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We consider an ultrarelativistic particle traveling on axis in an infinitely long cylindrical metallic beam pipe with azimuthally varying conductivity. For a circular geometry, a semianalytical solution is obtained using the Green functions and applying approximate boundary conditions for conductors. The theory predicts an image current distribution on the pipe walls practically independent of the azimuth, at least in the frequency range relevant for future machines such as the LHC. Numerical electromagnetic simulations and bench measurements confirm the theoretical predictions. Implications for the beam-induced heating in the copper-coated, welded LHC beam screen are also addressed.