A standing wave is formed when two waves of the same frequency and amplitude travel in opposite directions and interfere. Unlike a traveling wave, it does not appear to move: certain points (nodes) remain stationary while others (antinodes) oscillate with maximum amplitude.
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How to measure it in class
With the FizziQ app, it is possible to study the resonance frequencies associated with standing waves.
Steps:
- Use a guitar string or an elastic band stretched between two fixed points
- Pluck the string and measure the fundamental frequency with the FizziQ frequency meter
- Touch the string at its midpoint to produce the 2nd harmonic (octave)
- Touch at one-third for the 3rd harmonic, at one-quarter for the 4th harmonic
- Verify that the frequencies are indeed multiples of the fundamental
- Use the spectrogram to visualize the harmonics present
Scientific activities on this topic
- Study string harmonics: https://www.fizziq.org/en/activities/tubular-melodies/
- Measure tube resonances: https://www.fizziq.org/en/activities/tube-effect/
- Build a simple string instrument and analyze its modes
Learn more
Formation of standing waves:
On a string fixed at both ends, a wave emitted at one end reflects at the other. The outgoing wave and the returning wave interfere. At resonance frequencies, this interference creates a stable pattern: the standing wave.
Nodes and antinodes:
- Nodes: points where the amplitude is always zero (permanent destructive interference)
- Antinodes: points where the amplitude is maximum (constructive interference)
The fixed ends of a string are always nodes.
Vibration modes:
| Mode | Nodes | Antinodes | Frequency |
|---|---|---|---|
| 1 (fundamental) | 2 (at the ends) | 1 | f1 |
| 2 | 3 | 2 | 2f1 |
| 3 | 4 | 3 | 3f1 |
| n | n+1 | n | nf1 |
Musical applications:
String instruments (guitar, violin, piano) and wind instruments (flute, clarinet) produce sounds through standing waves. The timbre of the instrument depends on the relative amplitude of the different modes.
Formula
For a string of length L fixed at both ends:
f_n = n x v / (2L) = n x f1
Fundamental frequency of a string:
f1 = (1/2L) x sqrt(T/mu)
where:
- f_n: frequency of mode n (Hz)
- v: wave velocity on the string (m/s)
- L: length of the string (m)
- T: tension of the string (N)
- mu: linear mass density of the string (kg/m)
Application examples
- A 65 cm guitar string with f1 = 330 Hz has harmonics at 660, 990, 1320 Hz…
- Chladni figures visualize standing waves on vibrating plates
- In a microwave oven, standing waves create “hot spots” and “cold spots”
- Vocal cords also produce standing waves
FAQ
Q: Why is it called “standing” when it vibrates? A: The nodes and antinodes do not move: the oscillation pattern remains fixed in space. This is different from a traveling wave that transports energy from one point to another.
Q: How do you create a standing wave on a string? A: By exciting the string at a resonance frequency. The emitted wave and the reflected wave then interfere constructively.
Q: Do organ pipes work the same way? A: Yes, but with pressure waves in air. A pipe open at both ends has pressure antinodes at the extremities; a closed pipe has a node at the closed end.
Related concepts
Resonance - Harmonics - Fundamental frequency - Interference - Vibrating string - Vibration modes