acoustic branch
Picture a long line of people holding hands and gently swaying. If everyone leans the same way at almost the same moment, the whole line drifts together like a slow wave passing down a stadium crowd. The acoustic branch is the family of crystal vibrations that behave like this: neighbouring atoms move in the same direction, nearly in step.
These are the low-frequency vibrations of a solid. Because neighbouring atoms barely move relative to each other, the springs between them are hardly stretched, so very little energy is needed — that is why their frequency is low and, for the longest waves, drops all the way to zero. At those long wavelengths the motion is exactly ordinary sound travelling through the material, which is where the name 'acoustic' comes from. Every crystal has acoustic branches, regardless of what it is made of.
The acoustic branch matters because it carries sound and, at ordinary temperatures, most of the heat through a solid. A point worth getting straight: 'acoustic' here is a label for how the atoms move together, not a claim that you can hear it — most acoustic phonons vibrate far faster than any sound your ear can detect.
An earthquake sends pulses through solid rock that arrive at distant cities seconds apart. Those seismic waves are giant acoustic-branch vibrations of the Earth's crust — the very longest, slowest cousins of the same atomic swaying that carries the tap of a fingernail through a tabletop.
Seismic waves are acoustic-branch vibrations on a planetary scale.
Acoustic and optical branches come in pairs only when the crystal's repeating unit holds more than one atom. A crystal with a single atom per cell has acoustic branches alone — there is no second group of atoms to move against.