# Ripple Tank

Surface · z = f(x, y, t)

`z = sin(3 · hypot(x−3,y) − 4t) + sin(3 · hypot(x+3,y) − 4t)`

[Open in the app](https://www.wavelace.com/app#p=33) · [This page](https://www.wavelace.com/presets/ripple-tank)

### What it draws

Two point sources sit on the `x` axis, at `(−3, 0)` and `(3, 0)`. `hypot(x − 3, y)` is the distance from a point of the sheet to the right-hand source, so each term is a circular wave spreading out from its own source. The height is the sum of the two.

The 3 in front of each distance is the wavenumber: crests are `2π/3 ≈ 2.09` apart. The 4 on `t` is the angular frequency, so each set of rings takes `2π/4 ≈ 1.57` seconds to repeat. Both sets travel outward at `4/3 ≈ 1.33` units per second at `Speed` 1. Neither term fades with distance, so the far corners heave as hard as the middle.

### Why the pattern

Where the two distances differ by a whole number of wavelengths the waves arrive in step and the height doubles. Where they differ by half a wavelength more, they cancel and the sheet stays flat for all time. Each of those conditions is a constant difference of distances to two fixed points, which is the definition of a hyperbola with the sources as its foci. The sources are 6 apart, or 2.86 wavelengths, so five antinodal hyperbolae fan out from the pair, with six still nodal curves between them. The middle antinodal one is the `y` axis itself, a straight line.

### Try

- `Top` in the deck: the classic ripple-tank photograph, with the still nodal curves fanning out from between the sources.
- Lengthen the waves, `sin(2 · hypot(x − 3, y) − 4t) + sin(2 · hypot(x + 3, y) − 4t)`: fewer wavelengths between the sources, so fewer nodal curves.
- Bring the sources closer, to `x − 1.5` and `x + 1.5`, and the fan opens up.
- Drop one term and the hyperbolae vanish: a single source is plain circular rings.
- Raise `Span of x, y` to see the far field, where the hyperbolae straighten into rays.

### Read more

- [Ripple tank](https://en.wikipedia.org/wiki/Ripple_tank)
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)
- [Double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment)
- [Huygens–Fresnel principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle)
- [Hyperbola](https://en.wikipedia.org/wiki/Hyperbola)