# Crest Doubling

Wave · y = f(x, t)

`y = sin(x − t) + a · sin(2x − 2t)`

[Open in the app](https://www.wavelace.com/app#p=129) · [This page](https://www.wavelace.com/presets/crest-doubling)

### What it draws

Both terms are travelling sines. `sin(x − t)` is the fundamental: its wavenumber is 1, so its crests stand `2π ≈ 6.28` apart, and it slides right at one unit per second at `Speed` 1. `a·sin(2x − 2t)` is the second harmonic, at twice the wavenumber and twice the frequency, so its crests are only `π ≈ 3.14` apart.

Their speeds are the ratios `1/1` and `2/2`, both 1, so the sum carries its shape along unchanged. The curve at any moment is the curve at the start slid sideways by `t`, to the last digit. The letter `a` is the only thing that alters the shape, and it opens at 1.

### Where the second crest comes from

Write `u = x − t`, since only that combination matters. The slope of the sum is `cos(u) + 2a·cos(2u)`, and where the fundamental crosses zero going down, at `u = π`, it reads `−1 + 2a`. Below `a = 0.5` that is negative, the curve is still falling there, and each period holds one crest.

Above 0.5 the slope at that crossing turns positive, and a second maximum exists. The count per period goes from one to two at exactly `a = 0.5`, where the curve is flat to second order at the crossing, and never falls back. The arithmetic runs well ahead of the eye. The new crest is 0.1% of the main one at `a = 0.51`, 3% at 0.6 and 12% at 0.8, which is about where a reader first sees it as a crest rather than a kink. At the opening `a = 1` it stands at 0.37 against 1.76, and by `a = 10` the pair is nearly level, 9.30 against 10.71. The sign of `a` is immaterial, since −0.6 draws the same two crests mirrored left to right.

### Try

- Drag `a` down to 0.4: one crest per period again, a plain wave leaning forward, with no trace of the shoulder.
- Drag `a` to 0.8: the second crest at last reads as one, 0.19 against the main crest's 1.57.
- Drag `a` to 10: two crests of almost equal height, though the peak has gone from 1.76 to 10.71, so lower `Amplitude` to 0.1 to bring it back inside the frame.
- Change `2x − 2t` to `3x − 3t`: the third harmonic splits the crest itself into a matched pair, and leaves a small bump in the trough rather than a new crest. The threshold drops to `1/9 ≈ 0.11`, so `a` at 0.1 gives one crest and 0.2 gives two.
- `Top` in the deck: looking straight down, every ridge in the ribbon's recent past is a straight diagonal line, which is what rigid travel looks like from above.

### Read more

- [Harmonic](https://en.wikipedia.org/wiki/Harmonic)
- [Fourier series](https://en.wikipedia.org/wiki/Fourier_series)
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle)