[en] The paper describes a new model of alternating-phase focusing (APF) dynamics applicable to ion linacs with short independently controlled superconducting cavities. The equations of motion are derived for a cylindrically symmetric electric field represented by a traveling wave with continuous periodic phase modulation. solutions are obtained and analyzed for both the linear and nonlinear particle motion. Problems of linear stability and overall longitudinal acceptance are solved using standard mathematical techniques for periodic systems; analytical results are obtained. It is shown that the main beam dynamical aspects of APF are adequately described by four parameters: equilibrium synchronous phase, phase modulation amplitude, length of APF period, and incremental energy gain. The model can be applied to study the feasibility of realizing APF in a low-β section of a proton linac