whole gradient of the micro-deformation tensor plays a role (instead of only its Curl). We show that also the classical micromorphic continuum is not suitable to account for the decription of band-gaps. Finally, a particular second gradient material is obtained as limit case of the proposed classical micromorphic continuum by letting $\mu_{e}\rightarrow\infty$ and $\mu_{c}\rightarrow\infty$. Also in this case, we will show that the existence of band-gaps cannot be accounted for.

We can summarize by saying that the proposed relaxed micromorphic model (6 elastic coefficients) accounts for the description of frequency band-gaps in microstructured media which are “switched on” by the only parameter $\mu_{c}$ related to relative rotations of the microstucture with respect to the solid macroscopic matrix. These band gaps cannot be predicted by means of classical micromorphic or Cosserat and second gradient media, as it will be extensively pointed out in the body of the paper. Moreover, band gaps are impossible to observe in the purely one-dimensional situation, since, among others, the Curl operator vanishes in this simplified case. This absence of band gaps is confirmed by the results found in [@berezowski] in which a one-dimensional micromorphic model is presented together with a detailed analysis of wave propagation.

We 