Sound Waves: Calculate the Frequency Difference

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 Concept

When two sound waves with slightly different frequencies are played together, they interfere and create a pulsing effect called beats.

The beat frequency is:

f(beat) = |f₁ − f₂|

Where:

  • f₁ = reference frequency- a common or easy one to use is A440 (440 Hz)
  • f₂ = unknown frequency- should be close to the reference frequency, such as a slightly flat violin A string, or an electronic tone produced steadily (have an adult set it so you don't know the frequency of it)
  • f(beat) = beat frequency (Hz)

    So:
        •    if tones are off by 2 Hz → you hear 2 beats per second 
        •    if tones are off by 0.5 Hz → very slow pulsing 
        •    if perfectly in tune → no beats at all

Overall goal

To determine the frequency difference between two close tones by measuring beat frequency and solving for an unknown frequency.

Need

  • Two stable, sustained sound sources with similar pitch: 
    (examples: tuning app + instrument, two tone generators, tuning fork + app tone, two instruments)
  • Quiet environment
  • Stopwatch
  • A tuner or tuner app that will measure a frequency in Hz 

Safety

  • Keep sound levels at a comfortable listening volume

Procedure

  1. Produce two steady tones that are close in pitch.
    (One is treated as the reference frequency, the other as the unknown.)
    Our family used a computer-produced A440 and a note from a slightly detuned violin A string
  2. Listen for the pulsing “beats” created by interference. Listen for a 'wobble' in the sound rather than a distinct or separate sound. 
  3. Count the number of beats over a fixed time interval (recommended: 10 seconds). It may help to have a second person run the timer so you can focus on the beats.
  4. Repeat at least 2–3 times and average your results.
  5.  Calculate beat frequency:

    f(beat) = (number of beats) ÷ time (seconds)

  6.  Solve for the unknown frequency: 

    f₂ = f₁ ± f(beat)  

    7. Use a tuner app to measure the unknown frequency and see how close you were. 

Observations / Data Table 

  • Reference frequency (f₁): ______ Hz
  • Beats in 10 seconds (trial 1): ______
  • Beats in 10 seconds (trial 2): ______
  • Beats in 10 seconds (trial 3): ______
  • Average beat frequency: ______ Hz
  • Calculated frequency (f₂): ______ Hz or _______ Hz 
    (depending on if it is higher or lower than the reference sound)
  • Actual unknown frequency (do not measure until the end): ______ 

Math Check

Using your measured values:

f(beat) = |f₁ − f₂|

Interpret your result:

  • The math tells you the size of the difference but not whether the unknown is higher or lower — you may have two possible answers.
  • How consistent were your beat counts across trials?
  • What made the beats easier or harder to hear?

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