The complicated resonance modes of a 2-dimensional metal plate are neatly illustrated by putting freely moving sand on the surface.
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Chladni plates are an excellent tools for visualizing resonant patterns in two dimensions. Understanding how they work is much easier than understanding the theory and mathematics behind the patterns themselves, so that is what we will concentrate on here.
Let's begin by thinking of a simple 1-dimensional example. If a piece of string were made to vibrate at a special resonant frequency (determined by the string's length, density, and tension), the standing wave would have places along the length where the string didn't seem to be moving at all. These places are called "nodes". Everywhere else along the string is vibrating, moving up and down, back and forth very quickly. If we were able to balance some sand on the string, the sand away from the nodes would be flung off into space while the sand on the node would sit still happily where it is.
While calculating the resonant frequencies of a 1D string is fairly straightforward, the theoretical details in two dimensions become much more difficult. But we can easily see the patterns using the method described below. The Chladni plate is a rigid, thin piece of metal, usually of a regular shape but not necessarily so. The plate is attached to a source of vibration. The source must be strong enough to vibrate the plate enough to shake the sand into the lines of zero amplitude (the nodal lines). Ideally, the source should also be able to sweep through a large frequency range so investigate all possible patterns.
These patterns, especially at higher frequencies, show amazing complexity and symmetry. If one can stand the noise, it is easy to spend hours finding new patterns in the sand.