Calculus
Fourier Series
Any periodic wave is a sum of pure sines. Stack them as rotating circles (epicycles) and the tip traces out a square, sawtooth or triangle wave as you add harmonics.
Each circle is one harmonic: its radius is that sine's amplitude and it spins at an integer multiple of the base frequency. The stacked tip draws the wave on the right.
What to observe
- A single circle traces a pure sine. Each harmonic you add sharpens the corners, and the flat parts flatten, closing in on the target wave.
- Near every jump a small overshoot refuses to disappear however many harmonics you add: that is the Gibbs phenomenon.
- The triangle wave's amplitudes fall off like 1/k², far faster than the square's 1/k, so it converges to a clean shape with only a handful of circles.
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