Longitudinal coupling impedance for particle beams of parabolic transverse charge istribution moving in a resistive culindrical beam-pipe of finite wall thickens

The longitudinal coupling impedance for particle beams of parabolic transverse charge distribution moving in aresistive cylindrical beam-pipe of finite wall thicken studied theoretically. The particle beam is moving at constant speeddown a resistive cylindrical pipe of fintie wall thickness. The wave equations for the lectromagnetic fields induced by thebeam motion inside the cylindrical pipe have been derived and solved. The space charge and the wall impedances have been obtained analyticaaly and then analyzed numberically for diferent beam energies, diferent wall conductivities and diferent wave mode frequencies. The real part for the wall impedance is found to be a positive resistance while the imaginary part is a positive reacetance (inductive). At low beam energies we observed no differences between the impedances of quadratic and uniform beams. At high beam energies and for all wave mode frequencies, differences between the two impedance are observed frow all thickness bellow the skin penetration depth s. The impedance converges for large wall thicknetowared a small value of the order of the sufface impedance of a thick conducting wall.