Breathing Gauss
y = exp(−0.4(x − 3 · sin(t · 0.5))²) · 2
Open in the app The dials and keys named below are the app's.
What it draws
exp(−0.4·u²) is a Gaussian, the bell curve, and the trailing 2 is its height. The 0.4 sets its width: the bump is down to half height at u = √(ln2 / 0.4) ≈ 1.32, so it is 2.63 wide across the half-height points. The matching standard deviation is 1/√0.8 ≈ 1.12.
Everything else is in u = x − 3·sin(0.5t), which says where the bump is centred. The centre is at 3·sin(0.5t), sliding between −3 and 3 and back once every 2π/0.5 = 4π ≈ 12.6 seconds at Speed 1. The shape itself never changes.
The swing
The centre is a sine of the clock, so the bump moves the way a pendulum bob does. Its speed is 1.5·cos(0.5t), fastest through the middle at 1.5 units per second and momentarily still at each end, where it seems to hang before turning back. That is simple harmonic motion, and the ridge of peaks running back through the ribbon's recent past is a piece of that same sine.
Nothing here is a wave. A travelling wave carries a shape at a fixed speed, and a wave packet made of many frequencies spreads out as it goes. This bump is one rigid shape whose position is dictated by hand, so it neither disperses nor decays. It is a good picture of what a wave packet would look like if only it behaved.
Try
- Top in the deck: looking straight down, the ribbon is the graph of the centre against time, one arch of a sine.
- Change 0.4 to 0.1: the same swing under a bump twice as wide, half height at √(ln2 / 0.1) ≈ 2.63 from the centre.
- Make it breathe for real. exp(−0.4·x²)·(1 + cos(t)) pins the bump at the origin and pumps its height between 0 and 2 every 2π ≈ 6.28 seconds.
- Change 3·sin(t·0.5) to t for a bump that runs off to the right at one unit per second and never comes back.
- Raise Ribbon depth to its top: the ribbon then holds most of a full swing, and the ridge bends back on itself.