Lattice Dynamics and Melting of a Nonequilibrium Pattern
  We present a new description of nonequilibrium square patterns as a harmonically coupled crystal lattice. In a vertically oscillating granular layer, different transverse normal modes of the granular square-lattice pattern are observed for different driving frequencies ($f_d$) and accelerations. The amplitude of a mode can be further excited by either frequency modulation of $f_d$ or reduction of friction between the grains and the plate. When the mode amplitude becomes large, the lattice melts (disorders), in accord with the Lindemann criterion for melting in two-dimensions. 
Daniel I. Goldman and M. D. Shattuck and Sung Joon Moon and J. B.   Swift and Harry L. Swinney