$$\begin{aligned}
\ddot{u}_{1} & =c_{p}^{2}u_{1,11}-\frac{2\mu_{e}}{\rho}\, P_{,1}^{D}-\frac{3\lambda_{e}+2\mu_{e}}{\rho}P_{,1}^{S},\nonumber \\
\nonumber \\
\ddot{P}^{D} & =\frac{4}{3}\,\frac{\mu_{e}}{\eta}u_{1,1}+c_{g}^{2}P_{,11}^{D}-\omega_{s}^{2}P^{D},\label{Longitudinal-1}\\
\nonumber \\
\ddot{P}^{S} & =\frac{3\lambda_{e}+2\mu_{e}}{3\eta}u_{1,1}+c_{g}^{2}P_{,11}^{S}-\omega_{p}^{2}P^{S},\nonumber\end{aligned}$$

-   Two sets of three equations only involving transverse quantities in the $k$-th direction, with $\xi=2,3$:

$$\begin{aligned}
\ddot{u}_{\xi} & =c_{s}^{2}u_{\xi,11}-\frac{2\mu_{e}}{\rho}\, P_{\left(1\xi\right),1}+\frac{\eta}{\rho}\omega_{r}^{2}P_{\left[1\xi\right],1},\nonumber \\
\nonumber \\
\ddot{P}_{\left(1\xi\right)} & =\frac{\mu_{e}}{\eta}\, u_{\xi,1}+c_{g}^{2}P_{(1\xi)}{}_{,11}-\omega_{s}^{2}P_{(1\xi)},\label{Transverse-1}\\
\nonumber \\
\ddot{P}_{\left[1\xi\right]} & =-\frac{1}{2}\omega_{r}^{2}u_{\xi,1}+c_{g}^{2}P_{\left[1\xi\right]}{}_{,11}-\omega_{r}^{2}P_{\left[1\xi\right]},\nonumber\end{aligned}$$

-   One equation only involving the variable $P_{\left(23\right)}$:

$$\ddot{P}_{\left(23\right)}=-\omega_{s}^{2}P_{\left(23\right)}+c_{g}^{2}P_{\left(23\right),11},\label{Shear-1}$$

-   One equation only involving the variable $P_{\left[23\right]}$ :

$$\ddot{P}_{\left[23\right]}=-\omega_{r}^{2}P_{\left[23\right]}+c_{g}^{2}P_{\left[23\right],11},\label{Rotations23-1}$$

-   One equation only involving the variable $P^{V}$:

$$\ddot{P}^{V}=-\omega_{s}^{2}P^{V}+c_{g}^{2}P_{,11}^{V}.\label{VolumeVariation-1}$$

Linear waves in micromorphic media
==================================

In this section we study the dispersion relations for the considered relaxed micromorphic continuum, as well as for the Classical micromorphic continuum. We also consider dispersion relations for Cosserat media obtained as a degenerate limit case of our relaxed model. Finally, dispersion relations for a second gradient continuum obtained as a limit case of the classic micromorphic continuum are also presented. It will be pointed out that only our relaxed micromorphic model can disclose the presence of band-gaps.

Micro-oscillations
------------------

We start studying a particular solution of the set of introduced differential equations by setting $$\begin{gathered}
u_{i}=0,\ i=1,2,3,\quad P_{(23)}=Re\left\{ \alpha_{(23)}e^{i\omega t}\right\} ,\quad P_{[23]}=Re\left\{ \alpha_{[23]}e^{i\omega t}\right\} ,\quad P^{V}=Re\left\{ \alpha^{V}e^{i\omega t}\right\} ,\\
\\
P_{(1\xi)}=Re\left\{ \alpha_{(1\xi)}e^{i\omega t}\right\} ,\quad P_{[1\xi]}=Re\left\{ \alpha_{[1\xi]}e^{i\omega 