Building Waves

Differential Equations · Axiom Academy

Discover how any repeating wave — no matter how sharp or jagged — is really just simple sine waves, added together. Every periodic wave — a musical tone, a heartbeat, a radio signal — starts from the same building block: a pure sine wave, the fundamental . Add more sine waves at whole-number multiples of that frequency (the harmonics ), each with the right amplitude, and you can build almost any repeating shape. Watch a plain sine wave grow into something far more complex. Watch the fundamental sine wave gain harmonics one at a time — 1st, then 3rd, then 5th, then 7th — each shrinking in amplitude as its frequency climbs. By the end, the smooth curve has folded itself into the sharp steps of a near-square wave. Each harmonic you see appear is a separate sine wave — , then , and so on — added straight on top of what's already there. Turn Harmonics On and Add Them Up Turn on harmonics 1, 3, 5, 7, and 9 — the odd ones — and set each amplitude to 1/n . Watch the jagged steps of a square wave emerge, and see each harmonic's contribution as a bar in the spectrum below. Notice: only ODD harmonics (1, 3, 5, 7…) build a square wave. Try turning on an even one (2, 4…) — the symmetry breaks and the shape stops looking square. Every Wave Shape Has Its Own Recipe Different wave shapes need different harmonic recipes. Try each preset and watch how many harmonics it takes to get close to the true shape — and watch for the little overshoot ("ringing") at the sharp corners.

This is the written version of the interactive lesson above. See the full Differential Equations course.