Chapter 2Adding Waves Together

A single sine wave is simple. But add a couple together and you can already get shapes that don't look sine-like at all. This is the seed of the whole idea: complexity from simple ingredients.

The formula

Watch these numbers move as you drag the dials above:

Nothing fancy — at every instant in time, just add the two heights together. That's it. Do this at every point along the timeline and a new curve appears.

Things worth noticing

• Same frequency, matching phase — set both frequencies equal and phase to 0°: the waves line up crest-to-crest and reinforce each other (constructive interference). Set Wave B's phase to 180° and they cancel out instead.
• Different frequencies — try Wave A at 1 Hz and Wave B at 3 Hz. The sum is no longer a clean sine shape — it has a bump, because the faster wave rides on top of the slower one.
• More waves, more detail — this is the whole trick used later: stack enough sine waves at the right frequencies, amplitudes, and phases, and you can build any repeating shape, however jagged.