equations as different uncoupled sets of equations, namely:

-   A set of three equations only involving longitudinal quantities:

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

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

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

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

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

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

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

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

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

These 12 scalar differential equations will be used to study wave propagation in our relaxed micromorphic media.

It can be checked that, in order to guarantee positive definiteness of the potential energy (\[DefPos\]) , the characteristic velocities and frequencies introduced in Eq.(\[Definitions\]) cannot be chosen in a completely arbitrary way. In the numerical simulations considered in this paper, the values of the elastic parameters are always chosen in order to guarantee positive definiteness of the potential energy according to Eqs.(\[DefPos-1\]).

The classical micromorphic continuum
------------------------------------

For the classical micromorphic continuum, introducing the new characteristic velocity $$c_{g}=\sqrt{\frac{\alpha_{g}}{\eta}},$$ we get the following simplified one-dimensional equations

-   Longitudinal

