Standing waves with a variable wavelength

Demonstration

Three different experiments show standing waves having more than a single wavelength. To explain why each system behaves as it does, students need to understand both the factors affecting wave speed in a medium and also the relationship between wave speed and wavelength, for a given frequency.

Apparatus and Materials

• Signal generator
• Vibrator
• Xenon stroboscope
• 4mm leads
• rubber cords of different thickness...
• Rubber cord, 0.5 m long, 3 mm square cross-section
• Light rubber cord, 0.5 m long (e.g. dressmaking elastic)
• hanging chain...
• Length of light chain
• rubber strip of varying width...
• V-shaped sheet of rubber, about 0.5 m long, maximum width about 0.1 m
• G-clamp
• Wooden blocks to crimp the wider end of the rubber sheet

Health & Safety and Technical Notes

Read our standard health & safety guidance

Rubber cords of different thickness: Tie the thick rubber cord to the thin elastic, and fix one of them to the vibrator. Since the wave velocity depends on the square root of the mass per unit length, an effective demonstration requires cords having a mass ratio of at least four. Good lighting is important.

Hanging chain: Chain sold for securing bath plugs is suitable. It is easiest to swing the top round in a small circle to generate a wave, but it will be clearer that a standing wave is involved if the top is oscillated sideways.

Rubber strip of varying width: Rubber cot sheet is a suitable material. Cut the sheet with a razor blade along previously marked lines, while it is being held down and lightly stretched. A piece 0.5m long, tapering from 100 mm to 10 mm, is about right. A line drawn down the middle helps to make the motion clear, especially as the edges of the strip tend to flap. Stroboscopic illumination along the length of the strip is very effective. Use as large an amplitude of oscillation as can be managed.

Safety note: Using the xenon stroboscope, teachers should be aware that frequencies around 7 Hz have been known to cause epileptic fits in certain people. Ask your students if any know that they are susceptible to this response.

Procedure

rubber cords of different thickness...

1. Adjust the frequency of the signal generator until you get standing waves of large amplitude.
2. Questions for students to answer:
3. "Do both cords vibrate with the same frequency?"
4. "In which cord is the wavelength shorter?"
5. "In which cord does the wave travel more slowly? How do you know? Explain why the wave travels more slowly in that cord.
hanging chain..."

6. Hold the chain at one end and generate a standing wave.
7. Questions for students to answer:
8. "Where is the wavelength large, and where small? Where is the curvature of the chain large and where small, i.e., where does it change direction quickly, and where is it more nearly straight?"
9. "Where does the wave travel quickly, and where more slowly? How do you know?"
10. "Explain why the wave speed varies."
rubber strip of varying width....

11. Clamp the V-shaped sheet of rubber at its wider end and vibrate it at its narrow end. Observe its motion carefully. Where, along the length of the sheet, is the wavelength longest and where is it shortest? Can you explain this pattern in terms of a variable wave speed?
12. It is worth looking at the lowest frequency of vibration: the standing wave with one single loop, as shown below.

13. The peak is not in the middle, but near the wide end. Near that end, the rubber curves more sharply than at the other, just as it would have to do if there were many short waves at that end and few longer ones at the other.

Teaching Notes

• The first experiment shows that wavelength depends on thickness of cord.
• The second experiment shows that wavelength depends on tension.
• The top of the hanging chain carries more load than the bottom. Tension and hence velocity and wavelength are therefore greater at the top.
• The third experiment shows that wavelength depends on the mass per unit length. Where waves travel slowly, on the more massive wide part, the wavelength is short. The wavelength increases slowly along the strip, being long at the narrow end where waves travel fast.
• In every case, the velocity varies from place to place. The frequency is the same at all places, so the wavelength must vary from place to place. These are standing waves in which a wavelength, adjusted to the proper value for each place, must fit into a finite region.
• The standing waves should be viewed stroboscopically, as well as by eye, to give students a clear understanding of their nature.

This experiment has yet to undergo a health and safety check.