# Huygens Slits

Surface · z = f(x, y, t)

`z = ∫₋₃⁻² cos(2 · hypot(x − u, y) − 2t) du + ∫₂³ cos(2 · hypot(x − u, y) − 2t) du`

[Open in the app](https://www.wavelace.com/app#p=99) · [This page](https://www.wavelace.com/presets/huygens-slits)

### What it draws

The height is two integrals, taken numerically at every mesh point: `z = ∫₋₃⁻² cos(2·hypot(x − u, y) − 2·t) du + ∫₂³ cos(2·hypot(x − u, y) − 2·t) du`. Each is a slit on `y = 0`, from `x = −3` to −2 and from 2 to 3: width 1, centres 5 apart. `hypot(x − u, y)` is the distance to the source at `(u, 0)`, so each integrand is a circular wave and the sheet sums every source of both slits.

The 2 in front of the distance is the wavenumber, so crests are `π ≈ 3.14` apart. The 2 on `t` is the angular frequency, a period of `π ≈ 3.14` seconds at `Speed` 1, and crests travel at speed 1. Nothing fades and the sources radiate both ways, so the pattern repeats below the slits.

### Why the pattern

Two effects multiply. The slits interfere like two ripple-tank sources: where the distances to them differ by a whole wavelength the wavelets add, half a wavelength more and they cancel. Bright fringes leave the pair where `sin φ = m·λ/d`, with `φ` from the `y` axis and `λ/d ≈ 0.63`. That is a central fringe, a second maximum near 39° each side, and dark rays near 18° and 70°. At span 8 the sheet ends ≈ 11 units out, where the dark rays sit ≈ 1° wider.

Each slit alone diffracts as a slit of width 1, the fringes' envelope. Its first null would sit at `sin φ = λ/1 ≈ 3.14`, beyond any direction, so every fringe carries through: the second maximum is ≈ 0.94 of the central one. Near the slits the sum is the rippled near field of Fresnel.

### History

Thomas Young argued for the wave nature of light before the Royal Society in 1801, the fringes of two slits his evidence, in the Bakerian lecture printed in 1804. His sources were the wavelets Huygens had proposed in 1678. Fresnel showed in 1818 that summing them with their phases, as this formula does, accounts for fringes and diffraction alike.

### Try

- `Top`: the fringes leave the pair as straight rays, a central one and one on each side near 39°, mirrored below the slits.
- Move the slits apart, `integral(cos(2·hypot(x − u, y) − 2·t), u, −4, −3) + integral(cos(2·hypot(x − u, y) − 2·t), u, 3, 4)`. Centres 7 apart crowd the fringes: maxima near 27° and 64°, two on each side instead of one.
- Drop the second integral, `integral(cos(2·hypot(x − u, y) − 2·t), u, 2, 3)`: one slit of width 1, no fringes and no null, a fan still ≈ 0.84 of its axis height at 90°.
- Halve the wavelength, both 2s in front of hypot to 4: `λ/d` halves to ≈ 0.31 and three maxima fit each side, the last near 70°.

### Read more

- [Double-slit experiment](https://en.wikipedia.org/wiki/Double-slit_experiment)
- [Wave interference](https://en.wikipedia.org/wiki/Wave_interference)
- [Huygens–Fresnel principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle)
- [Diffraction](https://en.wikipedia.org/wiki/Diffraction)
- [Thomas Young (scientist)](https://en.wikipedia.org/wiki/Thomas_Young_(scientist))