us to determine so-called dispersion relations $\omega=\omega\left(k\right)$ for the longitudinal and transverse waves in the relaxed micromorphic conyinuum. As it will be better explained in the next section, the eigenvalues $\omega(k)$ solutions of (\[Dispersion\]) are associated both to optic waves and to acoustic waves.

Planar wave propagation in the classical micromorphic continuum
---------------------------------------------------------------

We want to deduce in this subsection the dispersion relations for the classical micromorphic model, i.e. for the dispersion relations associated to the energy (\[KinPot-1-1\]). If we replace the wave-form solution (\[WaveForm2\]) in the longitudinal and transverse equations (\[Longitudinal-1\]) and (\[Transverse-1\]) we get $$\mathbf{B}_{1}\cdot\boldsymbol{\beta}=0,\qquad\mathbf{B}_{\xi}\cdot\boldsymbol{\gamma}^{\xi}=0,\qquad\xi=2,3,\label{AlgSys-1}$$ where $$\begin{gathered}
\mathbf{B}_{1}=\left(\begin{array}{ccc}
-\omega^{2}+c_{p}^{2}\, k^{2} & \, i\: k\:2\mu_{e}/\rho\  & i\: k\:\left(3\lambda_{e}+2\mu_{e}\right)/\rho\\
\\
-i\: k\,\frac{4}{3}\mu_{e}/\eta & -\omega^{2}+k^{2}c_{g}^{2}+\omega_{s}^{2} & 0\\
\\
-\frac{1}{3}\, i\, k\:\left(3\lambda_{e}+2\mu_{e}\right)/\eta & 0 & -\omega^{2}+k^{2}\, c_{g}^{2}+\omega_{p}^{2}
\end{array}\right),\\
\\
\\
\mathbf{B}_{2}=\mathbf{B}_{3}=\left(\begin{array}{ccc}
-\omega^{2}+k^{2}c_{s}^{2}\  & \, i\, k\,2\mu_{e}/\rho\  & -i\, k\,\frac{\eta}{\rho}\omega_{r}^{2},\\
\\
-\, i\, k\,2\mu_{e}/\eta, & -2\omega^{2}+2\, k^{2}c_{g}^{2}+2\omega_{s}^{2} & 0\\
\\
i\, k\,\omega_{r}^{2} & 0 & -2\omega^{2}+2\, k^{2}c_{g}^{2}+2\omega_{r}^{2}
\end{array}\right).\end{gathered}$$ In order to have non-trivial solutions of the algebraic systems (\[AlgSys-1\]), one must impose that $$\mathrm{det}\,\mathbf{B}_{1}=0,\qquad\mathrm{det}\,\mathbf{B}_{2}=0,\qquad\mathrm{det}\,\mathbf{B}_{3}=0,\label{Dispersion-1}$$ which allow us to determine so-called dispersion relations $\omega=\omega\left(k\right)$ for the longitudinal and transverse waves in the classical micromorphic continuum.

As for the uncoupled waves, the dispersion relations associated 