We derive dispersion relations for a system of identical particles confined
in a two-dimensional annular harmonic well and which interact through a Yukawa
potential, e.g., a dusty plasma ring. When the particles are in a single chain
(i.e., a one-dimensional ring) we find a longitudinal acoustic mode and a
transverse optical mode which show approximate agreement with the dispersion
relation for a straight configuration for large radii of the ring. When the
radius decreases, the dispersion relations modify: there appears an
anticrossing of the modes near the crossing point resulting in a frequency gap
between the lower and upper branches of the modified dispersion relations. For
the double chain (i.e., a two-dimensional zigzag configuration) the dispersion
relation has four branches: longitudinal acoustic and optical and transverse
acoustic and optical.Comment: 10 pages, 8 fugure