Dispersion
y = Σ_(n=1)⁸ ((sin(n · x − n^(a) · t))/n)
Open in the app The dials and keys named below are the app's.
What it draws
Stop the clock and the sum is sin(x) + sin(2x)/2 + … + sin(8x)/8, the first eight harmonics of a sawtooth with the nth weighted by 1/n. One tooth takes 2π ≈ 6.28 in x, so a little under two of them cross the plot, and the ramp peaks at ≈ 1.67.
The clock enters as n^a·t. Rewriting the nth term as sin(n·(x − n^(a−1)·t)) says what that costs: the term has wavenumber n and frequency n^a, so it slides at n^(a−1). At the opening value a = 2 the fundamental moves one unit a second at Speed 1 and the eighth harmonic moves eight, so the fine ripple races out of the ramp it belongs to.
The dispersion relation
The letter is an exponent in a law, ω(k) = k^a, which is what a physicist calls the dispersion relation of a medium. It fixes the phase speed of every component at ω/k = k^(a−1), and a = 1 is the single exponent that gives them all the same speed.
A narrow band of wavenumbers travels as a packet at dω/dk = a·k^(a−1), which is a times the phase speed. At a = 2 an envelope therefore outruns its own crests two to one, the classic sign of a dispersive medium.
Rigid, reviving, or neither
At a = 1 the curve at time t matches the starting curve shifted by t to machine precision: the sawtooth simply translates. At any whole a of zero or more every frequency n^a is itself a whole number, so every term has period 2π in t and the smeared shape reassembles exactly every 2π ≈ 6.28 seconds. That return is a revival. Give a a fraction and the eight frequencies share no common period, so the shape never comes back at all.
Try
- Drag a to 1: every harmonic takes the same speed and the teeth slide sideways without changing shape, which is the motion the preset Saw Cascade draws in closed form.
- Drag a to 3: the eighth harmonic now runs at 8² = 64 units a second, a blur on top of a ramp that still rebuilds itself every 2π.
- Drag a to 1.5: the same violent smearing, but nothing ever returns. Compare the curve a few seconds apart and no two are alike.
- Drag a to 0: every term now has frequency 1, so the whole sum oscillates with one period while the nth harmonic creeps along at 1/n.
- Press Rewind, then Play, and count six seconds at Speed 1: the revival lands just after, at 2π.