The Alfve´n‐ion‐cyclotron instability is studied for a model where the ions are confined axially by a quadratic electrostatic well. Comparison between the homogeneous plasma dispersion relation, an effective dispersion relation including the effect of axially bouncing ions, and the complete eigenvalue solution shows that the first two approximate solutions are quite accurate for plasma lengths on the order of the ion gyroradius. The eigenmode with odd electric field symmetry is found to be the most unstable. Comparison is made with a previously reported particle simulation which imposes linear polarization for the electric field. It is found that the linear polarization constraint can give the unphysical result of instability at vanishingly small plasma &bgr;⊥for certain parameters.