Chapter 1Sine Waves 101
Every Fourier idea starts with one shape: the sine wave. Before combining them, get a feel for the three knobs that define a single one.
A sine wave is what you get when you track the height of a point spinning around a circle, over time. That's the whole picture on the left below — the dot spins, and its height traces the wave on the right.
Click and drag anywhere in the circle — the arm snaps to your cursor, setting amplitude and phase. Click and drag the wave itself — stretch it side to side to change its frequency.
Rotating point / wave
Connecting line (same instant, same height)
The formula
Every point on that curve comes from one equation — watch the numbers below update live as you drag:
Three numbers, three jobs:
- Amplitude ($A$) — how tall the wave is. It's the radius of the circle: a bigger circle, a bigger swing.
- Frequency ($f$) — how fast the point spins, in cycles per second (Hz). Higher frequency means the wave repeats itself more often in the same span of time.
- Phase ($\varphi$) — where the point starts on the circle. It shifts the whole wave left or right without changing its shape.
Why circles? This connection matters later: a "frequency" is really just a spinning speed, and in Chapter 4 you'll see a whole Fourier series as nothing more than several circles spinning at once, each one's tip riding on the last.
Try it
Drag each slider on its own and watch which part of the wave changes. Notice that frequency and phase change the timing of the wave, while amplitude changes its size — the wave's basic shape (a smooth S-curve) never changes.