Granular materials, or for short "sand", exhibit an extraordinary variety of different behaviors. The motion of individual grains must be maintained by external driving, and depending on its strength sand can assume a gaseous, fluid, or solid state. A great challenge lies in the development of continuum equations, which could serve as a basis for analytical descriptions.
Eggers and Riecke (1999) develop a phenomenological model for the formation of standing waves on a thin layer of sand on a vibrating plate. One of the most curious features of sand is its tendency to spontaneously form more dense regions or clusters. This phenomenon is exhibited by a granular assembly in a box, kept in a gaseous state by external vibrations. The box is separated by a wall safe for a small hole. As the driving is lowered below a critical value, a phase separation occurs in which the particles preferentially settle on one side of the box. Eggers (1999) gives an analytical description of this clustering phenomenon.