Earthquakes: An Application of Mathematics
The October 13, 2010 earthquake triggered a large buzz across Oklahoma University’s Norman campus, startling students walking to class or sitting in the middle of calculus. This magnitude 4.3 earthquake was 10 miles southeast of Norman and was felt as far north as Stillwater. To many students, the few seconds of shaking was unrecognizable as an earthquake because Oklahoma’s seismicity is not well known. In reality, Oklahoma experiences multiple earthquakes, almost daily; they are usually just not strong enough to be felt. While understanding how and why earthquakes occur in Oklahoma involves knowledge of faults and rock types, the methods involved in locating the epicenter of an earthquake and determining its magnitude require an awareness of the mathematical principles involved
The focus of an earthquake is the spot within the earth where the earthquake originates, where the fault is first ruptured. The epicenter of an earthquake is the location on the earth’s surface directly above the focus. When attempting to locate the epicenter of an earthquake, an understanding of seismic waves is crucial. There are two types of seismic waves, body waves and surface waves. Love and Raleigh waves are surface waves that travel across the earth’s surface and are responsible for a majority of the shaking felt during an earthquake. The two types of body waves, p and s-waves are more useful in determining information about earthquakes. Body waves travel throughout the interior of the earth, propagating from the focus of the earthquake eventually reaching seismograph stations. The arrival times of p and s-waves are used to locate the epicenter of the earthquake as the waves are recorded onto seismographs. Seismographs are devices that detect ground shaking, and record seismic waves. They are constructed using principles of inertia. Simply put, seismographs involve a pen suspended with its point on a roll of moving paper. As the earth shakes, the pen moves back and forth, recording the waves. Seismographs more complicated, and no longer constructed using pens, but the idea remains the same as the shaking of the ground is reflected by the lines drawn by the seismograph
P-waves, or compressional waves, are the first waves to arrive at the seismograph, as they travel the fastest. The p-wave velocity can be calculated using the equation:
This equation simply illustrates the fact that the velocity of p-waves is dependent on density (ρ), shear modulus (μ), and bulk modulus (κ) where a modulus is essentially a measure of how rigid the rock is that the waves pass through. P-waves travel through solids, liquids, and gas and vibrate in the direction in which they propagate. Because p- waves alternate from compression to expansion, they produce a volume change in the material they pass through.
S-waves, or shear waves, are like sine waves. They vibrate perpendicular to the direction in which they are...