strains and velocities of the axial and transverse displacements, namely, $$\pressurestrain:=\derivative{u_{1}}{X_{1}},\ \shearstrain:=\derivative{u_{2}}{X_{1}},\ \pressurevel:=\derivative{u_{1}}t,\ \shearvel:=\derivative{u_{2}}t.\label{eq:derivs}$$ In the limiting case of a simple shear $\left(u_{1}=0\right)$, the quantity $\shearstrain$ is called the amount of shear [@ogden97book], where in the limiting case of a uniaxial deformation $\left(u_{2}=0\right)$, the quantity $\pressurestrain$ is related to the axial stretch. We use Eq.$\ $ to carry out a reduction of order and rewrite Eq.$\ $ as $$\left(\begin{array}{c}
\pressurestrain\\
\shearstrain\\
\refden\pressurevel\\
\refden\shearvel
\end{array}\right)_{,t}+\left(\begin{array}{cccc}
0 & 0 & -\frac{1}{\refden} & 0\\
0 & 0 & 0 & -\frac{1}{\refden}\\
-\piolaonebyP & -\beta & 0 & 0\\
-\gamma & -\delta & 0 & 0
\end{array}\right)\left(\begin{array}{c}
\pressurestrain\\
\shearstrain\\
\refden\pressurevel\\
\refden\shearvel
\end{array}\right)_{,X_{1}}=\left(\begin{array}{c}
0\\
0\\
0\\
0
\end{array}\right),\label{eq:conservativeSystem}$$ where

$$\piolaonebyP=\derivative{\piolacomp{11}}{\pressurestrain},\;\piolaonebyQ=\derivative{\piolacomp{11}}{\shearstrain},\;\piolatwobyP=\derivative{\piolacomp{21}}{\pressurestrain},\;\piolatwobyQ=\derivative{\piolacomp{21}}{\shearstrain}.\label{eq:defineCoeff}$$

For hyperelastic materials we have that $\piolaonebyQ\equiv\piolatwobyP$ owing to the equality of mixed partials of $\hyper$.

The eigenvalues of Eq.$\ $ are the characteristic wave velocities at each material point, and are functions of the phase and deformation at that point, given by

$$c=\pm\sqrt{\frac{1}{2\refden}\left[\piolaonebyP+\piolatwobyQ\pm\sqrt{\left(\piolaonebyP-\piolatwobyQ\right)^{2}+4\piolaonebyQ\piolatwobyP}\right]}\eqqcolon\pm c_{\pm},\label{eq:wavespeeds}$$

where $c_{+}$ (resp.$\ $$c_{-}$) corresponds to the plus (resp.$\ $minus) sign of the inner square root. In the limit of linear elasticity, $c_{+}$ and $c_{-}$ are the velocities of pressure and shear waves, respectively [@DAVISON1966249]. Since in finite elasticity the 