## Leonhard Euler

Math and Earthquakes

Ceyda Toprak

Do you know the relation between earthquakes and math? If you would like to know, let’s learn together.

On December 26, 2004, an immense earthquake occurred beneath the Indian Ocean, resulting in a series of significant tsunamis. This event set records, marking it as the deadliest tsunami worldwide and the fourth-largest earthquake in terms of its magnitude. The initial measurement on the Richter scale was an exceptionally high 9.0.

In order to comprehend the cause of the earthquake, scientists aimed to identify its epicenter. The epicenter is situated directly above the earthquake's focus. By employing mathematical techniques, experts determine the epicenter and magnitude of earthquakes, enabling them to assess the severity of the event. This system of measurement, consisting of epicenter and magnitude, is utilized to classify and organize earthquakes.

The Epicenter

Seismometers serve as instruments that record and track the arrival times of seismic waves generated by an earthquake. The initial wave detected by the seismometer is known as the primary (P) wave, followed by the secondary (S) wave. To determine the distance between the seismometer and the epicenter, seismologists calculate the time interval between the S and P waves. By employing three separate seismometers, experts can utilize the three distinct distances obtained to locate the epicenter through a process called triangulation.

Triangulation involves utilizing the three seismometers as the centers of three distinct circles. Each circle possesses a radius equivalent to the measured distance between the seismometer and the epicenter. By drawing these three circles, each centered around a seismometer, the point where the circles intersect corresponds to the epicenter of the earthquake.

The following illustration shows the intersection of three circles at the epicenter, just south of San Francisco.

Magnitude

The 2004 Indian Ocean Earthquake, which struck the Indonesian coast, was recorded as a 9.0 magnitude earthquake. Magnitude represents the amount of energy released during an earthquake and is typically measured on the Richter scale, which ranges from 2 to 9. Earthquakes with a magnitude of 8.0 or higher are exceptionally rare and have the potential to cause complete destruction near the epicenter. The Richter scale measurements are logarithmic with a base of 10, indicating that an earthquake with a magnitude of 9.0 is ten times stronger than one with a magnitude of 8.0. Similarly, a magnitude 9.0 earthquake is 104 times more powerful than an earthquake measuring 5.0.

To determine the magnitude of an earthquake, experts utilize the S-P interval time and the maximum amplitude, both of which are recorded on a seismograph. By employing logarithms with a base of 10, specialists can calculate the earthquake's magnitude.

The Richter scale is standardized, where an earthquake that can be felt 100 kilometers away with an amplitude of 1 millimeter is assigned a magnitude of 3.0. This serves as the reference point, and all other magnitude measurements are compared to this standard. Consequently, if an earthquake is located 100 kilometers away but has an amplitude measurement of 10 millimeters, it would be assigned a magnitude of 4.0. The following graph illustrates this relationship and highlights the reference standard at magnitude 3.0.

To determine the magnitude of the earthquake, a direct line is drawn connecting the distance measurement and the amplitude measurement on each of the three seismographs. These three lines intersect at a single point on the magnitude scale, providing a reading for the earthquake's magnitude.

Conclusion

By integrating all of these mathematical elements, earthquakes can be classified based on their severity. This approach offers a tangible method for analyzing earthquakes and enables experts to convey information about earthquakes to the wider population using numerical data. Consequently, individuals can comprehend the level of severity of earthquakes worldwide, while experts can make predictions by observing historical patterns of earthquakes. This systematic approach enhances understanding and facilitates effective communication about earthquakes.

Now, let’s look from a more detailed perspective and look at some equations.

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Earthquakes

The Richter Scale

The magnitude of an earthquake on the Richter scale is connected to the amount of energy released, denoted as E, measured in joules (J), through a specific equation:

The 1906 San Francisco earthquake had a magnitude of 8.2 on the Richter scale. Applying the aforementioned equation, the amount of energy released during the earthquake was calculated to be:

The Richter Scale is frequently misconceived by people, often due to a lack of comprehension regarding its logarithmic nature. A common source of confusion is understanding the impact of a one-unit difference in magnitude on the released energy. To illustrate this, let's consider two earthquakes: one with a magnitude of and another with a magnitude of + 1. In such a scenario, we observe

As a result, and . We then have, and . Therefore, we can see that an increase in 1 of the magnitude of an earthquake results in an earthquake 31.623 times as strong.

References

Earthquakes, www.math.wichita.edu/~richardson/earthquake.html. Accessed 25 May 2023.

Glydon, Natasha. “Earthquakes and Math.” Earthquakes and Math - Math Central, mathcentral.uregina.ca/beyond/articles/earthquakes/earthquakes.html. Accessed 25 May 2023.