Abstract Relativistic electrons (RE) in the magnetosphere are considered as an important constituent of the radiation environment. The origin of RE sporadic fluxes remains rather enigmatic. The particular spatio–temporal structure of electromagnetic disturbances in the outer magnetosphere is significant in evaluation of RE acceleration opportunities there. The corresponding disturbances have the form of transverse-small-scale Alfvén resonant waves. To be localized in local time, the waves must have large azimuthal wave numbers. The effect of these waves on the electrons would be most significant on and near the poloidal resonance surfaces. The case of non-monochromatic oscillations is analysed. A short-lived source acting in a relatively narrow L interval is assumed. It is located in the nightside sector, in the vicinity of the boundary between tail-like and dipole-like field lines, and is assumed to correspond to a substorm activation in that region. The associated time evolution of the whole spatial structure of the wave field is analysed. The resonant response is shown to demonstrate fast decay associated with phase mixing of spectral components, rather than with ionospheric absorption. Initial results of a numerical analysis of the problem are presented.