the waves associated to the three considered variables $P_{\left(23\right)}$, $P_{\left[23\right]}$ and $P^{V}$ are so-called optic waves. Moreover, we also notice that the dispersion relation for the variables $P_{\left(23\right)}$ and $P^{V}$ are the same: this means that the wave-form solutions for these variables coincide modulo a scalar multiplication factor (see Eqs. (\[WaveForm1\])).

We now want to study harmonic solutions for the differential systems (\[Longitudinal\]) and (\[Transverse\]). To do so, we introduce the unknown vectors $\mathbf{v}_{1}=\left(u_{1},P^{D},P^{S}\right)$ and $\mathbf{v}_{\xi}=\left(u_{\xi},P_{(1\xi)},P_{[1\xi]}\right),\ \xi=2,3$ and look for wave form solutions of equations (\[Longitudinal\]) and (\[Transverse\]) in the form $$\mathbf{v}_{1}=Re\left\{ \boldsymbol{\beta}e^{i(kX-\omega t)}\right\} ,\qquad\mathbf{v}_{\xi}=Re\left\{ \boldsymbol{\gamma}^{\xi}e^{i(kX-\omega t)}\right\} ,\ \xi=2,3\label{WaveForm2}$$ where $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\beta_{3})^{T}$ and $\boldsymbol{\gamma}^{\xi}=(\gamma_{1}^{\xi},\gamma_{2}^{\xi},\gamma_{3}^{\xi})^{T}$ are the unknown amplitudes of considered waves. Replacing this expressions in equations (\[Longitudinal\]) and (\[Transverse\]) one gets respectively $$\mathbf{A}_{1}\cdot\boldsymbol{\beta}=0,\qquad\mathbf{A}_{\xi}\cdot\boldsymbol{\gamma}^{\xi}=0,\qquad\xi=2,3,\label{AlgSys}$$ where $$\begin{gathered}
\mathbf{A}_{1}=\left(\begin{array}{ccc}
-\omega^{2}+c_{p}^{2}\, k^{2} & \, i\: k\:2\mu_{e}/\rho\  & i\: k\:\left(3\lambda_{e}+2\mu_{e}\right)/\rho\\
\\
-i\: k\,\frac{4}{3}\,\mu_{e}/\eta & -\omega^{2}+\frac{1}{3}k^{2}c_{m}^{2}+\omega_{s}^{2} & -\frac{2}{3}\, k^{2}c_{m}^{2}\\
\\
-\frac{1}{3}\, i\, k\:\left(3\lambda_{e}+2\mu_{e}\right)/\eta & -\frac{1}{3}\, k^{2}\, c_{m}^{2} & -\omega^{2}+\frac{2}{3}\, k^{2}\, c_{m}^{2}+\omega_{p}^{2}
\end{array}\right),\\
\\
\\
\mathbf{A}_{2}=\mathbf{A}_{3}=\left(\begin{array}{ccc}
-\omega^{2}+k^{2}c_{s}^{2}\  & \, i\, k\,2\mu_{e}/\rho\  & -i\, k\,\frac{\eta}{\rho}\omega_{r}^{2},\\
\\
-\, i\, k\,2\mu_{e}/\eta, & -2\omega^{2}+k^{2}c_{m}^{2}+2\omega_{s}^{2} & k^{2}c_{m}^{2}\\
\\
i\, k\,\omega_{r}^{2} & k^{2}c_{m}^{2} & -2\omega^{2}+k^{2}c_{m}^{2}+2\omega_{r}^{2}
\end{array}\right).\end{gathered}$$ In order to have non-trivial solutions of the algebraic systems (\[AlgSys\]), one must impose that $$\mathrm{det}\,\mathbf{A}_{1}=0,\qquad\mathrm{det}\,\mathbf{A}_{2}=0,\qquad\mathrm{det}\,\mathbf{A}_{3}=0,\label{Dispersion}$$ which allow 