corresponding modes involve both axial and transverse displacements, *i.e.*, there are no pure modes, we refer to those associated with $\shearwavespeed$ and $\pressurewavespeed$ as *quasi-*shear and *quasi-*pressure waves, respectively [@bland1965plane; @ZIV2019mom].

The eigenvectors of the matrix in Eq.$\ $ which are associated with the generated waves are

$$r_{1}=\left(\begin{array}{c}
1\\
\eta_{+}\\
\refden c_{+}\\
\eta_{+}\refden c_{+}
\end{array}\right),\;r_{2}=\left(\begin{array}{c}
-\eta_{-}\\
1\\
-\eta_{-}\refden c_{-}\\
\refden c_{-}
\end{array}\right),\ r_{3}=\left(\begin{array}{c}
-\eta_{-}\\
1\\
\eta_{-}\refden c_{-}\\
-\refden c_{-}
\end{array}\right),\;r_{4}=\left(\begin{array}{c}
1\\
\eta_{+}\\
-\refden c_{+}\\
-\eta_{+}\refden c_{+}
\end{array}\right),\label{eq:eigenvectors}$$

where $$\eta_{+}=\frac{\refden c_{+}^{2}-\piolaonebyP}{\beta},\quad\eta_{-}=\frac{\beta}{\alpha-\refden c_{-}^{2}},\label{eq:eta}$$ and the corresponding characteristic velocities are $-\pressurewavespeed,\,-\shearwavespeed,\,\shearwavespeed$ and $\pressurewavespeed$, respectively. Through algebraic manipulation and using the fact that $\alpha,\beta,$ and $\delta$ are derived from the same scalar potential, we find that $\text{\ensuremath{\eta_{+}}}$ is identical to $\eta_{-}$ and independent of $\rho_{L}$. The components of the eigenvectorsand specifically $\eta_{\pm}$are differential relations between the strain and velocity fields. When the initial conditions are piecewise-constant with a single jump discontinuity and the medium is homogeneous, the problem is called a Riemann problem, which can be solved by a variation of the method of characteristics. The method exploits the fact that the solution can be expressed as function of a single variablehere $X_{1}/t$in which case the corresponding waves are called simple waves [@davison2008book]. The 