t}\right\} ,\ \xi=2,3.\end{gathered}$$ Replacing these expressions for the unknown variables in each of Eqs.(\[Longitudinal\])-(\[VolumeVariation\]), noticing that the space derivatives are vanishing, one gets that the first of Eqs.(\[Longitudinal\]) and (\[Transverse\]) are such that the frequency calculated for the waves $u_{1}$ and $u_{\xi}$ is vanishing, i.e. $$\omega=0.\label{Acoustic}$$ Moreover, from the remaining equations one gets the values of frequency $\text{\ensuremath{\omega}}$ for the waves $P^{D}$, $P^{S}$, $P_{(1\xi)}$, $P_{[1\xi]}$, $P_{(23)}$, $P_{[23]}$ and $P^{V}$ respectively $$\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{p}^{2}\quad\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{r}^{2},\quad\omega^{2}=\omega_{s}^{2},\quad\omega^{2}=\omega_{r}^{2},\quad\omega^{2}=\omega_{s}^{2}.\label{Optic}$$ It is clear that one gets exactly the same result when considering the complete Mindlin-Eringen model, since the only difference is on the space derivatives of $\mathbf{P}$ which do not intervene when studying micro-oscillations. The characteristic values of the frequencies given in Eq.(\[Optic\]) are fixed once the material parameters of the system are specified (see Eqs.(\[Definitions\])) and they represent the limit values for $k\rightarrow0$ of the eigenvalues $\omega(k)$ associated to propagative waves, as it will be better shown in the next section. This preliminary study of micro-oscillations allows a precise classification of waves which can propagate in a micromorphic medium. More particularly, we can distinguish two types of propagative waves:

-   **acoustic waves**, i.e. waves which have vanishing frequency when the wavenumber $k$ is vanishing,

-   **optic waves,** i.e. 