We examine indetail the first-order vibration modes of a bunched beam then apply the results to the analysis of halo formation using the particle-core model. The three-dimensional envelope equations are derived from first principle in order to avoid some discrepancies currently found in the literature. From the full envelope equations, we employ perturbation analysis to compute analytic expressions for the natural oscillation frequencies and corresponding oscillation modes of a beam bunch. Moreover, the mode excitations due to a periodic accelerator structure can be estimated. We then use the results of the envelope analysis as a starting point in the study of halo formation in bunched beams. An overview of the current nonlinear analysis using the particle-core model is presented. The collected results of this study are to be published at a later time.