Longitudinal collective modes of a bunched beam with a repulsive inductive impedance (the space charge below transition or the chamber inductance above it) are analytically described by means of reduction of the linearized Vlasov equation to a parameterless integral equation. For any multipolarity. the discrete part of the spectrum is found to consist of infinite number of modes with ... https://www.mariameetsanna.com/product-category/leave-in-treatment/