Bessel-ish
y = cos(x · x · 0.2 − t) / sqrt(1 + abs(x)) · 2
Open in the app The dials and keys named below are the app's.
What it draws
Two parts make the curve: a cosine whose phase is quadratic in x, and an envelope that shrinks it towards the edges.
The phase is 0.2x² − t. Differentiating it in x gives the local wavenumber, 0.4·x, which grows with distance from the origin. Crests are 2π/(0.4x) apart, so 5.24 at x = 3 and 2.62 at x = 6. They keep bunching up the further out one looks, and a signal whose frequency slides like that is a chirp. The envelope is 2/√(1 + |x|): height 2 at the middle, 1 at x = ±3. The absolute value makes the whole curve even in x, symmetric at every instant, with a small kink at the origin where |x| turns the corner.
Crests from the middle
Follow one crest. It keeps its phase, so 0.4x times its speed must equal 1, which gives 2.5/x units per second. Crests well up at the origin and stream outward both ways, quickly at first, then slower as they crowd together far out. At the origin itself the curve pumps up and down as 2·cos(t), once every 2π ≈ 6.28 seconds at Speed 1.
The name is a caution as much as a claim. The Bessel function J₀ is the radial profile of a struck drumhead. For large x it behaves like √(2/(πx))·cos(x − π/4), so its amplitude fades as 1/√x, exactly the rate the envelope here imitates. Its crests, though, are evenly spaced 2π apart, because its phase is linear. This preset borrows the decay and replaces the phase with a chirp, so it looks like a cross-section of spreading ripples without solving anything.
Try
- Make it honest: cos(abs(x) − t)·2/√(1 + abs(x)) is the large-x shape of J₀, with evenly spaced crests moving outward at one unit per second.
- Flatten the chirp with cos(x²·0.05 − t)/√(1 + abs(x))·2: at x = 3 the spacing goes from 5.24 to 20.9.
- Steepen the fade by dropping the root, cos(x²·0.2 − t)/(1 + abs(x))·2: the edges sink to 0.29.
- Push Span of x out to 12 to reach the far field, where the crests are barely 1.3 apart and creep outward at a fifth of a unit per second.