Sinc Splash
y = sin(4x − t) /(0.4 + abs(x)) · 2
Open in the app The dials and keys named below are the app's.
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.