Envelope Modes of a Mismatched Bunched Beam

Frequencies of the 3 mismatched envelope modes of a bunched beam are calculated as a function of: - the beam aspect ratio a = γZ₀/R₀, R₀ and Z₀ being the matched transverse and longitudinal envelope sizes of the beam and γ being the Lorentz factor of the beam centroid in the lab ref., - the transverse space-charge tune depression η_t = k_t/k_t0, k_t and k_t0 being the transverse phase advance per unit length of particle in the homogenous beam respectively with and without space-charge forces, - the external force aspect ratio α = k_l0/k_t0, k_l0 and k_t0 being the phase advance per unit length of particles respectively in the transverse and the longitudinal direction. The calculated wave numbers of the 3 modes are then compared to those obtained from the Fourier analysis of the envelope of mismatched bunched beam computed numerically. A good Agreement is found. As with a continuous beam, some particles always feel 1/2 order resonances with this envelope modes, and can moreover be excited by a first order resonance. (author)