The patterns probably depend on the stiffness of the plate/membrane as well as its shape, and the grain size and density of the sand.
It should depend on stiffness, density and shape. The speed of sound in a material is sqrt(stiffness/density). The partial differential equation for waves depends on these two constants, and the amplitude of forcing. The units are a little tricky (and they are different depending on the number of dimensions) I have been engrossed by this idea, since I read it on the list :) I'm sure you'll have a lot to research to make this work, and I really hope you make something cool!
By analogy with the Karplus-Strong vibrating string, which is a one-dimensional CA, usually the stiffness of the string is ignored. Also, the grain size would be the same as the pixel size and the density would be ignored. I see it as being like 2D version of the KS waveguide, with the superimposed grains moving in each time step towards the neighbour whose vertical acceleration is the lowest among 8 neighbours. Apart from the difficulty of doing a 2D KS, there is the further complication of an external frequency forcing (maybe introduced at the edges of the plate?).
Martin
I see the edges as being different kinds of boundary conditions, Dirichlet, Neumann, and Robin. Dirichlet -> amplitude is zero at the boundary (reflected waves are 180 deg out of phase) Neuman -> 1st deriviative is zero at the boundary (reflected waves are in phase with incoming waves) Robin -> 1st order differential equation (specifies a constant phase difference between incoming/reflected waves)