# Sinc Splash

Wave · y = f(x, t)

`y = sin(4x − t) /(0.4 + abs(x)) · 2`

[Open in the app](https://www.wavelace.com/app#p=6) · [This page](https://www.wavelace.com/presets/sinc-splash)

### What it draws

`sin(4x − t)` sets the ripple. Its wavenumber is 4, so crests are `2π/4 ≈ 1.57` apart with a zero every `π/4 ≈ 0.79`. The phase `4x − t` carries them right at `1/4` unit per second at `Speed` 1, one full cycle at a fixed point every `2π ≈ 6.28` seconds.

Dividing by `0.4 + abs(x)` and doubling gives the height allowed at each place: `5` in the middle, `1.43` at `abs(x) = 1`, `0.31` at `abs(x) = 6`. The middle of the plot swings sixteen times as far as the ends, which is the splash of the name.

### Why the pattern

An envelope falling like `1/abs(x)` is the signature of a sinc, `sin(x)/x`: the same slow decay, side lobes shrinking as a harmonic series rather than dying away. A sinc is what a single slit makes on a distant screen, and what a rectangular pulse looks like once its frequencies are laid out. Its lobes are stubborn for the same reason a Fourier series rings at a jump: a sharp edge in one variable costs a long tail in the other.

Two things here are not a sinc, and both are visible. The `0.4` in the denominator keeps the centre finite instead of a pole, and it also puts a corner there, since `abs(x)` has a kink at zero that the smooth `sin(x)/x` does not. The `−t` sits inside the sine and not in the envelope, so the ripple slides right underneath a standing shape. The tall centre stays put and pumps up and down while the crests file past it.

### Try

- `Front` in the deck: the `1/abs(x)` outline and its corner at the middle are clearest straight on.
- Round the peak with `sin(4x − t) / (0.4 + x²) · 2`: no corner, and the lobes die away far faster.
- Ask for a true sinc, `sin(4x) / x · 2`: it stands still, and at `x = 0` the formula reads `0/0`, so the curve breaks wherever a sample lands there.
- Halve the ripple with `sin(2x − t) / (0.4 + abs(x)) · 2`: crests `π ≈ 3.14` apart, and only a few lobes across the plot.
- Turn `ground` off: the part of the curve below the plane stops being hidden and the tall centre reads as a full swing.

### Read more

- [Sinc function](https://en.wikipedia.org/wiki/Sinc_function)
- [Envelope (waves)](https://en.wikipedia.org/wiki/Envelope_(waves))
- [Diffraction](https://en.wikipedia.org/wiki/Diffraction)
- [Rectangular function](https://en.wikipedia.org/wiki/Rectangular_function)