The paper presents a novel spectral approach, accompanied by an asymptotic model and numerical simulations for slender elastic systems such as long bridges or tall buildings. The focus is on asymptotic approximations of solutions by Bloch waves, which may propagate in a infinite periodic waveguide. Although the notion of passive mass dampers is conventional in the engineering literature, it is not obvious that an infinite waveguide problem is adequate for analysis of long but finite slender elastic systems. The formal mathematical treatment of a Bloch wave would reduce to a spectral analysis of equations of motion on an elementary cell of a periodic structure, with Bloch–Floquet quasi-periodicity conditions imposed on the boundary of the cell. Frequencies of some classes of standing waves can be estimated analytically. One of the applications discussed in the paper is the “dancing bridge” across the river Volga in Volgograd.