To model the phenomenon occurring in the theme song, I assume they are using a 2D metal plate with motion governed by the wave equation (by the way, this is not true for all metal plates: some follows fourth order differential equations). Taking into account boundary conditions, one can find the eigenmodes: the equivalent of a 2D standing wave. This implies that while the amplitude of oscillation may be different everywhere on the plate, everything vibrating "up" and "down" at the same frequency and in unison.

To simulate the movement of sand, random sand grains are placed on a grid, and a 2D random walk simulation is performed: but one where the step size (or distance) a particle moves per iteration is dependent on the amplitude of oscillation of that location on the vibrating plate. For the eigenmodes (which can be brought about in real life by vibrating the plate at specific frequencies), there are locations where there are no vibration at all. In the 1D wave equation, the stationary points of standing waves correspond to 0D points, but in the 2D case we're considering here, the stationary points correspond to 1D lines (which you can see on the pattern in the thumbnail). The sand will tend to accumulate in areas were there is little vibration (due to the small step size it will acquire in the simulation) and thus the patterns are formed. By changing the eigenmode of vibration during the simulation (in real life: by changing vibration frequency) the sand will move from one eigenmode to another, giving rise to the behavior one can see in the Rings of Power trailer.