Recently, the propagation of an electron beam through an ion channel with periodically varying ion density has been proposed as a method of transporting or modulating an electron beam. A theoretical treatment indicates that, in the absence of an external magnetic field, cylindrically symmetric radial oscillations of the beam electrons are excited by the channel, with resonant excitation occurring when the wavelength of the ion density variation equals the betatron wavelength. An analysis of the ion focusing regime (IFR) ion hose instability indicates that thee‐folding length of this instability increases when the variations of the electron density are increased. In the absence of an external magnetic field and damping, moderate electron density variations (caused, for example, by a periodic ion channel or the nonrigidity of the beam and channel) result in a wavelength ande‐folding length on the order of the beam‐averaged betatron wavelength.