These images are top views, looking down onto a granular layer 8 glass
beads deep and confined inside a cylinder. The whole system is vibrated
vertically up and down and is illuminated from the side. The white regions are
ridges of grains, the dark areas lie lower (in the shadow regions) and thus
contain fewer grains. To image these states (from top as well as from the
side) we use high-speed video imaging (up to 1000 frames per second).
Typically, square patterns appear at low driving frequencies, and stripes at higher frequencies. All wave states are parametric instabilities in the sense that they oscillate back and forth between a pattern and the inverse pattern. At low drive accelerations this leads to a situation where the pattern responds with f/2 when the plate is oscillated at frequency f. In the case of the squares and stripes the inverse pattern is just a shifted version of the original one. But there are other cases such as, at higher drive accelerations, triangles that change into hexagons and back. Or resonance-like states such as the one shown below that are a superposition of several square states with a five-fold symmetric defect in the center:
The two images above are snapshots taken one drive period apart. They
clearly show how the pattern on the left has changed into its inverse on
the right. Another drive period later the pattern on the left would re-emerge.
For a very general and easily understandable description of theses surface wave states and how they are produced, check out Paul Umbanhowar's article for the APS in connection with the March Meeting 1996.
There is a host of different kind of wave states in addition to the ones described above, including traveling waves and even possibly chaotic states. Some of these are described in more detail in our recent reviews: