# Standing Wave

Wave · y = f(x, t)

`y = sin(2x) · cos(2t) · 1.5`

[Open in the app](https://www.wavelace.com/app#p=8) · [This page](https://www.wavelace.com/presets/standing-wave)

### What it draws

The formula is a product of one factor in `x` and one in `t`, and the two never mix. `sin(2x)` fixes the shape. The `2` is the wavenumber, so crests are `2π/2 = π ≈ 3.14` apart and the curve crosses zero every `π/2 ≈ 1.57`. Across `±6` that puts seven crossings on the plot, at `0, ±1.57, ±3.14, ±4.71`, with eight peaks and troughs alternating between them.

`cos(2t)` fixes the timing. Every point returns to where it started after `2π/2 = π ≈ 3.14` seconds at `Speed` 1, and the trailing `1.5` is the height of the tallest crest.

### Why the pattern

Because `x` and `t` sit in separate factors, nothing travels. The curve swells and collapses in place, and the seven crossings stay pinned at zero for all time. Those are the *nodes*, and the eight points of greatest swing are the *antinodes*.

The identity `sin(2x)·cos(2t) = ½sin(2x − 2t) + ½sin(2x + 2t)` says where the stillness comes from. This one curve is two identical waves passing through each other in opposite directions, each moving at `2/2 = 1` unit per second. A node is where the two are always out of step, an antinode where they always agree. Twice per period, at `t = π/4, 3π/4, …`, `cos(2t)` is zero and the whole line is momentarily straight, with all of its energy in the motion rather than the shape. A plucked string, an organ pipe and a particle in a box stand still for this reason.

### History

Michael Faraday described standing waves in 1831, on the surface of a liquid in a shaking vessel. Franz Melde produced them on a string driven by a tuning fork in 1859 and gave them the name *stehende Welle*.

### Try

- `Front` in the deck: straight on, the textbook figure, with the nodes as the points that never leave the line.
- Write the two travelling waves out in full, `sin(2x − 2t)·0.75 + sin(2x + 2t)·0.75`: the picture does not change.
- Then drop one of them. `sin(2x − 2t)·0.75` alone slides steadily to the right and the nodes are gone.
- Change `sin(2x)` to `sin(3x)` for eleven nodes, `π/3 ≈ 1.05` apart, at the same rate of flapping.
- Raise `Ribbon depth` until the ribbon's recent past covers one whole cycle, and its far edge repeats its near one.

### Read more

- [Standing wave](https://en.wikipedia.org/wiki/Standing_wave)
- [Melde's experiment](https://en.wikipedia.org/wiki/Melde%27s_experiment)
- [Node (physics)](https://en.wikipedia.org/wiki/Node_(physics))
- [Superposition principle](https://en.wikipedia.org/wiki/Superposition_principle)
- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation)