  We investigate the phonon spectra of two-dimensional liquid dusty plasmas on
a one-dimensional periodic substrate using numerical simulations. The
propagation of the waves across the potential wells of the substrate is
inhibited due to the confinement of the dust particles by the substrate minima.
If the substrate wells are narrow or deep, one-dimensional chains of particles
are formed in each minima, and the longitudinal motion of an individual chain
dominates the propagation of waves along the potential wells of the substrate.
Increasing the width or decreasing the depth of the substrate minima allows the
particles to buckle into a zig-zag structure, and the resulting spectra develop
two branches, one for sloshing motion and one for breathing motion. The
repulsion between neighboring dust particles produces backward propagation of
the sloshing wave for small wave numbers.
