finally remark that there are some similarities between dispersion relations for micromorphic continua (including the proposed relaxed model) and dispersion relations of various type of plates models, see [@Victor1] where Kirchhoff, Mindlin, Reissner types of plates including consideration of rotatory inertia are investigated. In particular, for almost all models of plates there are optical branches of dispersion relations and linear asymptotes when the wave number tends to infinity. On the other hand, in the theory of plates there are no band gaps discovered within the considered models. Also porous media with strong contrast between the solid and fluid phases show a marked dispersive behavior (see e.g. [@Steeb; @Steeb1]), but band gaps are not observed in such media.

Notations
---------

Let $\mathbf{x},\mathbf{y}\in\mathbb{R}^{3}$ be two vectors and $\mathbf{X},\mathbf{Y}\in\mathbb{R}^{3\times3}$ be two second order tensors of components $x_{i},y_{i}$ and $X_{ij},Y_{ij}$, $i,j=\{1,2,3\}$, respectively. We let[^2] $\text{<}\mathbf{x},\,\mathbf{y}>_{\mathbb{R}^{3}}=x_{i}y_{i}$ and $\text{<}\mathbf{X},\,\mathbf{Y}>_{\mathbb{R}^{3\times3}}=X_{ij}Y_{ij}$ denote the scalar product on $\mathbb{R}^{3}$ and $\mathbb{R}^{3\times3}$ with associated vector norms $\left\Vert \mathbf{x}\right\Vert _{\mathbb{R}^{3}}^{2}=\text{<}\mathbf{x},\,\mathbf{x}>_{\mathbb{R}^{3}}$ and $\left\Vert \mathbf{X}\right\Vert _{\mathbb{R}^{3\times3}}^{2}=\text{<}\mathbf{X},\,\mathbf{X}>_{\mathbb{R}^{3\times3}}$, respectively. In what follows, we omit the subscripts $\mathbb{R}^{3}$ and $\mathbb{R}^{3\times3}$ if no confusion can arise. We define the standard divergence, gradient and curl operators for vectors and second order tensors respectively as[^3] $$\begin{gathered}
\mathrm{Div}\,\mathbf{x}=x_{i,i},\quad\left(\,\nabla\,\mathbf{x}\,\right)_{ij}=x_{i,j},\quad\left(\,\mathrm{Curl}\,\mathbf{x}\,\right)_{i}=x_{a,b}\,\epsilon_{iab},\\
\\
\left(\,\mathrm{Div}\,\mathbf{X}\,\right)_{i}=X_{ij,j},\quad\left(\,\nabla\,\mathbf{X}\,\right)_{ijk}=X_{ij,k},\quad\left(\,\mathrm{Curl}\,\mathbf{X}\,\right)_{ij}=X_{ia,b}\epsilon_{jab},\end{gathered}$$ where 