Square Fourier
y = sin(x−t) + sin(3(x−t))/3 + sin(5(x−t))/5 + sin(7(x−t))/7
Open in the app The dials and keys named below are the app's.
What it draws
Every term is a sine of the same quantity x − t, so the shape is rigid and slides right at exactly one unit per second at Speed 1. It repeats every 2π ≈ 6.28 units and, at a fixed point, every 6.28 seconds. What the four terms build is a square wave. The fundamental has wavenumber 1, then the third harmonic at a third the height, the fifth at a fifth, the seventh at a seventh, and no even harmonics at all. Each one steepens the rise and flattens the top a little further.
Why the pattern
Taking the odd harmonics on for ever, sin(u) + sin(3u)/3 + sin(5u)/5 + … sums to exactly π/4 ≈ 0.785 for every u between 0 and π, and to −π/4 on the other half. That is a flat step which jumps at x − t = 0 and ±π. Four terms already give the flat stretches and the near-vertical sides, and the visible wobble on the plateau is the tail of the series that has been left out.
The dents and horns beside each jump are worth a look. The tallest ripple, just beside the step, reaches 0.930 against a plateau of 0.785, an overshoot of eighteen per cent. Adding terms pushes that horn closer to the jump and makes it narrower, but never shorter: with six terms it still stands at 0.928. This is the Gibbs phenomenon, and it is why a square pulse pushed through a filter comes out ringing.
History
Fourier's claim that any shape is a sum of sines came with the heat theory he published in 1822. Henry Wilbraham noticed the overshoot in 1848. Josiah Willard Gibbs described it in a note to Nature in 1899, and Maxime Bôcher analysed it and gave it Gibbs's name in 1906.
Try
- Front in the deck: the flat tops and the horns beside each jump read best straight on.
- Add the next harmonic, sin(x − t) + sin(3(x − t))/3 + sin(5(x − t))/5 + sin(7(x − t))/7 + sin(9(x − t))/9: flatter plateau, same horn.
- Go the other way with sin(x − t) + sin(3(x − t))/3 and watch the square dissolve into a wobble.
- Halve Speed and the rigid shape crawls right without changing at all.
- Raise Ribbon depth until the ribbon's recent past covers a whole period, and its far edge repeats its near one.