17.3

Standing Waves

Student Notes

This simulation shows the standing waves that occur when waves reflecting from each end of a string or air column interfere constructively. This is what happens in musical instruments, for example, although a musical note is generally composed of a fundamental (lowest frequency standing wave) and several harmonics, unlike the single-frequency situation shown here. This string is driven from the left end at the chosen frequency, and the standing wave pattern builds up gradually as reflections occur at the two ends. Here are some things to investigate:

1. | With the string fixed at both ends, the fundamental frequency is 2 Hz. What is the relationship between L, the length of the string, and l, the wavelength?
2. | With the string fixed at both ends, what happens when the frequency is 1 Hz? Why?
3. | With the string fixed at both ends, see if you can determine a relationship between the length of the string and the wavelengths of the waves that produce standing waves.
4. | Repeat with the string free at both ends. Is the relationship the same, or different? Why?
5. | Repeat with the string free at one end only. What is the frequency of the fundamental now? What is the relationship between the length of the string and the wavelengths of the standing waves?

Self-test

1. | The lowest frequency standing wave is known as the fundamental frequency. In this simulation, the fundamental frequency of the string when it is fixed at both ends is 2 Hz. What is the fundamental frequency when the string is free at both ends? What about when the string is free at only one end? Explain why this is true.
2. | Harmonics are standing waves that occur at frequencies higher than the fundamental frequency. If the fundamental frequency is denoted by f, at what frequency does the next harmonic occur? (e.g., 2f? 3f?). Does this depend on whether the ends of the string are both fixed, both free, or have just one end free?
3. | Nodes are positions of zero amplitude oscillations, while anti-nodes are positions of maximum amplitude oscillations. When the string is being driven at a standing-wave frequency, do the positions of the nodes and anti-nodes change? (This is why these patterns are known as standing waves!)
4. | In a standing wave, is a fixed end a node or an anti-node? What about a free end?

Copyright © 2006 John Wiley & Sons, Inc. All rights reserved.