The effect of a rectilinear current on Alfve´n waves in a bounded medium is investigated. An infinite spectrum of frequencies corresponds to each Alfve´n frequency obtained without the current. Therefore theI = 0case is considered as degenerate since eigenfrequencies tend to the Alfve´n frequency in this limit. A radial component of the disturbed field will always be present, even though none need be present for the current‐free case. This component is a discontinuous function of the current (discontinuous at zero). A formula connecting the radial and azimuthal components of the field is obtained. This formula carries through when theI →0limit is taken. For theI = 0case the azimuthal and radial fields are completely unconnected, hence the discontinuity. In particular the azimuthal component of the magnetic field will vanish on the boundaries for all finite currents and in theI → 0limit, even though this is not true of theI = 0case. The boundary conditions on the azimuthal component of the field are therefore seen to be discontinuous.