# AM Carrier

Wave · y = f(x, t)

`y = sin(20x − 10t) · (1 + 0.8 · sin(x − t))`

[Open in the app](https://www.wavelace.com/app#p=18) · [This page](https://www.wavelace.com/presets/am-carrier)

### What it draws

A fast wave is multiplied by a slow one. The *carrier* is `sin(20x − 10t)`. Its wavenumber is `20`, so crests are `2π/20 ≈ 0.314` apart. Its angular frequency is `10`, so each point repeats every `0.628` seconds at `Speed` 1. Together the two carry a crest right at `10/20 = 0.5` units per second.

The *envelope* is `1 + 0.8·sin(x − t)`, one long wave `2π ≈ 6.28` across that runs right at `1` unit per second. It swings between `0.2` and `1.8`, so the loud parts stand nine times as tall as the quiet ones. The `0.8` is the modulation index, and below `1` the line never falls silent.

### The sidebands

Multiplying two sines is the same as adding two more, and that is the whole of amplitude modulation. Expanding the product gives `sin(20x − 10t) + 0.4·cos(19x − 9t) − 0.4·cos(21x − 11t)`: the carrier untouched, plus one wave just below its wavenumber and one just above. Those are the sidebands, and they are where a radio signal's information sits. The carrier itself never changes. Their spacing from it, `1` in wavenumber and `1` in frequency, is the envelope's own.

### Two speeds

Watch a single crest and it slides through the bulge that carries it. Crests travel at `0.5` and the bulge at `1`, so a crest is born at the back of a loud patch, crosses it and dies at the front. The first speed is the phase velocity `ω/k`, the second the group velocity `dω/dk`, and the three components give it directly: `(11 − 9)/(21 − 19) = 1`. A medium in which those two differ is dispersive, and this formula is the plainest picture of what that means.

### History

Amplitude modulation carried the first voices over radio. Reginald Fessenden sent speech about a mile at Cobb Island in December 1900, and on Christmas Eve 1906 made the first AM entertainment broadcast.

### Try

- `Front` in the deck: side on, where the envelope reads as the outline of the ink.
- Overmodulate by changing `0.8` to `1.2`: the envelope dips below zero, the carrier flips phase where it does, and the outline crosses itself instead of pinching.
- Hold the carrier still with `sin(20x)·(1 + 0.8·sin(x − t))`: the crests stop dead and only the bulge travels, which separates the two motions.
- Halve the envelope's wavelength, `sin(20x − 10t)·(1 + 0.8·sin(2x − 2t))`: twice as many bulges, and sidebands at `18` and `22`.

### Read more

- [Amplitude modulation](https://en.wikipedia.org/wiki/Amplitude_modulation)
- [Sideband](https://en.wikipedia.org/wiki/Sideband)
- [Group velocity](https://en.wikipedia.org/wiki/Group_velocity)
- [Modulation index](https://en.wikipedia.org/wiki/Modulation_index)
- [Reginald Fessenden](https://en.wikipedia.org/wiki/Reginald_Fessenden)