ANALYTICAL TREATMENT OF RESISTIVE WAKE POTENTIALS IN ROUND PIPE.pdf

TESLA Report 2003-25 1 ANALYTICAL TREATMENT OF RESISTIVE WAKE POTENTIALS IN ROUND PIPE M. Ivanyan, V. Tsakanov Center for the Advancement of Natural Discoveries using Light Emission -CANDLE, 375040 Yerevan, Armenia Abstract The modified analytical form for the longitud inal and transverse resistive wake potentials of point- like charge moving parallel to the axis of round pipe with frequency independent walls conductivity is obtained. The short range wake potentials are presented by uniformly converged series. For the frequency dependent conductivity, the resonator term of the longitudinal monopole wake potential is presented in analytical form. The diffusion term of the potential is modified to simple integral form. E-mail: ivanian@asls.candle.am TESLA Report 2003-25 2 1. INTRODUCTION The longitudinal and transverse impedances of infinitely resistive round pipe with frequency independent conductivity have been obtained by Chao [1]. The further treatment of the impedances both for frequency independent (DC-direct current) and frequency dependent (AC-alternate current) conductivities has been performed in Ref. [2]. In particular, the analytical extension of longitudinal impedance (monopole term) to the complex plane has been derived as [2]: ( ) 1 2 0 2 2 ? ?? ? ?? ? ?= κ κ λ κ i cb s Z , (1) where b is the pipe radius, 3120 )2( πσcbs = is the characteristic distance, c is the velocity of light, σ is the conductivity of the walls and 0ks=κ is the complex dimensionless wavenumber. For the DC conductivity, σ is the static quantity and the parameter λ is given by ( )i+= 1κλ . (2) For the AC case, the conductivity is given by )1(0 ωτσσ i?= with 0σ the static conductivity, ω the frequency and τ the relaxation time of the metal. The parameter λ is then given by [2]: ( ) [ ]λλκκλ tti ?++Γ+= ? 111