Travelling Ripple
y = sin(x − t) /(1 + 0.15x²) · 3
Open in the app The dials and keys named below are the app's.
What it draws
sin(x − t) is the wave. Its wavenumber is 1, so crests are 2π ≈ 6.28 apart. The phase depends only on x − t, so the whole shape slides right at exactly one unit per second at Speed 1. A fixed point of the line rises and falls every 6.28 seconds.
Everything else is the window the wave is seen through. 3/(1 + 0.15x²) is a bell that peaks at 3 in the middle and falls to half of that at x = ±√(1/0.15) ≈ ±2.58. By x = ±6 it is down to 0.47.
Why the pattern
The two pieces answer different questions. The sine says where the crests are, the bell says how tall a crest is allowed to be where it stands. Because the bell is a function of x alone it does not move, so nothing here is a travelling pulse.
A crest is born small at the left edge, swells as it climbs the bell, reaches 3 as it crosses the middle and dies away to the right. The ribbon's recent past draws that whole life at once, each slice showing the crest one step back along the same road. A curve 1/(1 + x²) falling off as 1/x² is the Lorentzian, the same profile as a resonance line and the Cauchy distribution.
Try
- Front in the deck: straight on, the bell is the outline the crests never cross.
- Change sin(x − t) to sin(x + t): the same picture running left instead.
- Widen the window with sin(x − t) / (1 + 0.02x²) · 3: half height is now at ±7.07, past the edge, so the crests barely fade at all.
- Raise Span of x to 12: nearly four crests fit, and the bell shrinks them hard at both ends.