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It is known from both theory and numerical simulations that a current pulse suffers apparent attenuation as it propagates along a vertical perfect conductor of uniform, nonzero thickness (e.g., a cylinder) above perfectly conducting ground, excited at its bottom by a lumped source. The associated electromagnetic field structure is non-transverse electromagnetic (TEM), particularly near the source region. On the other hand, it has been shown analytically by Thottappillil et al. (2001, 2004) that no attenuation occurs and the electromagnetic field structure is pure transverse electromagnetic (TEM) if the conductor thickness and source size are assumed to be infinitesimal. The goal of this paper is to examine the mechanism of current attenuation as it propagates along a nonzero thickness conductor, based on the scattering theory and on a nonuniform transmission line approximation. In applying the scattering theory, we decompose the "total" current in the conductor into two components that we refer to as the "incident" and "scattered" currents. The "incident" current serves as a reference (no attenuation), specified disregarding the interaction of resultant electric and magnetic fields with the conductor, while the "scattered" current, found here using the finite-difference time-domain (FDTD) method, can be viewed as a correction to account for that interaction. The scattered current modifies the incident current so that the resultant total current pulse appears attenuated. Thus, the current attenuation is likely to be due to field scattering that does not occur in the case of zero thickness conductor. The attenuation of the total current pulse is accompanied by the lengthening of its tail, such that the total charge transfer is independent of height. Approximation of the vertical conductor above ground by a nonuniform transmission line whose characteristic impedance increases with increasing height is shown to reasonably reproduce the current pulse attenuation predicted by the scattering theory. In this approximation, the apparent current attenuation with height can be attributed to waves reflected back to the source. The results have important implications for development and interpretation of lightning models.