  Phonon spectra of a two-dimensional (2D) solid dusty plasma modulated by 2D
square and triangular periodic substrates are investigated using Langevin
dynamical simulations. The commensurability ratio, i.e., the ratio of the
number of particles to the number of potential well minima, is set to 1 or 2.
The resulting phonon spectra show that propagation of waves is always
suppressed due to the confinement of particles by the applied 2D periodic
substrates. For a commensurability ratio of 1, the spectra indicate that all
particles mainly oscillate at one specific frequency, corresponding to the
harmonic oscillation frequency of one single particle inside one potential
well. At a commensurability ratio of 2, the substrate allows two particles to
sit inside the bottom of each potential well, and the resulting longitudinal
and transverse spectra exhibit four branches in total. We find that the two
moderate branches come from the harmonic oscillations of one single particle
and two combined particles in the potential well. The other two branches
correspond to the relative motion of the two-body structure in each potential
well in the radial and azimuthal directions. The difference in the spectra
between the square and triangular substrates is attributed to the anisotropy of
the substrates and the resulting alignment directions of the two-body structure
in each potential well.
