Two travelling waves, running opposite ways on a fixed string, add to a shape that does not travel — it stands, and only its amplitude breathes. Because both ends are tied down, the string can only ring at whole-number harmonics: fn = n·v / 2L. Down the center, data flows: tension and length go in, the engine finds the modes, the harmonic ladder comes out. The blue team builds and defends it; the red team tries to break it.
source M. Mersenne, Harmonie universelle (Paris, 1636) — first statement of the string laws; B. Taylor, Phil. Trans. 28 (1713) gave the fundamental's shape. Scan: archive.org/details/bub_gal_ark_12148_bpt6k5471093v (AMBER — 17th-c. facsimile, no canonical DOI). Rendered, not quoted.
A rightward wave sin(kx−ωt) meets a leftward wave sin(kx+ωt) reflected off the far clamp. Their sum is an exact trig identity:
sin(kx−ωt) + sin(kx+ωt)
= 2·sin(kx)·cos(ωt)
The sin(kx) half fixes the shape; the cos(ωt) half only rescales it in time. So the zeros — the nodes — never move. Both fixed ends must be nodes, which forces a whole number of half-wavelengths and quantises the modes.
For the current mode, the node ledger (should be zero displacement at each):
| node j | x = jL/n | 2·sin(kx) |
|---|
Two waves standing still, born of the fixed-end condition, are counter-propagating solutions of the-wave-equation — the same utt = c²uxx, now boxed between two clamps.
Because the boundary allows only n = 1, 2, 3…, the string emits an integer stack of frequencies. That stack is the-overtone-series — the physics standing behind every plucked note. Each sphere is the next one's premise.
The blue team's live check: recompute the harmonic ladder and the end-node condition from the engine's own functions and confirm them. If red makes the modes inharmonic, this badge is where it shows.
A stretched string carries waves at speed v = √(T/μ) — tension T over linear density μ. Clamp it at both ends a distance L apart and you feed the panel three numbers:
| symbol | quantity | sets |
|---|---|---|
| T | tension (N) | speed ↑ |
| μ | mass / length (kg/m) | speed ↓ |
| L | clamp spacing (m) | pitch ↓ |
Pick the harmonic n too. That is the whole input — and it is what you feed the panel below.
The shape is 2·sin(kx)·cos(ωt) computed per-pixel; nodes marked, antinodes ringing. Display time is slowed for the eye — the frequencies at right are the real physics.
Change any control — v, the ladder and the nodes are computed on the spot from √(T/μ) and n·v/2L, never looked up.
What the machine produces, proven: the wave speed v = √(T/μ), and a stack of resonances at exactly integer multiples of the fundamental — fn = n·v/2L, with the nth mode carrying n−1 interior nodes and a clamp-node at each end.
The blue team's witness (left) confirms the ladder is integer live; the red team (right) tries to detune it.
The model also assumes fixed ends, no damping, small amplitude, and a uniform string. Break any of those and the modes shift, decay, or couple. The ideal is exact only in its own limit — which is precisely the tamper the red team will exploit below.
"The string vibrates at one frequency." Cut. A plucked string rings the whole ladder at once; timbre is the recipe of harmonic amplitudes.
"Higher harmonics have longer wavelength." Cut. Backwards: λn = 2L/n shrinks with n; frequency fn = n·v/2L rises.
"Nodes are where the string moves most." Kept, corrected. Nodes are the still points; the antinodes between them swing widest.
The red team's move: detune the ladder to fn = n1.02·f1 — inharmonic. Now the far clamp is no longer a node: the whole-number-of-half-wavelengths condition is violated. The blue team's witness (window 7) is watching.
Detune and the modes are no longer integer multiples, the boundary node at x=L breaks — the witness recomputes, disagrees with the harmonic law, and turns red. Nothing is faked; the attack is real and it is caught.