It is well known that an undamped periodic structure displays passbands and stop bands, which are frequency bands over which wave motion, respectively, can and cannot propagate. For narrow-band excitation the response of the structure will be minimized by a design which places the excitation band within a stop band. For wideband excitation the effect of the structural periodicity is not obvious, as the response will be enhanced within a passband and suppressed within a stop band. The wideband performance of a periodic structure is investigated here by considering the wave transmission coefficient of an N-bay embedded periodic system. Simple closed-form expressions are derived for the passband and wideband-averaged transmission coefficient in the absence of damping; these expressions depend only upon the amplitude of the transmission coefficient of a single joint within the system. Damping is then considered and it is shown that the wideband efficacy of a damping treatment can be significantly enhanced by structural periodicity, and a simple approximate formula for the effect of damping is derived. The analytical results are illustrated by application to an example system which represents a beam on multiple rotational spring supports.