We investigate theoretically how sheath radio-frequency (RF) oscillations relate to the spatial structure of the near RF parallel electric field E∥ emitted by ion cyclotron (IC) wave launchers. We use a simple model of slow wave (SW) evanescence coupled with direct current (DC) plasma biasing via sheath boundary conditions in a 3D parallelepiped filled with homogeneous cold magnetized plasma. Within a ‘wide-sheath’ asymptotic regime, valid for large-amplitude near RF fields, the RF part of this simple RF + DC model becomes linear: the sheath oscillating voltage VRF at open field line boundaries can be re-expressed as a linear combination of individual contributions by every emitting point in the input field map. SW evanescence makes individual contributions all the larger as the wave emission point is located closer to the sheath walls. The decay of |VRF| with the emission point/sheath poloidal distance involves the transverse SW evanescence length and the radial protrusion depth of lateral boundaries. The decay of |VRF| with the emitter/sheath parallel distance is quantified as a function of the parallel SW evanescence length and the parallel connection length of open magnetic field lines. For realistic geometries and target SOL plasmas, poloidal decay occurs over a few centimeters. Typical parallel decay lengths for |VRF| are found to be smaller than IC antenna parallel extension. Oscillating sheath voltages at IC antenna side limiters are therefore mainly sensitive to E∥ emission by active or passive conducting elements near these limiters, as suggested by recent experimental observations. Parallel proximity effects could also explain why sheath oscillations persist with antisymmetric strap toroidal phasing, despite the parallel antisymmetry of the radiated field map. They could finally justify current attempts at reducing the RF fields induced near antenna boxes to attenuate sheath oscillations in their vicinity