  Small amplitude dipolar oscillations are considered in artificial spin ice on
a square lattice in two dimensions. The net magnetic moment of each elongated
magnetic island in the spin ice is assumed to have Heisenberg-like dynamics.
Each island's magnetic moment is assumed to be influenced by shape anisotropies
and by the dipolar interactions with its nearest neighbors. The magnetic
dynamics is linearized around one of the ground states, leading to an $8\times
8$ matrix to be diagonalized for the magnetic spin wave modes. Analytic
solutions are found and classified as antisymmetric and symmetric with regard
to their in-plane dynamic fluctuations. Although only the leading dipolar
interactions are included, modes similar to these may be observable
experimentally.
