3D Sine Wave as a Phasor Helix: Projection onto Walls and Fourier Synthesis

Setting up the bench...

Presets

Live readouts

Graphs

Line spectrum: amplitude of each harmonic against frequency

Gibbs phenomenon: peak of the partial square-wave sum against N (red: the limit 1.179)

RMS error of the truncated series against the number of harmonics kept (grey: Parseval tail)

Try this

Square wave with 49 harmonics: the overshoot does not shrink (Gibbs). Beats: set f2/f near 1.1. Phase 90 degrees: sine becomes cosine.

Numerical checks

running...

How to use

The glowing curve is the tip of the phasor chain traced against time; the blue trace on the back wall is its vertical shadow (the sine), the orange trace on the floor its horizontal shadow (the cosine). Drag to orbit. Raise the number of harmonics to see a square wave form and the overshoot at the corners stay. Switch on the second tone for beats. Space pauses, R rewinds.

The bench, screens and phasor disc were built in Blender; all dynamics run in JavaScript.

The science

The sine curve. A sinusoid is y(t) = A sin(2 pi f t + phi) + c: amplitude A, frequency f (period T = 1/f), phase phi and offset c. Its slope is A w cos(w t + phi) with w = 2 pi f, its mean is c and its rms value (without the offset) is A / sqrt(2).

Phasors. The sine is the vertical shadow of a point moving on a circle of radius A with angular speed w. Several sines add as arrows placed tip to tail (the chain on the left): the signal is the height of the last tip. In three dimensions the same chain, drawn against time, is a helix and the sine is its projection onto a wall.

Fourier series. Any periodic wave is a sum of harmonics sum a_n sin(n w t + phi). The page uses square: a_n = 4A / (pi n), n odd sawtooth: a_n = (2A / (pi n)) (-1)^(n+1) triangle: a_n = (8A / (pi^2 n^2)) (-1)^((n-1)/2), n odd Parseval's theorem says the mean square of the signal equals sum a_n^2 / 2, so the rms error of a series cut at N is the square root of the omitted tail. Near a jump the partial sums overshoot by about 8.95 % of the jump (the Gibbs phenomenon) however many terms are kept; for a square wave of amplitude A the peak tends to (2/pi) Si(pi) A = 1.179 A.

Honest simplifications. The time axis scrolls and the amplitudes are dimensionless; the beat envelope is only meaningful when the two frequencies are close. RMS and mean are measured over ten periods of the fundamental, which is exact for a single periodic wave and approximate for two incommensurate tones. The Gibbs plot always uses a unit square wave.

Checks run in the page. The numerical rms of a sine equals A / sqrt(2); a numerical integral of one half cycle equals 2A / w; a central-difference derivative equals A w cos(w t + phi); Parseval's sum equals the measured power for the current wave; the partial square-wave peak at N = 199 equals 1.17898.