In light of a recent demonstration of the general nonexistence of a singularity at the Alfv¿n resonance in cold, ideal, linearized magnetohydrodynamics, we examine the effect of a small density gradient parallel to uniform, open ambient magnetic field lines. To lowest order, energy deposition is quantitatively unaffected but occurs continuously over a thickened layer. This effect is illustrated in a numerical analysis of a plasma sheet boundary layer with perfectly absorbing boundary conditions. Consequences of the results are discussed, both for the open field line approximation and for the ensuing closed field line analysis. ¿ American Geophysical Union 1994