Fourier Series
Every repeating wave is a stack of pure sines. Add harmonics one at a time and watch a square wave assemble itself — bumps, ringing, and all.
Targets repeat every 2π. At jumps the target reads 0 — the midpoint the series converges to.
Try this
- Take the square wave from 1 harmonic to 25 and watch the corners sharpen.
- Toggle parts on and see which sine does the heavy lifting.
- Drag the probe into a jump and compare target vs sum.
How Fourier series work here
Any periodic wave can be written as a sum of sines and cosines — one per harmonic. This lab uses the exact textbook formulas for three classic waves, so each harmonic you add is the mathematically correct next term.
Near a jump, the sum always overshoots by about 9% no matter how many harmonics you add. That is the Gibbs phenomenon — a theorem about partial sums, not a rendering bug. The marker shows it; the lab never hides it.
Hear the math behind the music
ManySpace turns ideas like this into guided lessons.