Chirp Wave
y = sin(x² + t) · cos(3x − t)
Open in the app The dials and keys named below are the app's.
What it draws
The formula is a product of two waves. In sin(x² + t) the phase grows like x², so the local wavenumber is its slope, 2x, and crests are π/x apart. That is 3.14 near x = 1 and 0.52 out at x = 6. That widening and narrowing is what a chirp is. The second factor, cos(3x − t), is a plain wave with crests 2π/3 ≈ 2.09 apart, sliding right at 1/3 unit per second.
Why the pattern
A product of a sine and a cosine is a sum of two waves: sin(x² + t)·cos(3x − t) = ½sin(x² + 3x) + ½sin(x² − 3x + 2t). The first half has no t in it at all, so half of what is on screen is frozen. Its local wavenumber is 2x + 3, which vanishes at x = −1.5. There the still half stretches into one long slow bend, while at x = 6 its crests are only 2π/15 ≈ 0.42 apart.
The second half carries all of the motion. Its crests hold x² − 3x + 2t fixed, so each one slides at 2/(2x − 3) towards x = 1.5 from both sides. At x = 6 that is 0.22 unit per second leftward, and at x = −6 it is 0.13 rightward. The eye reads the two halves as one restless texture that streams inward and never repeats across the plot.
Where chirps turn up
A signal whose frequency sweeps is the working tool of radar and sonar. A long chirp carries the energy of a long pulse and, matched against a copy of itself, the timing of a short one. Bats and dolphins sweep their calls for the same reason.
Try
- Front in the deck: straight on, the crowding of the crests towards the right edge is easiest to read.
- Type the split out in full, 0.5 · sin(x² + 3x) + 0.5 · sin(x² − 3x + 2t): the picture does not change.
- Then keep the frozen half alone, 0.5 · sin(x² + 3x). Nothing moves, and the long bend at x = −1.5 stands out.
- Change cos(3x − t) to cos(6x − t): the still point moves to x = −3 and the inward drift to x = 3.
- Drop Speed to 0.3 and watch one crest walk inward instead of the whole texture shimmering.