Damped Pulse
y = exp(−0.1x²) · sin(6x − 4t)
Open in the app The dials and keys named below are the app's.
What it draws
sin(6x − 4t) is the wave inside the pulse. Its wavenumber is 6, so crests sit 2π/6 ≈ 1.05 apart. The 4t makes a fixed point of the line rise and fall every 2π/4 ≈ 1.57 seconds at Speed 1. The phase 6x − 4t holds still for a crest moving right at 4/6 ≈ 0.67 unit per second.
exp(−0.1x²) is the Gaussian bell that shapes it. It is 1 at the middle and half height at x = ±2.63, down to 1/e at ±3.16 and to 0.027 by ±6. Five crests fit inside the half-height width, and beyond it the line lies flat.
Why the pattern
The bell holds only x, so it is nailed to the plot: the pulse is a window, not a parcel. Crests are made at the left edge of the bell, grow as they climb it, cross the middle at full height and shrink away on the right. All of them travel at the same steady 0.67 unit per second.
That is the difference between this and a real pulse on a string or in a fibre, where the carrier and its envelope travel together and the shape moves as one body. Writing exp(−0.1 · (x − 0.67t)²) for the bell puts the envelope on the same journey as the crests and turns the picture into a genuine travelling packet.
Try
- Front in the deck: the bell is the outline the crests grow into and out of.
- Send the window along with the wave, exp(−0.1 · (x − 0.67t)²) · sin(6x − 4t): one pulse crosses the plot and leaves.
- Tighten the bell to exp(−0.4x²) · sin(6x − 4t): half height at ±1.32, and only a couple of crests survive.
- Flip the sign of the 4t to send the crests left through the same window.
- Raise Ribbon depth: the ribbon's recent past fills with the diagonal stripes the crests leave behind as they cross the fixed bell.