Damping beam displacements through phase mixing - An illustrative model

We develop a simple model of a beam transported in a hard wall channel. The extremely anharmonic nature of the potential damps coherent trans verse displacements of the beam via phase mixing. For small uniform displacements of the beam, we can write down by inspection the analytical form of the motion of the beam centroid. This model allows quantitative evaluation of the effects of discrete, thin scattering cells (foils), apertures, and focussing cells (thin lenses) upon the damping process and upon emittahce growth.