Build periodic waveforms from sine and cosine harmonics. Visualize square, sawtooth, and triangle wave approximations term by term.
The Fourier Series Visualizer builds a periodic curve by adding a finite number of sine and cosine harmonics. Square, sawtooth, and triangle presets expose how each additional term changes the partial sum, including its edges, symmetry, and fine detail. Harmonics occur at integer multiples of the fundamental frequency, while their coefficients determine amplitude and sign. The most common surprise appears near a square or sawtooth discontinuity: adding terms makes the affected region narrower, but the peak overshoot does not shrink to zero. This is the Gibbs phenomenon.
At each horizontal coordinate x, the visualizer evaluates a partial sum of the form S_N(x) = a0/2 + sum from n = 1 through N of [a_n cos(n x) + b_n sin(n x)]. The constant term sets the mean level. Each a_n or b_n controls the cosine or sine contribution at the nth multiple of the fundamental.
The waveform presets differ in which harmonics they include and how quickly their coefficients decrease. In the usual centered forms, square and triangle waves use odd harmonics, while a sawtooth uses every positive integer harmonic. Square and sawtooth magnitudes decrease like 1/n; triangle magnitudes decrease like 1/n^2. Signs and sine-versus-cosine placement depend on the chosen phase origin.
Truncating a Fourier series produces oscillation near a discontinuity. Adding harmonics confines the strongest ripples to a narrower region and improves the approximation away from the jump, but the limiting overshoot remains a fixed fraction of the jump size. This Gibbs behavior is expected and does not, by itself, indicate incorrect harmonic coefficients.
No. A Fourier series represents a periodically repeated function. You can treat a finite interval as one period, but its left and right endpoints are then joined when the interval repeats. If those endpoints do not meet, the periodic extension contains a jump, and a finite partial sum will show ringing near that join.
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