waves which have non-vanishing frequency when the wavenumber $k$ is vanishing.

Indeed, we will see in the following, that a third type of wave may exist in relaxed micromorphic media under suitable hypothesis on the constitutive parameters:

-   **standing (or evanescent) waves,** i.e. waves which have imaginary wavenumber $k$ corresponding to some frequency ranges. These waves do not propagate, but keep oscillating in a given, limited region of space.

According to the performed study of micro-oscillations we can conclude that

-   For the **uncoupled waves** $P_{(23)}$, $P_{[23]}$ and $P^{V}$, only optic waves are found with cutoff frequencies $\omega_{s}$, $\omega_{r}$ and $\omega_{s}$. Moreover, the structure of the governing equations for $P_{(23)}$ and $P^{V}$ are formally identical (see Eqs. (\[Shear\]) and (\[VolumeVariation\])) so that the associated dispersion relations will give rise to superimposed curves. In the limit $\mu_{c}\rightarrow0$ one has from Eq. (\[Definitions\]) that $\omega_{r}\rightarrow0.$ This means that in this limit case one has one acoustic waves and two superimposed optic waves with cutoff frequency $\omega_{s}$.

-   For the **longitudinal waves** $u_{1}$, $P^{D}$, $P^{S}$, we can identify one acoustic wave and two optic waves with cutoff frequencies $\omega_{s}$ and $\omega_{p}$. In the limit $\mu_{c}\rightarrow0$, which implies that $\omega_{r}\rightarrow0$, the situation for longitudinal 