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Objective
To explore constructive and destructive interference of sine waves and plot a complex wave.

• copies of the "Sine Wave Science" student handout (PDF or HTML)
• copies of the "About Sine Waves" student handout (PDF or HTML)
• copies of the "Combining Sine Waves" student handout (PDF or HTML)
• pencils with erasers
• rulers
1. The seismic signals from volcanoes are complex waves that result from multiple single waves that occur simultaneously. To help students understand how a complex wave is formed, tell them that in this activity they will be combining two simple sine waves.

2. Organize students into teams and distribute copies of the "Sine Wave Science," "About Sine Waves," and "Combining Sine Waves" student handouts and other materials to each team member.

3. Review with students what a sine wave is and how sine waves are combined, as described on the "Combining Sine Waves" student handout. Explain to students that each complex wave point is determined one at a time. Have students plot the point for the sum of sine waves 1 and 2 before connecting their points into a wave. Have each student in the team plot her or his own wave, but allow team members to help each other understand the activity concepts.

4. After they have plotted all the summative points, have students connect the points to create the resulting complex wave.

5. Once students have successfully drawn their complex wave, have them answer the questions listed on the student handout. What features do they observe about each of the waves they worked with?

6. As an extension, have student teams plot new complex waves by picking different values for wavelength and amplitude. Students may notice that the closer the original sine wave wavelengths are, the longer the resulting complex wave wavelength will be.

Small earthquakes within volcanoes, which occur as magma rises to the surface of a volcano, create sine waves that seismographs can record. The complex sine waves Bernard Chouet studied were a combination of multiple sine waves. Although students create a complex wave by summing only two simple sine waves, the concept of constructive and destructive interference is readily grasped. When the waves being summed reinforce each other, it is constructive interference; when the waves reduce each other, it is destructive interference.

Students experience complex waves every day. Anyone who has swum at an ocean beach has experienced large waves created by constructive interference and waited through the lulls caused by destructive interference. But have any of your students ever timed the arrival of the large waves? What might they predict based on their observation of the complex sine wave they created? If they expect that the time between the arrival of large waves would be constant, they would be right.

Here are almost two cycles of the resulting complex sine wave.

Students will plot almost two complete cycles of the complex wave on their student handout. They will be able to see that the wave repeats after 24 baseline units.

The first wave has a wavelength of 8 units, the second, 12 units. The complex wave has a wavelength of 24 units. Many middle and high school students will realize that 24 is the Lowest Common Multiple (LCM) of the two sine waves. This is a characteristic of complex waves; the wavelength of a complex wave is equal to the LCM of the wavelengths of the constituent sine waves.

The LCM correctly suggests that there are mathematical relationships that can be used to reveal the sine waves imbedded in a seismic record. Today, seismic records are digitized, allowing computerized analysis.

Books

Bruce, Victoria. No Apparent Danger: The True Story of Volcanic Disaster at Galeras and Nevado del Ruiz. New York: HarperCollins, 2001.
Tells the story of volcanologists' struggles to predict the eruptions of Nevado del Ruiz and Galeras in Colombia.

Williams, Stanley and Fen Montaigne. Surviving Galeras. Boston, MA: Houghton Mifflin, 2001.
Recounts Williams' escape from the eruption of Galeras and discusses the study of volcanology.

Web Sites

www.pbs.org/nova/volcano/
Provides program-related articles, interviews, interactive activities, and resources.

Global Volcanism Program
www.volcano.si.edu/gvp/
Archives weekly and monthly volcanic activity reports on volcanoes around the world.

Volcano World
volcano.und.nodak.edu/
Displays satellite imaging of volcanoes around the world and outlines the steps toward becoming a volcanologist.

The "Plotting the Spots" activity aligns with the following National Science Education Standards and Principles and Standards for School Mathematics:

 Science Standard B:Physical Science

Transfer of energy

• Energy is a property of many substances and is associated with heat, light, electricity, mechanical motion, sound, nuclei, and the nature of a chemical. Energy is transferred in many ways.

 Mathematics Standard: Algebra

 Science Standard B:Physical Science

Interactions of energy and matter

• Waves, including sound and seismic waves, waves on water, and light waves, have energy and can transfer energy when they interact with matter.

 Mathematics Standard: Algebra