Fourier Series Builder Simulation

Any repeating wave, however sharp its corners, can be written as a sum of plain sine waves whose frequencies are whole-number multiples of the wave's own. Which sines, and how loud, is set by the shape: a square wave needs only the odd harmonics with amplitudes falling as 1/k, a sawtooth needs all of them with the sign flipping each time, and a triangle needs the odd ones falling much faster, as 1/k². Adding terms always reduces the average error, and each harmonic removes exactly its own share of it. But at a jump the sum never settles down: the ripple beside the discontinuity gets narrower as terms are added while staying about 17.9% taller than the wave itself — 8.95% of the full jump. That is the Gibbs phenomenon, and it is a property of the series, not of the drawing. Students can change harmonics and target wave and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSF.TF.B.5, CCSS.MATH.CONTENT.HSF.BF.A.1.B

• Add harmonics one at a time to a square wave and watch each new sine flatten a bit more of the top
• Push the harmonic count to thirty and check whether the spike at the corner has got any shorter
• Switch to the triangle wave and see the overshoot vanish, because a corner is not a jump
• Compare the RMS error of the triangle and the square at the same harmonic count, and link the gap to 1/k² against 1/k
• Watch the chain of circles and check that the height of the last one is always the value of the wave

1. 1Predict how changing harmonics will affect the highest harmonic, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask