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Analytical solutions are derived for both transient and steady state gradient distributions in the traveling wave (TW) accelerating structures with arbitrary variation of parameters over the structure length. The results of the unloaded and beam loaded cases are presented. Finally, the exact analytical shape of the rf pulse waveform was found in order to apply the transient beam loading compensation scheme during the structure filling time. The obtained theoretical formulas were cross-checked by direct numerical simulations on the CLIC main linac accelerating structure and demonstrated a good agreement. The proposed methods provide a fast and reliable tool for the initial stage of the TW structure analysis

Analytical solutions for transient and steady state beam loading in arbitrary traveling wave accelerating structures