Free linear solutions of the barotropic shallow-water equations are computed for a variety of basin shapes and topographies relevant to the Pacific Ocean. Attention is drawn to the 2-day to 6-day period band within which large-scale coherency is observed in Pacific tide-gauge records of sea level. In a particular model basin with realistic topography, the fundamental planetary mode of the Pacific occurs with a period of 3 days and four other planetary-topographic modes populate the 4–6 day period band. We argue that the frequencies and structures of these modes are representative of the largest-scale vorticity oscillations of the Pacific and may help account for the observed broad-bandedness of sea-level in the 4–6 day period band. These predictions will guide future analyses of sea-level and bottom-pressure observations from which a direct estimate of the frictional damping of barotopic Rossby waves may ultimately be obtained.