FACTS · 吉布斯現象 The Gibbs Phenomenon

Every mathematical statement on screen and in the voice, and where it comes from. All numbers on screen are computed by film.py from the same values that draw the picture.

1. The square wave's Fourier series

On screen: f(t) = 4/π Σ_{k=1}^{N} sin((2k−1)ωt)/(2k−1); circles of radius aₙ = 4/(nπ) for odd n (1.273, 0.424, 0.255, 0.182, …); voice: "three times as fast, a third the size", "odd ones only".

• A. V. Oppenheim, A. S. Willsky, Signals and Systems, 2nd ed., Prentice Hall 1997, §3.3 (Fourier series of a square wave).
• Wikipedia, "Square wave", section Fourier analysis: https://en.wikipedia.org/wiki/Square_wave
• The ideal square wave of the opening has the same frequency (1 Hz) and amplitude (±1 V) as the series.

2. The overshoot (Gibbs phenomenon)

On screen: Δ = 0.179 V above 1.000 V, peak 1.179 V; "8.95 % of the 2 V jump"; "overshoot → 8.95 % for every N, even ∞"; voice: "It never gets shorter. About 9%, every time."

• Next to the jump, the partial sums' first maximum tends to (2/π)·Si(π) ≈ 1.17898 for a square wave of amplitude 1, an overshoot of 0.17898, i.e. 8.949 % of the jump (2), for every N; the maximum moves towards the jump (at ωt = π/2N) and narrows, but its height does not go to zero.
• Wikipedia, "Gibbs phenomenon": https://en.wikipedia.org/wiki/Gibbs_phenomenon
• E. Hewitt, R. E. Hewitt, "The Gibbs–Wilbraham phenomenon: an episode in Fourier analysis", Archive for History of Exact Sciences 21 (1979), 129–160.
• Computed in the film (peak(N) in film.py; tools/gibbs.py prints the table): N = 25 → 1.17913 (8.96 %); N = 100 → 1.17900 (8.95 %); N = 400 → 1.17898 (8.95 %). Shown at three decimals (0.179 V, 1.179 V), which stay the same from N = 25 on; the share of the jump appears once N = 400 (8.95 %).

3. What is drawn and what is said

• At the end the sum's upper limit is written ∞; the curve drawn is the 400-term partial sum (at that zoom it is indistinguishable from the limit), and the readout says N = ∞.
• Scales: 0.50 V/div and 250 ms/div at zoom 1 follow from 200 world units per volt, 400 per second and 100-unit divisions; the TIME/DIV and V/DIV readouts follow the camera's zoom.