in micromorphic media
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In our study of wave propagation in considered micromorphic media, we will limit ourselves to the case of plane waves travelling in an infinite domain. With this end in mind, we can suppose that the space dependence of all the introduced kinematical fields is limited only to the component $X$ of $\mathbf{X}$ which we also suppose to be the direction of propagation of the considered wave. It is immediate that, according to the Cartan-Lie decomposition for the tensor $\mathbf{P}$ (see Eq. (\[CartanLie\])), the component $P_{11}$ of the tensor $\mathbf{P}$ itself can be rewritten as $P_{11}=P^{D}+P^{S}$ where we set $$P^{S}:=\frac{1}{3}\left(P_{11}+P_{22}+P_{33}\right),\qquad P^{D}:=\left(\mathrm{dev\: sym}\,\mathbf{P}\right)_{11}.\label{SpherDev}$$ We also denote the components 12 and 13 of the symmetric and skew-symmetric part of the tensor $\mathbf{P}$ respectively as $$\left(\mathrm{sym}\:\mathbf{P}\right)_{1\xi}=P_{(1\xi)},\qquad\left(\mathrm{skew}\:\mathbf{P}\right)_{1\xi}=P_{\left[1\xi\right]},\quad\xi=1,2.\label{SymSkew}$$ We finally introduce the last new variable $$P^{V}=P_{22}-P_{33}.\label{VolumeVar}$$

The relaxed micromorphic continuum
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We want to rewrite the equations of motion (\[eq:bulk-mod-3-1\]) in terms of the new variables (\[SpherDev\]), (\[SymSkew\]) and, of course, of the displacement variables $u_{i}$. Before doing so, we introduce the quantities[^6] $$\begin{gathered}
c_{m}=\sqrt{\frac{\alpha_{c}}{\eta}},\qquad c_{s}=\sqrt{\frac{\mu_{e}+\mu_{c}}{\rho}},\qquad c_{p}=\sqrt{\frac{\lambda_{e}+2\mu_{e}}{\rho}},\nonumber \\
\nonumber \\
\omega_{s}=\sqrt{\frac{2\left(\mu_{e}+\mu_{h}\right)}{\eta}},\qquad\omega_{p}=\sqrt{\frac{\left(3\lambda_{e}+2\mu_{e}\right)+\left(3\lambda_{h}+2\mu_{h}\right)}{\eta}},\qquad\omega_{r}=\sqrt{\frac{2\mu_{c}}{\eta}},\label{Definitions}\\
\nonumber \\
\omega_{l}=\sqrt{\frac{\lambda_{h}+2\mu_{h}}{\eta}},\qquad\omega_{t}=\sqrt{\frac{\mu_{h}}{\eta}}.\nonumber\end{gathered}$$ With the proposed new choice of variables and considering definitions (\[Definitions\]) we are able to rewrite the governing 