# Radial Pond

Surface · z = f(x, y, t)

`z = sin(r · 2 − 3t) /(1 + 0.2r) · 3`

[Open in the app](https://www.wavelace.com/app#p=32) · [This page](https://www.wavelace.com/presets/radial-pond)

### What it draws

Everything in the formula sits inside `r = hypot(x, y)`, the distance from the middle of the sheet. Nothing depends on `x` and `y` separately, so the height is the same all the way round a circle and the sheet is a set of concentric rings.

Inside the sine, the 2 on `r` is the radial wavenumber, so rings sit `2π/2 = π ≈ 3.14` apart. The 3 on `t` is the angular frequency, so a ring takes `2π/3 ≈ 2.09` seconds to be replaced by the next one. Every ring travels outward at `3/2 = 1.5` units per second at `Speed` 1.

### Why it fades

The factor `3/(1 + 0.2r)` is the envelope. It is 3 at the middle, `1.5` at `r = 5` and 1 at `r = 10`, so the swell shrinks as it spreads. Real circular waves fade because the energy a ring carries is spread over a circumference that grows with `r`. That makes the true height fall off like `1/√r` far from the source. Here `1/(1 + 0.2r)` is a tamer stand-in that stays finite at `r = 0`, where `1/√r` would blow up. The middle is the source: the height there is `3 · sin(−3t)`, pumping between `±3`, and the surface meets it in a point rather than a smooth cap.

### Try

- `Top` in the deck: the rings from above, evenly spaced and paling toward the rim as the envelope drops.
- Use the real fall-off, `3 · sin(2r − 3t)/√(1 + r)`: the near rings tower and the far ones flatten faster.
- Shorten the waves, `sin(4r − 3t)/(1 + 0.2r) · 3`: rings `π/2 ≈ 1.57` apart, now creeping out at `3/4` of a unit per second.
- Send them inward, `sin(2r + 3t)/(1 + 0.2r) · 3`: the rings run to the middle, where the envelope makes them tallest.

### Read more

- [Wave equation](https://en.wikipedia.org/wiki/Wave_equation)
- [Phase velocity](https://en.wikipedia.org/wiki/Phase_velocity)
- [Bessel function](https://en.wikipedia.org/wiki/Bessel_function)
- [Capillary wave](https://en.wikipedia.org/wiki/Capillary_wave)