# Dispersion

Wave · y = f(x, t)

`y = Σ_(n=1)⁸ ((sin(n · x − n^(a) · t))/n)`

[Open in the app](https://www.wavelace.com/app#p=132) · [This page](https://www.wavelace.com/presets/dispersion)

### What it draws

Stop the clock and the sum is `sin(x) + sin(2x)/2 + … + sin(8x)/8`, the first eight harmonics of a sawtooth with the nth weighted by `1/n`. One tooth takes `2π ≈ 6.28` in `x`, so a little under two of them cross the plot, and the ramp peaks at `≈ 1.67`.

The clock enters as `n^a·t`. Rewriting the nth term as `sin(n·(x − n^(a−1)·t))` says what that costs: the term has wavenumber `n` and frequency `n^a`, so it slides at `n^(a−1)`. At the opening value `a = 2` the fundamental moves one unit a second at `Speed` 1 and the eighth harmonic moves eight, so the fine ripple races out of the ramp it belongs to.

### The dispersion relation

The letter is an exponent in a law, `ω(k) = k^a`, which is what a physicist calls the dispersion relation of a medium. It fixes the phase speed of every component at `ω/k = k^(a−1)`, and `a = 1` is the single exponent that gives them all the same speed.

A narrow band of wavenumbers travels as a packet at `dω/dk = a·k^(a−1)`, which is `a` times the phase speed. At `a = 2` an envelope therefore outruns its own crests two to one, the classic sign of a dispersive medium.

### Rigid, reviving, or neither

At `a = 1` the curve at time `t` matches the starting curve shifted by `t` to machine precision: the sawtooth simply translates. At any whole `a` of zero or more every frequency `n^a` is itself a whole number, so every term has period `2π` in `t` and the smeared shape reassembles exactly every `2π ≈ 6.28` seconds. That return is a revival. Give `a` a fraction and the eight frequencies share no common period, so the shape never comes back at all.

### Try

- Drag `a` to 1: every harmonic takes the same speed and the teeth slide sideways without changing shape, which is the motion the preset Saw Cascade draws in closed form.
- Drag `a` to 3: the eighth harmonic now runs at `8² = 64` units a second, a blur on top of a ramp that still rebuilds itself every `2π`.
- Drag `a` to 1.5: the same violent smearing, but nothing ever returns. Compare the curve a few seconds apart and no two are alike.
- Drag `a` to 0: every term now has frequency 1, so the whole sum oscillates with one period while the nth harmonic creeps along at `1/n`.
- Press `Rewind`, then `Play`, and count six seconds at `Speed` 1: the revival lands just after, at `2π`.

### Read more

- [Dispersion relation](https://en.wikipedia.org/wiki/Dispersion_relation)
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Group velocity](https://en.wikipedia.org/wiki/Group_velocity)
- [Sawtooth wave](https://en.wikipedia.org/wiki/Sawtooth_wave)