waves remains unchanged.

-   For the **transverse waves** $u_{\xi}$, $P_{(1\xi)}$, $P_{[1\xi]}$, ($\xi=1,2$) we identify one acoustic wave and two optic waves with cutoff frequencies $\omega_{s}$ and $\omega_{r}$. In the limit $\mu_{c}\rightarrow0$ which implies that $\omega_{r}\rightarrow0$, we have two acoustic waves and one optic wave with cutoff frequency $\omega_{s}$.

Planar wave propagation in the relaxed micromorphic continuum
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We now look for a wave form solution of the previously derived equations of motion. We start from the uncoupled equations (\[Shear\])-(\[VolumeVariation\]) and assume that the involved unknown variables take the harmonic form $$P_{\left(23\right)}=Re\left\{ \beta_{\left(23\right)}e^{i(kX-\omega t)}\right\} ,\quad P_{\left[23\right]}=Re\left\{ \beta_{\left[23\right]}e^{i(kX-\omega t)}\right\} ,\quad P^{V}=Re\left\{ \beta^{V}e^{i(kX-\omega t)}\right\} ,\label{WaveForm1}$$ where $\beta_{\left(23\right)}$, $\beta_{\left[23\right]}$ and $\beta^{V}$ are the amplitudes of the three introduced waves. It can be remarked that the variables $P_{\left(23\right)}$, $P_{\left[23\right]}$ and $P^{V}$ respectively represent transverse (with respect to wave propagation) micro-shear, transverse micro-rotation and transverse micro-deformations at constant volume. Replacing this wave form in Eqs. (\[Shear\])-(\[VolumeVariation\]) and simplifying one obtains the following dispersion relations respectively: $$\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{m}^{2}},\qquad\omega(k)=\sqrt{\omega_{r}^{2}+k^{2}c_{m}^{2}},\qquad\omega(k)=\sqrt{\omega_{s}^{2}+k^{2}c_{m}^{2}}.\label{Eigen1}$$ We notice that for a vanishing wave number ($k=0$) the dispersion relations for the three considered waves give non-vanishing frequencies which correspond to the ones calculated in the previous subsection for the same waves. This is equivalent to say that 