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Unformatted text preview: 21 Superposition Recommended class days: 3 Background Information The principle of superposition distinguishes waves from particles. Waves can be superimposed, particles cannot. Superposition is essential to an understanding of topics ranging from standing waves and beats to optical interference, but limited research finds superposition to be another concept with which students have great difficulty. Wittman et al. (1999) found that many students were unable to draw the shape of a string as two wave pulses passed through each other. Student difficulties in optics are due not to a lack of understanding of electric and magnetic fields but to a lack of understanding of waves and, particularly, of how interference arises. Students find it hard to believe that a standing wave is the superposition of two traveling waves. After all, a simple observation of a standing wave gives no hint of the two counter-propagating waves. A graphical demonstration of adding two counter-propagating waves is more convincing than a mathematical analysis. The standard textbook picture of a standing wave, such as the one shown here, is confusing to many students. They’re not sure how to interpret this. As drawn, the wave appears to be simultaneously at its maximum and minimum displacements. Students need some help to see this picture as representing the extrema of the displacement. I like to use colored chalk to highlight the wave at a single instant of time. Another graphical source of confusion involves standing sound waves in gas-filled tubes. A common practice is to draw sinusoidal waves that fit exactly inside a “picture” of the tube, with the peak-to-peak displacement matching the tube’s diameter. Students already have a difficult time understanding the relationship between a longitudinal oscillation and a graph of the oscillation. Standing waves drawn to match the tube convey the impression that the amplitude of the wave is equal to the radius of the tube. As an example, I gave an exam problem to an honors class based on the figure shown here. The students had no difficulty recognizing that this is the m = 2 mode, calculating the frequency, and answering other questions that involved the usual standing-wave equations. But then I asked, “Can you determine the amplitude of the wave?” Approximately 75% of the class answered that the amplitude was 1 cm. As you can see in the textbook, I now draw sound standing waves with an amplitude larger than the picture of the tube. This practice reduces, but by no means eliminates, students’ tendency to associate the wave amplitude with the picture. Whenever discussing longitudinal waves, instructors need to have students verbalize the physical meaning of the quantity graphed on the vertical axis....
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- Spring '10