A speaker transmits the vibrations created by your voice to the center of the metal square. From there, they radiate outward, reflecting off the edges of the square, then overlapping to create patterns of stillness and vibration.
Sand bounces off the strongly vibrating places and collects in the still places, leaving a visual record of the vibrations in the square.
Different notes vibrate the square at different rates; some special notes are resonant, meaning they vibrate the square at one of its preferred or natural frequencies, leading to extra strong vibrations and a clear and definite pattern on the square.
Chladni patterns make surface vibrations visible: Neighboring clear zones are moving in opposite directions, and sand collects in areas where the surface is still. (click image to enlarge)
Physicist-musician Ernst Chladni came up with the idea to use sand to reveal the pattern of vibrations on a surface in 1787.
Today, some makers of violins, cellos, and similar musical instruments still use Chladni patterns to help them craft the belly and back of each instrument, the shapes of which are crucial to the instrument’s unique voice.
The mathematics underlying Chladni patterns also helped shed light on the quantum mechanical understanding of electron orbital patterns.