In cold magnetospheric plasmas, ultra-low frequency (ULF) waves occur in either the Alfvén or magnetosonic mode. Owing to the strong inhomogeneity of the near-Earth plasma, these waves can produce cavity-like or waveguide-like eigenmodes which are mutually coupled by wave-driven ionospheric currents and perpendicular plasma inhomogeneity. Here we present results from a numerical study of the ionospheric Alfvén resonator (IAR) and the ULF waves generated by wave-driven field-aligned currents (FACs) in the ionosphere. We demonstrate the excitation of cavity and waveguide eigenmodes of the IAR by a localized field-aligned current, and examine the polarization patterns of the resultant ground magnetic fields. We find that the distribution of ellipticity reflects the configuration of the magnetospheric source while the distribution of polarization angles depends primarily on the localization of the signal. The pattern of magnetosonic waves generated by ionospheric Hall currents is generally isotropic, but its configuration is representative of the structure of the incident FAC. Apparent preferences for wave propagation eastward and to lower latitudes are identified, and implications for the interpretation of ground magnetic signatures are discussed.