Thought Toys · Exhibit 19
String a row of little weights together with springs and pin both ends. Pluck it and it shudders in a tangle no one could write down. But that tangle is a fake: it's only a few pure shapes, each ringing at its own steady pitch, added together. Dial in one of them alone and the whole chain settles into a single, calm standing wave.
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The dots are equal masses; the lines between them are springs, and the two ends are bolted down. Each mass can only bob up and down, pulled by the springs to its left and right. That's the entire machine — no driving, no friction, just push-and-pull between neighbours.
Choose a pure mode and watch what "standing wave" means. The whole chain swings in lock-step to one frozen shape — it only grows and shrinks, like a guitar string's overtone. Certain masses, the nodes, never move at all; between them the antinodes swing hardest. The fundamental (mode 1) is a single smooth arch. Step the mode up and the shape adds one more node each time, ringing faster and faster — mode p has p humps and rings at a higher pitch. The bar chart below is lit on exactly one bar: the motion is pure.
Now press pluck the middle. You pull one point up and let go — and the chain breaks into a wobble that never settles into a shape. Look at the recipe: many bars light up at once. A pluck isn't a new kind of motion; it's a sum of those same pure modes, each ringing at its own pitch. Because the pitches differ, the modes drift in and out of step — they beat — so the overall shape keeps churning and never stands still. That hidden truth, that any motion is a stack of pure modes, is exactly what a Fourier series is — and why a plucked string sounds like a chord of one fundamental plus its overtones.
improve/verify/19-standing-waves.js):
the shapes are exact eigenvectors of the spring coupling, a pure mode obeys the wave equation
step for step, the sum reproduces any pluck exactly, and energy is conserved.
Counter-example: a single mode keeps its shape forever (a true standing wave),
but a two-mode mixture provably drifts out of shape — so a pluck is not a standing wave.
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 19.