Numerical modeling of magnetosphere-ionosphere coupling by Alfv¿n waves has proven to be a valuable tool in describing the propagation and evolution of field-aligned currents and the magnetic fields produced by these waves as observed on the ground. Although many models of this type have assumed that magnetic field lines penetrate the ionosphere vertically, this assumption is not valid at lower latitudes, where the dipole tilt is significant. This paper presents a new model of this interaction that takes the dipole geometry into account, while including an ionosphere that is radially stratified. Since the orthogonal dipole coordinates that have been used in previous studies do not come to a constant radial distance at the ionosphere, a nonorthogonal system is developed that reduces to the orthogonal system at high altitudes. This model is useful for many different problems of wave coupling and is applied here to the propagation of ducted Pc1 oscillations that can propagate thousands of kilometers across magnetic field lines.