Wave Packet
y = exp(−0.2x²) · sin(15x + 5t)
Open in the app The dials and keys named below are the app's.
What it draws
sin(15x + 5t) is the carrier. Wavenumber 15 puts its crests 2π/15 ≈ 0.42 apart, the finest ripple any of the wave presets draws. The +5t sends them left at 5/15 ≈ 0.33 unit per second at Speed 1, while a fixed point cycles every 2π/5 ≈ 1.26 seconds.
exp(−0.2x²) gathers them into a hump. It is at half height at x = ±1.86 and 1/e at ±2.24, and under a thousandth of full height by ±6. Nine crests live inside the half-height width.
The physics behind the shape
This is the picture a textbook draws for a particle that is somewhere rather than everywhere. A narrow band of wavenumbers around k = 15 is added together, cancelling away from the middle and reinforcing inside the hump. The narrower the hump in x, the broader the band of k it takes to build it, which is the whole content of the uncertainty relation.
What a real packet then does, this formula does not. Left to itself under the Schrödinger equation each wavenumber travels at its own speed, so the hump moves at the group velocity and slowly spreads while the crests inside it slip through at a different rate. Here the hump is a fixed function of x and only the crests move. Schrödinger looked for packets that keep their shape in 1926 and found them for the harmonic oscillator alone; a free packet always spreads.
Try
- Front in the deck, then Pause and Step: the crests slide left, the hump does not.
- Send the hump with them, exp(−0.2 · (x + 0.33t)²) · sin(15x + 5t): now the packet itself travels.
- Reverse the carrier with exp(−0.2x²) · sin(15x − 5t) to run the crests right instead.
- Raise Ribbon depth: the leaning stripes in the ribbon's recent past are the crests' path through the standing hump.
- Widen the hump with exp(−0.05x²) · sin(15x + 5t): more crests, and a shape closer to one pure wavenumber.