Theory of ion-hose instability

Solutions are derived for the partial-differential equations of the rigid model of ion hose with and without an applied axial magnetic field. An upper bound on the amplitude of the beam response, derived from these exact solutions, is compared with the upper bound derived by means of an asymptotic saddle-point calculation. In the exact solutions, explicit boundary conditions are retained. The details of the boundary conditions are significant in understanding the driven ion-hose instability in the experiments. Consideration of the effects of a magnetic field is important because in the ion-hose experiments at Sandia National Laboratories the beam is immersed in an applied axial-magnetic field