This visualiser was generated by Claude (Anthropic) as part of the Wireless 101 course materials. Every sample of the sum, the beat frequency, the carrier, the envelope, the combined peak amplitude and the period are computed live from the formulas printed at the foot of the page — the sum waveform is added point by point, never sketched. Three kinds of number appear, each labelled where it is used: computed (everything the plot and the readouts show), published figure (none — the only constant here is 2π), and illustrative (the on-screen frequencies are a few hertz so the carrier and the beat are both visible; the animation rate is for the eye). It does not measure anything and it does not call out to a server.
The trigonometric identity behind beats: sin(2πf₁t) + sin(2πf₂t) = 2 cos(2π·(Δf/2)·t) sin(2π·f̅·t), with f̅ = (f₁+f₂)/2 and the beat at Δf = |f₁−f₂|. Course source: M2-L3 superposition and beats.
Add two or three sinusoids and watch their sum. Slide two frequencies close together and a slow beat appears at |f₁−f₂|; set two equal tones in phase and they add, anti-phase and they cancel. The waveform, its envelope and every amplitude are read off the computed samples — nothing is drawn by hand.
Each component is one sine. They are summed sample by sample into one waveform; when two frequencies are close, the sum shows a beat whose envelope is drawn from the same samples. Every thumbnail is computed — nothing here is an icon.
The thin coloured lines are the individual components A sin(2πf t + φ); the thick line is their sum. When exactly two components are active and their amplitudes are equal, the dashed red curve is the beat envelope ±2A cos(2π·(Δf/2)·t), which touches the sum at every peak. The time axis is auto-ranged to show a few beat periods.