Devansh Mittal
Devansh Mittal
Physics Educator and Mentor • Lead Developer
Acoustic Beats: Superposition of Sound Waves
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1. How to use this App User Guide & Experimentation
  1. Choose Sound 1 Frequency: Select the base musical pitch for Sound 1 from the top row: do (262 Hz), re (293 Hz), mi (329 Hz), fa (349 Hz), or so (392 Hz).
  2. Configure Sound 2 Offset: In the middle row, choose the plus (+) or minus (−) sign and select the frequency difference (Δf) ranging from 0 Hz up to 20.0 Hz. Sound 2 frequency is automatically computed and displayed as f2 = f1 ± Δf.
  3. Audition Individual Tones: Click the "Sound 1 only" (blue) or "Sound 2 only" (red) button to listen to each pure tone in isolation for 3 seconds. Notice that when Δf is small (e.g., 1 Hz), the two pitches sound almost indistinguishable when played separately.
  4. Experience Acoustic Beats: Click the "Mixing sounds 1 & 2 (duration: 10 s)" (magenta) button to play both sound waves simultaneously. Listen to the rhythmic pulsing (waxing and waning) of loudness.
  5. Measure the Beat Frequency: Time the beat pulsations using a stopwatch. Notice that setting Δf = 1.0 Hz produces exactly 1 loudness maximum per second, while Δf = 2.0 Hz produces 2 beats per second, directly verifying the formula fbeat = |f1 − f2|.
2. General theory related to the Physics of the idea of the App Theory & Acoustics
Superposition of Two Harmonic Sound Waves:
y(t) = y1(t) + y2(t) = A sin(2π f1 t) + A sin(2π f2 t)
y(t) = [ 2A cos(2π ((|f1 − f2|) / 2) t) ] · sin(2π ((f1 + f2) / 2) t)

• The Phenomenon of Acoustic Beats: When two sound waves with identical amplitudes and slightly different frequencies (f1 and f2) travel in the same medium, they continuously shift in and out of phase with each other. Constructive interference produces a surge in sound intensity (loudness peak), followed by destructive interference where the waves cancel, causing a periodic waxing and waning of loudness called beats.

• Carrier Frequency vs. Modulated Envelope: The resulting acoustic wave consists of a rapid carrier vibration at the average frequency (fcarrier = (f1 + f2) / 2), which determines the overall musical pitch heard by the ear, contained inside a slowly oscillating amplitude envelope that fluctuates at half the frequency difference ((|f1 − f2|) / 2).

• Beat Frequency and Beat Period: Because sound intensity is proportional to the square of the amplitude, the perceived sound volume reaches a maximum twice in each envelope cycle (at both the positive and negative envelope peaks). Consequently, the beat frequency (number of intensity peaks per second) is:

fbeat = |f1 − f2| = Δf,   Tbeat = 1 / fbeat = 1 / |f1 − f2|

• Auditory Range & Real-World Applications:
– For small frequency differences (Δf < 10 Hz), the human ear hears a single smooth throbbing tone ("wah-wah-wah").
– For moderate differences (Δf between 10 Hz and 20 Hz), the rapid fluttering creates a rough, buzzy auditory sensation.
– Musical Instrument Tuning: Musicians rely heavily on beats to tune pianos, guitars, and orchestral instruments. When playing a reference pitch and an out-of-tune string simultaneously, the musician adjusts string tension until the beat frequency slows down and completely disappears (fbeat = 0 Hz), indicating exact unison.