Acoustic beats

This experience allows students to discover the phenomenon of acoustic beats and its musical applications. It develops their auditory perception while illustrating the principles of wave interference.

Musicians have used an elegant acoustic phenomenon to tune their instruments for centuries: when two notes of nearly identical pitch are played together, a rhythmic pulsation emerges in the combined sound. These pulsations, called acoustic beats, are not a product of imagination but a measurable physical phenomenon arising from wave superposition. The frequency of this pulsation equals the exact difference between the two source frequencies, providing musicians with an incredibly precise tuning tool. When the beats slow down and eventually disappear, the two instruments are in tune. The same principle underlies the tremolo effect in electronic music, where low-frequency oscillations modulate the amplitude of a carrier wave. In the world of piano tuning, professional tuners listen to beat frequencies between pairs of strings to achieve precise temperament adjustments. This experiment allows students to generate, hear, and measure acoustic beats using the FizziQ synthesizer, connecting the mathematical description of wave superposition with direct auditory and visual evidence.

Activity overview:

The student uses the FizziQ synthesizer to simultaneously produce two sounds of very close frequencies (440 Hz and 441 Hz) and observes the regular variations in sound intensity that result. Using the sound intensity sensor, it measures the period of the beat and verifies that its frequency corresponds to the difference between the two frequencies emitted. The experiment is repeated with different frequency deviations to establish the general rule.

Level:

High school

FizziQ

Author:

Duration (minutes) :

35

What students will do :

- Generate acoustic beats by superposing two sounds of close frequencies using the FizziQ synthesizer
- Measure the beat frequency and verify that it equals the difference between the two source frequencies
- Visualize the amplitude modulation pattern on the FizziQ sound level graph
- Establish the general relationship between frequency difference and beat frequency
- Connect the mathematical model of wave superposition to auditory perception

Scientific concepts:

- Wave interference, Sine wave superposition, Auditory perception, Frequency and period, Amplitude modulation

Sensors:

- Microphone (sound level meter)
- Sound frequency analyzer
- FizziQ synthesizer (sound generator)

Material needed:

- Smartphone with the FizziQ application, A quiet environment, Optionally headphones for better perception

Experimental procedure:

Open FizziQ and navigate to the Synthesizer tool. Set the first tone to 440 Hz (concert A).
Enable a second simultaneous tone and set it to 441 Hz, creating a frequency difference of 1 Hz.
Listen carefully. You should hear a single tone whose volume slowly pulses approximately once per second.
On a second device (or using FizziQ on the same device), open the Sound Level (dB) sensor and start recording.
Record for at least 15 seconds to capture multiple beat cycles. The graph should show regular oscillations in sound intensity.
Measure the period of the beats from the graph (time between consecutive maxima or minima). Verify that the beat frequency is approximately 1 Hz (period ≈ 1 second).
Now change the second frequency to 443 Hz (3 Hz difference). Listen to the faster pulsation and record again.
Measure the new beat period and verify that the beat frequency is now approximately 3 Hz.
Repeat with frequency differences of 5 Hz, 10 Hz, and 15 Hz, recording each time.
Create a table in your FizziQ notebook with columns: f₁, f₂, Δf = |f₁ - f₂|, measured beat frequency.
Plot a graph of measured beat frequency versus frequency difference (Δf). Verify that the relationship is linear with a slope of 1.
Finally, increase the difference to 20 Hz and then 30 Hz. Note at what point you stop perceiving discrete beats and begin hearing a rough or dissonant sound instead.

Expected results:

For a 1 Hz frequency difference (440 Hz + 441 Hz), students should clearly hear a slow pulsation with a period of about 1 second and observe corresponding oscillations of 3-6 dB on the sound level graph. As the frequency difference increases, the beat rate increases proportionally: 3 Hz difference produces 3 beats per second, 5 Hz gives 5 per second, etc. The graph of beat frequency versus frequency difference should be approximately linear with slope 1. Beyond about 15-20 Hz, individual beats become too fast to distinguish as separate pulsations, and the perception transitions to a rough or buzzing quality. At 30 Hz difference, most listeners will hear two distinct tones rather than beats. Measurement precision is typically ±0.5 Hz due to noise and the difficulty of identifying exact maxima on the sound level graph.

Scientific questions:

- Why does the beat frequency equal the difference between the two source frequencies rather than their sum?
- At what frequency difference do beats become too fast to perceive individually? Why does this happen?
- How do piano tuners use beats to tune strings? What do they listen for?
- What would happen if the two sounds had different amplitudes? Would beats still occur?
- How is the beat phenomenon related to the LFO (Low Frequency Oscillation) effect in electronic music?
- Can beats occur with non-sinusoidal (complex) sounds? How would the perception differ?

Scientific explanations:

The phenomenon of acoustic beats results from the superposition of two sound waves of slightly different frequencies. Mathematically, when two sinusoidal waves of frequencies f₁ and f₂ add together, the resulting wave can be expressed as a single wave whose amplitude varies periodically: 2A·cos[2π(f₁-f₂)t/2]·cos[2π(f₁+f₂)t/2].

This formula shows that the resulting wave has an average carrier frequency (f₁+f₂)/2, modulated by a frequency envelope (f₁-f₂)/2. The ear thus perceives a medium frequency sound whose intensity varies at the frequency |f₁-f₂|.

This phenomenon is widely used in music: for tuning instruments (the absence of beats indicates identical frequencies), and in electronic music where the LFO (Low Frequency Oscillation) effect creates rhythmic modulations. The perceptual limits are important: below 0.5 Hz, the beats are too slow to be perceived as a pulsation; beyond 20 Hz, they exceed the temporal resolution limit of the ear.

Extension activities:

Frequently asked questions:

Q: I can hear the beats but the sound level graph does not show clear oscillations. Why?
R: The sound level meter may be averaging too quickly or too slowly. Try adjusting the acquisition frequency in FizziQ settings. Also ensure the microphone is positioned close to the speaker and that background noise is minimal, as ambient sounds can mask the small amplitude variations.

Q: Why do I hear two separate tones instead of beats when the frequency difference is large?
R: Beats are only perceived when the two frequencies are close enough that the ear cannot resolve them as separate tones. This critical difference is around 15-20 Hz for frequencies near 440 Hz. Beyond this, the auditory system separates the two sounds.

Q: The measured beat frequency does not exactly match the frequency difference. Is this normal?
R: Small discrepancies of ±0.5 Hz are normal due to measurement uncertainty. Ensure you are measuring over multiple beat cycles and averaging. The synthesizer frequency may also have slight inaccuracies.

Q: Can I observe beats visually without the sound level meter?
R: Yes, you can use the FizziQ oscilloscope or waveform display to see the amplitude modulation envelope directly in the time-domain waveform.