Fourier Series: Convergence, Epicycles and the Gibbs Overshoot

A Fourier-series playground. Pick a target wave and watch its N-term partial sum converge to it, while a chain of rotating vectors (epicycles) draws exactly the same curve at its tip. Near a jump the partial sum always overshoots by the Wilbraham-Gibbs fraction (about 8.95 percent of the jump); raising N only makes the overshoot narrower, never smaller, which the convergence panel shows as a flat line. The smooth triangle has no jump and no overshoot. Parseval energy in the coefficients climbs to the function mean square. Away from a jump the partial sum converges, at a jump it sits at the average of the two sides, Parseval's energy is recovered, and the Wilbraham-Gibbs overshoot stays at about 8.95 percent of the jump no matter how many terms are added.

Figure 1. Any periodic shape is a sum of sines and cosines. Add more terms and the sum gets closer to the target, except right at a jump, where it always overshoots by about 8.9 percent of the jump no matter how many terms you add (the Gibbs phenomenon, the overshoot only gets narrower). The same sum is the tip of a chain of rotating vectors (epicycles), and Parseval's theorem says the energy in the coefficients equals the energy in the function. Method: analytic Fourier coefficients, partial sums and the Wilbraham-Gibbs constant; Canvas2D, deterministic.

target wave

number of terms N8

WHAT TO TRY

Vary each control and watch the rail readouts respond.

Compare the diagnostic plot against the live scene.