Crest Doubling
y = sin(x − t) + a · sin(2x − 2t)
Open in the app The dials and keys named below are the app's.
What it draws
Both terms are travelling sines. sin(x − t) is the fundamental: its wavenumber is 1, so its crests stand 2π ≈ 6.28 apart, and it slides right at one unit per second at Speed 1. a·sin(2x − 2t) is the second harmonic, at twice the wavenumber and twice the frequency, so its crests are only π ≈ 3.14 apart.
Their speeds are the ratios 1/1 and 2/2, both 1, so the sum carries its shape along unchanged. The curve at any moment is the curve at the start slid sideways by t, to the last digit. The letter a is the only thing that alters the shape, and it opens at 1.
Where the second crest comes from
Write u = x − t, since only that combination matters. The slope of the sum is cos(u) + 2a·cos(2u), and where the fundamental crosses zero going down, at u = π, it reads −1 + 2a. Below a = 0.5 that is negative, the curve is still falling there, and each period holds one crest.
Above 0.5 the slope at that crossing turns positive, and a second maximum exists. The count per period goes from one to two at exactly a = 0.5, where the curve is flat to second order at the crossing, and never falls back. The arithmetic runs well ahead of the eye. The new crest is 0.1% of the main one at a = 0.51, 3% at 0.6 and 12% at 0.8, which is about where a reader first sees it as a crest rather than a kink. At the opening a = 1 it stands at 0.37 against 1.76, and by a = 10 the pair is nearly level, 9.30 against 10.71. The sign of a is immaterial, since −0.6 draws the same two crests mirrored left to right.
Try
- Drag a down to 0.4: one crest per period again, a plain wave leaning forward, with no trace of the shoulder.
- Drag a to 0.8: the second crest at last reads as one, 0.19 against the main crest's 1.57.
- Drag a to 10: two crests of almost equal height, though the peak has gone from 1.76 to 10.71, so lower Amplitude to 0.1 to bring it back inside the frame.
- Change 2x − 2t to 3x − 3t: the third harmonic splits the crest itself into a matched pair, and leaves a small bump in the trough rather than a new crest. The threshold drops to 1/9 ≈ 0.11, so a at 0.1 gives one crest and 0.2 gives two.
- Top in the deck: looking straight down, every ridge in the ribbon's recent past is a straight diagonal line, which is what rigid travel looks like from above.