Eulerian formalism of linear beam-wave interactions

A detailed account of a formal mathematical description of the interaction of relativistic charged particle beams with electromagnetic waves, within the frame of classical electrodynamics, is presented. The standard system of eight equations (Maxwell and Lorentz gauge conditions and fluid dynamics) in the four-vector potential ${\mathit{A}}_{\mathrm{\mu }}$ and the four-vector current density ${\mathit{j}}_{\mathrm{\mu }}$ is reduced, after linearization, to a canonical system of four coupled partial differential equations in the electromagnetic field perturbation δ${\mathit{A}}_{\mathrm{\mu }}$. Both electromagnetic and dynamical quantities are treated as fields, according to the Eulerian formalism. This new system is very general, and different beam-wave interactions are characterized by different fluid equilibria and boundary conditions for δ${\mathit{A}}_{\mathrm{\mu }}$ and its derivatives. Finally, the equations are used, as an example, to study the dispersive characteristics of space-charge waves propagating in a cylindrical waveguide, along a relativistic electron beam confined by an axial magnetic field. The problem is treated in a fully relativistic way, for arbitrary values of the axial guide field and any degree of azimuthal symmetry.