Faraday waves
Use the FizziQ frequency generator with a loudspeaker to create spectacular standing waves on the surface of a liquid.
Faraday waves are standing waves that appear on the surface of a liquid when its container is subjected to vertical vibration above a certain amplitude threshold. They were first described by Michael Faraday in 1831 and remain an active area of research in fluid dynamics. Using the FizziQ frequency generator connected to a loudspeaker, you can create these stunning wave patterns on the surface of a shallow tray of water and observe how the geometry changes with frequency.
Activity overview:
The student places a small tray of colored water on a loudspeaker driven by the FizziQ frequency generator. By sweeping frequencies from 20 to 200 Hz, they observe the appearance of standing waves at specific resonant frequencies and photograph the geometric patterns.
Level:
High School
FizziQ
Author:
Duration (minutes) :
30
What students will do :
- Create standing waves on the surface of a liquid using a loudspeaker
- Identify the resonant frequencies of a vibrating system
- Observe and photograph the geometric patterns formed at different frequencies
- Measure the wavelength and relate it to the driving frequency
- Understand parametric resonance and subharmonic response
Scientific concepts:
- Standing waves
- Resonant frequency
- Faraday waves
- Parametric instability
- Chladni figures
- Subharmonic resonance
- Dispersion relation
Sensors:
- Frequency generator (built-in synthesizer)
Material needed:
- Smartphone or tablet with FizziQ
- A powerful loudspeaker (portable speaker or desktop speaker)
- A shallow tray or plate
- Water and food coloring
- Optional: cornstarch for non-Newtonian fluid
Experimental procedure:
Place a powerful loudspeaker flat, membrane facing up. A bass speaker or a round desktop speaker works well.
Place a small shallow tray (plate, plastic lid, small pie tin) on the speaker membrane. It should sit stably.
Fill the tray with a thin layer of water (3 to 5 mm). Add a few drops of food coloring or milk to make the waves visible.
Open FizziQ and use the frequency generator (synthesizer) to produce a sinusoidal sound. Connect the smartphone to the speaker via Bluetooth or cable.
Start with a low frequency (about 20 Hz) and moderate volume. Gradually increase the volume until you see the surface begin to vibrate.
Slowly sweep the frequencies from 20 Hz to 200 Hz. Observe the different patterns that appear at certain resonant frequencies.
At each resonant frequency, the waves form a regular geometric pattern: concentric circles, square grid, hexagons, or radial star shapes.
Note the frequencies at which patterns appear. Measure the wavelength λ (distance between two nodes or two antinodes).
Spectacular variant: replace the water with a cornstarch-water mixture (non-Newtonian fluid). At certain frequencies, the surface forms finger-like protrusions that dance on the speaker.
Present your observations and best photos in your FizziQ experiment notebook.
Expected results:
The water surface remains flat below a vibration amplitude threshold. Above the threshold, standing waves appear abruptly (bifurcation). The patterns become more complex and finer at higher frequencies. Typical resonant frequencies range from 30 to 150 Hz depending on the tray size and water depth. The wavelength decreases as the frequency increases, consistent with the dispersion relation.
Scientific questions:
- Why do Faraday waves vibrate at half the excitation frequency?
- How does the container geometry influence the patterns that form?
- Why is there a minimum amplitude threshold for waves to appear?
- What is the relationship between the observed wavelength and the vibration frequency?
- How do these patterns compare with Chladni figures on a solid plate?
- Why does a non-Newtonian fluid behave so differently from water?
Scientific explanations:
Faraday waves are an example of parametric instability: when a system parameter (here the vertical acceleration of the tray) oscillates above a threshold, the flat surface becomes unstable and standing waves spontaneously appear.
The frequency of Faraday waves is exactly half the vibration frequency of the tray. This is a subharmonic resonance phenomenon: the waves oscillate at f/2 when the tray vibrates at f.
Intuitively, the surface must rise AND fall to complete one wave cycle, which takes two vibration cycles of the tray. This 2:1 ratio is the hallmark of Faraday instability.
The geometric patterns correspond to the eigenmodes of vibration of the liquid surface, constrained by the container edges. The shape and size of the container determine which patterns are possible.
This phenomenon is described mathematically by the Mathieu equation, which models systems subjected to periodic parametric excitation. The stability analysis predicts the threshold amplitude above which waves appear.
The non-Newtonian fluid (cornstarch-water mixture) shows spectacular behavior because under rapid stress (vibration), it becomes rigid and can form solid-like structures that jump on the speaker.
The dispersion relation for surface waves in shallow water is ω² = gk tanh(kh), where k = 2π/λ is the wave number and h the water depth. This determines which wavelength corresponds to each frequency.
Chladni figures, observed with sand on a vibrating plate, are the solid analogue of Faraday waves. The sand collects at the nodes where the plate does not vibrate.
Extension activities:
- Why do Faraday waves vibrate at half the excitation frequency?
- How does the container geometry influence the patterns that form?
- Why is there a minimum amplitude threshold for waves to appear?
- What is the relationship between the observed wavelength and the vibration frequency?
- How do these patterns compare with Chladni figures on a solid plate?
- Why does a non-Newtonian fluid behave so differently from water?
Frequently asked questions:
Q: Nothing happens, the surface stays flat.
R: The volume is probably insufficient. Increase it gradually. Make sure the tray is well coupled to the speaker membrane and does not rattle.
Q: The patterns are irregular and messy.
R: Reduce the volume slightly. Too much amplitude creates turbulence. The clearest patterns appear just above the instability threshold.
Q: Can I use any loudspeaker?
R: You need a speaker with good bass response. Phone speakers are too weak. A portable Bluetooth speaker or desktop speaker with at least 5W works well.
Q: Why does food coloring help?
R: Transparent water makes the waves nearly invisible. Coloring or milk adds contrast, making the wave crests and troughs clearly visible.