time of the P and S waves can be used to calculate the distance of a seismometer from the focus of the earthquake. Given that P waves travel at 8 km/s and S waves travel at 5 km/s, how far away from the focus is the seismometer?
2. Now that we know the distance between the focus and the seismometer, how long did it take for the P and S waves to reach the seismometer? These are known as arrival times.

3. The same earthquake was recorded on 2 more seismometers. How far away are they from the focus? As above, the red and blue lines mark the times of the P and S waves respectively.

*scroll down to the bottom of the page for the answers

The larger waves that happen after the P and S waves are called surface waves, and they are the waves that cause the most damage in an earthquake. However, as we’ve seen above, we only need to know about the P and S waves to find the epicenter of the earthquake.

Now that we know the distances between the seismometers and epicenter, we can use trilateration to find where the epicenter is. Trilateration is locating a point in 3D space (x,y,z) based on 3 known distances. It is more common and important than you may realise. Ever wondered how Google maps can quickly generate a quickest route to your destination, or how you can share your location with your friends when you get separated? Location services rely on the global positioning system, or as it’s more commonly known GPS, which works using trilateration.

Whenever you want to find your location, 3 of the 24 GPS satellites orbiting Earth will send a signal to your phone, and your phone sends a signal back to the satellite. The time it takes for the GPS to receive the signal indicates the distance between the satellite and the phone. We can then draw a circle with a radius equal to this distance around each satellite. If we do this for all 3 satellites, we can find the unique intersection point, which is where your phone – and therefore you – are located!

The process is exactly the same for locating the epicenter of an earthquake, except we use seismometers instead of satellites. Try your hand at the puzzles below and see if you can put your new-found knowledge to use!

1. The seismometers are located as shown in the table and diagram below. Draw a circle around each seismometer, where the radii of each seismometer are the distances calculated in questions 2 and 4.
1. Each circle should intersect twice with the other circles. Draw a line between each pair of intersections. The intersecting point of the 3 lines is the epicenter of the earthquake.
2. Now that we have found the epicenter in question 6, we can measure the distance between the epicenter and the seismometer. This is not to be confused with the distances in question 2 and 4, which are the distances between the seismometers and the focus (see diagram below for clarification). Using your answers in question 2, 4, and 6, find the depth of the earthquake.
1. Here is some data from a seismometer 10080.25 km away from the epicenter (remember the red line represents the arrival of the P waves and the blue line the arrival of the S waves). Calculate its distance from the focus. Are the 2 distances similar? Why?

*scroll down to the bottom of the page for the answers

As the waves from an earthquake propagate around the world they do so without losing much energy. This is why scientists use seismometers around the world instead of just those close to the earthquake to calculate the epicenter. Furthermore, they are far enough away that we can treat the epicenter and focus as approximately the same point. This means the circles will also pretty much intersect at the epicenter, which saves us the effort of doing the steps in question 6!

Of course, in a real-world situation the calculations above will be done by a computer within seconds rather than by hand, which allows scientists to identify the epicenter much quicker. But, why do we want to know where the epicenter is? This is because knowing the location of the epicenter is very important for earthquake prevention systems and saving lives. An earthquake rarely happens by itself: there is always another one that precedes or succeeds it. This is where the terms foreshock and aftershock came from, and they describe earthquakes happening before and after the main earthquake. After detecting a forequake, scientists can notify nearby cities and help people prepare for the main earthquake and the following aftershocks. This is the secret to how the earthquake alerts on your phone work!

*scroll down for the answers to questions 1-4 and then 5-8

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Solutions to Questions 1-4

1. 1:40 or 100 seconds

2. d = 1333 km

3. vp = 167s, vs = 267s

4. LVC = 1897 km, NNA = 3136 km

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Solutions to Questions 5-8

7. 100 km (answer can vary given the accuracy of measurements)

8. The distances are basically the same because the horizontal distance is so large that the depth of the earthquake can be neglected.