Explore connections in mathematics and nature with this article on earthquakes.
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We think of earthquakes as disasters, and rightly so when they occur in populated areas; but they are also a part of nature – a result of the movement of the earth's crust. Over the past thirty years, scientists have developed the study of plate tectonics, working to explain the connection between movements in the earth's crust and phenomena such as volcanoes and earthquakes. Advances in plate tectonics have allowed scientists to predict where these events are likely to occur, and thanks to the work of American seismologist Charles Richter, scientists also have a way of measuring the magnitude of any given earthquake.

Charles Richter determined that the seismic waves emitted from an earthquake could be used to estimate its magnitude, and he developed a calibrated system for measuring these waves from earthquakes in California. From his research, Richter determined that the larger the earthquake, the larger the amplitude of ground motion at a given distance from the quake. His equation for magnitude is shown below:

R(x) = .67 • log(.37x) + 1.46

In this equation, x equals the energy of an earthquake in kilowatt-hours and R(x) equals the magnitude of that earthquake on the Richter scale. Since this equation uses logarithms, the difference in strength between earthquakes on either ends of the scale can be immense. For example, an earthquake of magnitude 3 on the Richter scale is roughly 31 times more powerful than one of magnitude 2. However, an earthquake of magnitude 7 is roughly 29 million times stronger than one of magnitude 2. As you can see, this scale easily accounts for either extreme. An earthquake of less than 3.5 can be recorded, but is not generally felt. Quakes under 6.0 can result in slight damage to well-constructed buildings, but major damage to poorly constructed ones. Quakes from 7.0-7.9 are considered major, and result in serious damage over large areas, but quakes of 8.0 magnitude or greater can cause serious damage over areas extending hundreds of kilometers. But where does this natural phenomenon come from?

The rigid outer shell of the earth's crust is called the lithosphere. The lithosphere is divided into a series of seven major plates. These plates "float" on top of the upper layer of mantle, called the aesthenosphere. Keep in mind that these plates include not only the continents, but also the ocean floors. As new crust is continuously generated by the welling up of volcanic material through great rifts in the ocean floor, the pressure from this new material causes the plates to shift and knock into one another.

Since the movement of plates across the earth's surface only amounts to millimeters per year, it is hard to tell when the strain in any particular zone will reach the breaking point, causing an earthquake or other seismic event. When an earthquake does occur, however, it creates two types of energy: deformation, which is static and results in permanent displacement of the ground; and seismic waves, which are dynamic sound waves radiating through the ground from the quake itself. The seismic waves are divided into two different types: P waves, which shake the ground in the direction they are moving; and S waves, which shake the ground perpendicularly, or transverse to the P waves. The ratio of speed of P to S waves is constant, so scientists can time the delay between them and estimate the distance of a given earthquake from their observation site. To do this, they simply multiply the time of S-P by a factor of 8 km/s to get the approximate distance in kilometers.

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