Standing Waves on a String

Pluck a string tied down at both ends and it doesn't ring at any pitch you want. It rings at one specific pitch plus a family of overtones, because the string's two ends force the wave to reflect back on itself over and over. The original wave and its reflection are the same wave moving in opposite directions, and when they overlap just right, they add up into a fixed pattern: some points sit perfectly still (nodes), others swing with maximum motion (antinodes), and the pattern itself never travels anywhere. That's a standing wave. The ends, fixed in place, must always be nodes, which is the constraint that throws out every frequency except a special set.

That constraint is geometric: the string's length L has to fit a whole number of half-wavelengths, because each half-wavelength spans node to node. The lowest frequency that fits, the fundamental, fits exactly one half-wavelength, so lambda1 = 2L and f1 = v/(2L), where v is the wave speed set by the string's tension and mass per length. The next harmonic fits two half-wavelengths, the one after that fits three, and so on, giving f_n = n times f1. Draw them out: the fundamental has a node-antinode-node shape, the second harmonic adds a node in the middle, the third harmonic has two internal nodes splitting the string into three equal lobes.

This is why a guitarist fretting a string changes its pitch: pinning a point partway along forces a node there, shortening the vibrating length and raising every harmonic by the same factor. It's also the string analog of the resonant air columns coming up in pipes, where open and closed ends play the same role fixed ends play here, just with pressure nodes instead of displacement nodes.