Beat Interference
y = sin(x − t) + sin(1.1x + 0.7t)
Open in the app The dials and keys named below are the app's.
What it draws
Two waves are added, and neither is remarkable alone. sin(x − t) has wavenumber 1, crests 2π ≈ 6.28 apart, and runs right at one unit per second at Speed 1. sin(1.1x + 0.7t) has wavenumber 1.1, crests 2π/1.1 ≈ 5.71 apart, and runs left at 0.7/1.1 ≈ 0.64 unit per second. They are close enough in wavelength that the sum is not a mess but a slow rhythm.
Why it pulses
Adding two sines gives a product of their half sum and half difference: sin(x − t) + sin(1.1x + 0.7t) = 2 · sin(1.05x − 0.15t) · cos(0.05x + 0.85t). The first factor is what the eye calls the wave, with crests 2π/1.05 ≈ 5.98 apart, creeping right at 0.15/1.05 ≈ 0.14 unit per second.
The second factor is the loudness. Its wavenumber is only 0.05, so across the plot it hardly changes at all, but its 0.85t runs fast. Every π/0.85 ≈ 3.70 seconds it passes through zero and the whole curve nearly flattens, falling from a height of 2 to 0.5, then filling out again.
Watch one point instead of the whole line and the classic beat appears. At x = 0 the two waves are tones of angular frequency 1 and 0.7, and their sum swings inside an envelope 2 · sin(0.15t). It is silent at t = 0, loudest near t = 10.5, and silent again at t ≈ 20.9 seconds. Two piano strings a hair apart in pitch do this, and a tuner listens for the beats to slow to nothing.
Try
- Front in the deck, then wait: the swell and collapse every 3.70 seconds is the whole point.
- Make the two waves match exactly, sin(x − t) + sin(x + t): the beats vanish and a standing wave takes their place, with nodes every π.
- Bring them closer instead, sin(x − t) + sin(1.02x + 0.7t): the crests are longer lived and drift faster.
- Raise Ribbon depth: the ribbon's recent past then holds a whole swell and collapse at once.
- Pause, then Step through a quiet moment to see the two waves cancelling rather than stopping.