What is the Relationship Between the Energy of an Earthquake and the Amplitude of a Wave?

Submitted by John Portle

Bloomington High School North

PURPOSE: To observe the relationship between the energy of an earthquake and the resulting seismograph.

PROCEDURE: Vary the energy of a simulated earthquake and observe the resulting seismograph, especially the amplitude.

It would be convenient if we could dial-up earthquakes of magnitudes that increase by a known factor each time while keeping the epicenter a constant. Since this is currently not possible, a miniature earthquake can be generated by dropping a cast iron ball from a known height and fixed distance from the seismometer. Since our seismometer is located on a concrete slab just above bedrock I choose to drop a ball approximately 8.0 cm in diameter with a mass of 1.8 kg. (17.6 N). I then varied the height from 0.2 m to 0.8 m by increments of 0.2 m., the floor is surprisingly elastic since the ball rebounded so much. I was able to catch the ball in the air after each rebound, and this is no easy task for a "elder" science teacher. The advantage of this rebound and catching is that there is only one pulse generated and the damping coefficient of the floor causes the resulting floor oscillations to be very small in amplitude and damp out quickly. We ran into a problem taking the data since the lower heights (and correspondingly lower energies) caused the peaks to be lost in the background noise. To help isolate and locate the pulses we first had to synchronize our watches to the GPS clock in the instrument. There is a delay before the waves are displayed due to signal processing and matching the peaks with the correct energy of the quake was a problem. Once you are synchronized, you can drop the ball at thirty-second intervals and record the correct times on your data sheet. This will make correlating the data of the drops and the seismographs easy. The data that we obtained is shown below.

Height of Drop Amplitude of Wave Potential Energy of the Earthquake

0.2 m 9,984 3.53 J

0.4 13,824 7.06

0.6 15,616 10.58

0.8 16,384 14.11

The potential energy was calculated from:

PE = m ag h

The data from the seismometer is shown below. You can easily see the four peaks and note that they are getting higher in amplitude. Most students will be surprised that although the fourth peak was caused by dropping it from four times the height of the first peak, it is NOT four times as tall. The next part of the lab is to determine this relationship between the quake and its resulting amplitude energy.

To obtain the relationship between the energy of the wave (the quake) and the amplitude we graphed the potential energy Vs the amplitude, as shown below.

The graph shows very clearly the result that the relationship between the energy and the amplitude is:

Energy = Amplitude2

This is unexpected by many students who assume that it would be a linear one. The consequences of this square relationship is that if a second earthquake has twice the energy of an initial earthquake and everything else is constant, the amplitude of the second quake would be the square root of two higher or 1.41 times as high. An earthquake ten times as powerful has an amplitude that is only square root of ten or 3.2 times as high.

CURVE FITTING:

We tried to fit a curve to this graph but with only four data points we weren’t able to obtain the correct result. The shape of the curve is of the form y = k x2. We believe that more data points would enable the computer to solve this problem.

THEORY: According to PHYSICS by Giancoli the energy carried by a wave is given by the equation:

E = 2 P 2 r S n f 2 A2

Where r = density of the medium, S = cross-sectional surface area through which the wave travels, f = frequency of the wave and A is the amplitude of the wave. The amplitude of the wave shown by the seismometer would also have to have an additional term added to the right side due to the sensitivity and amplification of the instrument. However it is easy to see that our equation from our data would agree with the equation from Giancoli if we lumped all of the additional terms into one constant.

VARIATIONS AND IMPROVEMENTS IN THE LAB:

1. More data points to obtain a more accurate graph. I wouldn’t want to drop the ball from more than one meter since I might damage the floor. You could easily add more points between the maximum and minimum.
2. If your seismometer is located in a vault than you would have to drop a considerable more mass. A fifty pound (sorry for the English units) weight from your weight room as school dropped onto a metal plate should give sufficient peaks to measure.
3. It would be interesting to see if there is a linear function with distance from the impact point to the seismometer.