Sprinkle fine sand on a plate of metal and draw a violin bow across its edge, and the sand jumps, and then it runs — always away from the places where the plate is moving most, and always towards the places where it is still. Within a second the plate has drawn its own mode in sand, and the figure is a picture of the wave standing on it.
A grain sitting on a vibrating plate is thrown about by the acceleration of the surface beneath it. Where the plate moves fast, the grain bounces and skips, wandering more or less at random. Where the plate is still, the grain simply sits.
So the sand is doing a gradient descent on the square of the displacement — walking downhill towards the places where the vibration is smallest. That is why the figures are so clean: the pattern is not painted on, it is the shape of the zeros of the mode function, and the sand finds them by brute force.
A square plate with free edges does not vibrate like a drum. The mode shapes are not simple sine patterns but combinations of them that satisfy the free-edge condition, and they are not exactly the same as the ones you get from a simple cosine product — which is what is drawn here.
What the cosine approximation gets right is the topology: the number of node lines, the symmetry, and the fact that the figures close on themselves. Ernst Chladni's own experiments, in the seventeen-nineties, produced exactly these patterns and he had no way at all to explain where they came from.