Radial Pond
z = sin(r · 2 − 3t) /(1 + 0.2r) · 3
Open in the app The dials and keys named below are the app's.
What it draws
Everything in the formula sits inside r = hypot(x, y), the distance from the middle of the sheet. Nothing depends on x and y separately, so the height is the same all the way round a circle and the sheet is a set of concentric rings.
Inside the sine, the 2 on r is the radial wavenumber, so rings sit 2π/2 = π ≈ 3.14 apart. The 3 on t is the angular frequency, so a ring takes 2π/3 ≈ 2.09 seconds to be replaced by the next one. Every ring travels outward at 3/2 = 1.5 units per second at Speed 1.
Why it fades
The factor 3/(1 + 0.2r) is the envelope. It is 3 at the middle, 1.5 at r = 5 and 1 at r = 10, so the swell shrinks as it spreads. Real circular waves fade because the energy a ring carries is spread over a circumference that grows with r. That makes the true height fall off like 1/√r far from the source. Here 1/(1 + 0.2r) is a tamer stand-in that stays finite at r = 0, where 1/√r would blow up. The middle is the source: the height there is 3 · sin(−3t), pumping between ±3, and the surface meets it in a point rather than a smooth cap.
Try
- Top in the deck: the rings from above, evenly spaced and paling toward the rim as the envelope drops.
- Use the real fall-off, 3 · sin(2r − 3t)/√(1 + r): the near rings tower and the far ones flatten faster.
- Shorten the waves, sin(4r − 3t)/(1 + 0.2r) · 3: rings π/2 ≈ 1.57 apart, now creeping out at 3/4 of a unit per second.
- Send them inward, sin(2r + 3t)/(1 + 0.2r) · 3: the rings run to the middle, where the envelope makes them tallest.