A relativistic electron beam propagating through an unmagnetized, underdense plasma exhibits a transverse instability due to the coupling of the beam centroid to plasma electrons at the "ion-channel" edge. The transverse wake field corresponding to this "electron-hose" effect is calculated in the "frozen-field" approximation for a low-current, cylindrical beam in a radially infinite plasma. The asymptotic growth of beam-centroid oscillations is computed, and the growth length is found to be very rapid, indeed much less than the betatron period of the beam. Results for a radially finite plasma and for a slab beam are noted. Damping and saturation mechanisms are discussed.PACS numbers: 52.40. Mj, 29.15.Dt, 52.50.Gj In recent years, the demands of the TeV-energy electron-positron collider [l] have spurred considerable interest in the transport of intense relativistic electron beams in the "ion-focused regime" (IFR). Proposed applications include the plasma lens [2], the continuous plasma focus [3,4], the plasma emittance damper [5], and plasma wake-field acceleration [6,7]. At the same time, coherent radiation from intense beams in the IFR has also been the subject of much theoretical [8][9][10][11] and experimental [12] work. These novel applications draw on a large body of work in beam-plasma physics [13][14][15] and extensive application of the IFR in accelerator and radiation research [16,17].Typically the IFR refers to propagation along a narrow plasma channel which is "underdense" (i.e., with charge density much less than that of the beam) and in addition has total plasma charge per unit length less than that of the beam. In this limit, all plasma electrons are ejected radially to large distances. However, for many novel applications, the plasma may initially extend to large radii, or a broad plasma may be created by beam and secondary ionization. In this Letter, we show that propagation in such a regime suffers from a previously unrecognized hose instability, similar in character to the "transverse two-stream" instabilities [18,19] (e.g., the "ion-hose" instability [15]). This instability results from the electrostatic coupling of transverse beam displacements to plasma electrons at the boundary between the ion channel and the surrounding quasineutral plasma, beyond the beam volume. We show that the growth length for the "electron-hose" instability is so short that IFR transport in this regime is problematic at best.To compute this growth length, we consider first equilibrium propagation of a relativistic electron beam in a uniform, unmagnetized, preionized plasma of density rie, and infinite radial extent. We assume unperturbed beam charge density of the form pboir,s)'= -enij(s)H(a -r), where H is the step function, -^ is the electron charge, rib is the beam density on axis, a is the beam radius ( Fig. 1), s-t -zic is the retarded time, t is time, z is axial displacement, and c is the speed of light. As the beam head propagates through the plasma, it expels plasma electrons from the beam volume on the sho...