The effect of tool joints on the passage of plane longitudinal and torsional waves along a drill pipe was studied. An approximate solution to the governing equations of motion found, and an idealized tool joint constructed. Calculations were made for the effect of the idealized joint on an example drill string. The results showed that tool joints had negligible effect for exciting frequencies of the same order as common rotary speeds and the drill pipe could be taken as a uniform pipe with negligible error. Equivalent systems of rigid inertias and massless springs were developed for a drill string with a uniform drill pipe and one drill collar section, for fixed-fixed or fixed-free end boundary conditions. Undamped responses of the equivalent systems were found for arbitrary periodic forces applied to any point of the drill string. Approximate responses were derived for small damping. Specific formulas were derived for responses at the top of the drill string for given periodic displacement or force at the bottom, for fixed-fixed or fixed-free boundary conditions.