Beats
About this app
Two almost equal tones: the loudness pulses Δf times per second, not Δf/2. Phasors, the envelope with a bracket T = 1/Δf, the spectrum with the auditory filter and a perception axis show when the pulsing tone turns rough and, beyond the critical bandwidth, into two separate tones. Eight acoustic examples (tuning a guitar, accordion musette, minor second, low and high third, fifth …), all audible.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
What a beat is
Two sine tones with frequencies f₁ and f₂ add up to sin(2πf₁t) + sin(2πf₂t) = 2·cos(πΔf·t)·sin(2πf̄t). If the frequencies are close, you hear one tone at the mean frequency f̄ = (f₁ + f₂)/2 whose loudness swells and fades. Because the ear hears the magnitude of the envelope, there are Δf = f₂ − f₁ loudness maxima per second, not Δf/2. The beat period is T = 1/Δf.
What the app shows
Four cards show the same superposition: a magnifier with a phasor picture in which tone 2 turns against tone 1, the sum signal with its envelope and a bracket T = 1/Δf, the spectrum with the auditory filter and the perceived pitch, and a perception axis with the zones “one tone, pulsing”, “rough” and “two tones”. You can listen to both tones; in the 2-second window picture and sound run together.
Acoustic examples to try
Eight examples lead through the zones: unison and cancellation, tuning a guitar 2 Hz apart, the accordion tremolo of musette tuning, the minor second as the roughest interval, the fifth as two separate tones, and the same major third low and high. “Slowly pull apart” increases Δf from 0 to 250 Hz in 24 seconds, so you hear the transition from pulsing through roughness to two tones.
Frequently asked questions
- How do you calculate the beat frequency?
- The beat frequency is the difference between the two frequencies: f_beat = |f₁ − f₂|. Two tones at 440 Hz and 442 Hz therefore produce 2 beats per second. The envelope 2·cos(πΔf·t) oscillates at Δf/2, but its magnitude, and so the loudness, has Δf maxima per second.
- When do you hear two tones instead of beats?
- Below a spacing of about 15 Hz you hear one tone with pulsing loudness. Above that it sounds rough as long as both tones fall into the same auditory filter. Only when the spacing exceeds the critical bandwidth of hearing, about 114 Hz at 440 Hz, do you hear two separate, smooth tones.
- What is the critical bandwidth?
- The inner ear splits sound into critical bands that act like band-pass filters. Their width is called the critical bandwidth; below 500 Hz it is about 100 Hz, above that it grows, to about 160 Hz at 1 kHz. The ear cannot hear tones within one critical band separately; they produce beats or roughness.
- What can I try in the app?
- You set the frequency of tone 1, the spacing Δf, and the loudness and phase of tone 2, listen to both tones together and see the envelope, phasors, spectrum and perception zone. Eight acoustic examples and a slow pull-apart show the transition from one tone to two.
Subject: School physics
More from School physics
- The exposure triangle
- Density and buoyancy
- The Doppler effect
- Additive and subtractive colour mixing
- Newton’s cradle and collisions
- Converging lenses and ray diagrams
- Projectile motion
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