Send a wave down a string tied at the far end and it comes straight back: two waves, opposite directions, one string.

Two waves, one string#

Why the sum stops travelling#

Add the two travelling waves and position and time come apart: $y = 2A\sin(kx)\cos(\omega t)$. Where $\sin(kx) = 0$ the string never moves: those points are the nodes. Halfway between them it swings widest, to $\pm 2A$: the antinodes.

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Fitting a wave into a fixed length#

The clamps force a node at each end, so only wavelengths that put a whole number of half-wavelengths between them fit:

$$L = n\frac{\lambda}{2}, \qquad \lambda_n = \frac{2L}{n}, \qquad n = 1, 2, 3, \dots$$

The medium fixes the speed and $\vel{v} = f\lambda$, so each allowed wavelength comes with one allowed frequency:

$$f_n = \frac{n\,\vel{v}}{2L}, \qquad \vel{v} = \sqrt{\frac{T}{\mu}}$$

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Louder is not higher#

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Pipes#

An organ pipe does the same with air. At a closed end the air cannot move through the wall: a displacement node. At an open end it sloshes freely in and out: an antinode. In a pipe the curve is a graph: the air moves along the pipe, and the curve draws that displacement sideways.

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A pipe closed at one end needs a node there and an antinode at the open end, so it fits odd numbers of quarter-wavelengths:

$$L = (2n-1)\frac{\lambda}{4}, \qquad f = \frac{(2n-1)\,\vel{v}}{4L}$$

What comes next#

You made a string swing in loops, tuned it by its length and by its tension, and capped a pipe. Two waves in one place added to a pattern that stands still. That was a special case: identical waves, opposite directions. Next the two waves have slightly different frequencies, and instead of a pattern fixed in space we get one that pulses in time: beats.