  This paper brings a comparative analysis between dynamic models of
couple-stress elastic materials and structured Rayleigh beams on a Winkler
foundation. Although physical phenomena have different physical origins, the
underlying equations appear to be similar, and hence mathematical models have a
lot in common. In the present work, our main focus is on the analysis of
dispersive waves, band-gaps and localized waveforms in structured Rayleigh
beams. The Rayleigh beam theory includes the effects of rotational inertia
which are neglected in the Euler--Bernoulli beam theory. This makes the
approach applicable to higher frequency regimes. Special attention is given to
waves in pre-stressed Rayleigh beams on elastic foundations.
