micromorphic media is affected by the presence of microstructure in a quite controllable manner. More precisely, we show that, if only longitudinal waves are allowed to propagate in the considered medium (no displacements and micro-deformations allowed in the direction orthogonal to the direction of wave propagation), then the presence of band-gaps can be forecasted only with 5 parameters. The main features of these longitudinal band gaps are seen to be directly related to the absolute rotations of the microstructure. Nevertheless, the proposed 5-parameters relaxed model is not able to account for the presence of band-gaps when also transverse waves are allowed to propagate in the considered material. In order to treat the more general case, we hence consider a slightly generalized relaxed model by introducing only one extra elastic parameter, the so-called *Cosserat couple modulus* $\mu_{c}$, which is related to the relative deformation of the microstructure with respect to the macroscopic matrix. We hence end up with a relaxed micromorphic model with only 6 parameters (5+1) which is able to account for the prediction of band-gaps when generic waves can travel in the considered medium.

Moreover, we consider the so-called “classical micromorphic medium”, i.e. a micromorphic continuum in which the 