\frac{\partial P}{\partial \beta}\bigg|_\beta \frac{\partial \alpha}{\partial e} \bigg|_n \bigg)^2 \geq 0 ,\end{aligned}$$ where $\beta = 1/T$, using the definition of the sound velocity (\[eq:cs2\]) and the thermodynamic properties $$\begin{aligned}
\frac{\partial P}{\partial \alpha}\bigg|_\beta &=& \frac{n}{\beta} , \label{eq:dpda}\ \
\frac{\partial P}{\partial \beta}\bigg|_\alpha = - \frac{h}{\beta} ,\end{aligned}$$ and $$\begin{aligned}
\frac{\partial \beta}{\partial n} \bigg|_e &=& -  \frac{\partial \alpha}{\partial e} \bigg|_n \label{eq:bnae}.\end{aligned}$$

Although it is possible to analytically solve the quartic equation, the general solutions are complicated. Here asymptotic forms at small $k$ are considered for more physical arguments. The propagating modes are, up to the leading order in real and imaginary parts, $$\begin{aligned}
\omega = \pm c_s k + i \frac{\kappa_V (c_s^2 c_2 - c_4 )}{2 c_s^2} k^2 ,\end{aligned}$$ and the non-propagating mode is $$\begin{aligned}
\omega = \frac{i}{\tau_V} ,\end{aligned}$$ aside from the trivial $\omega = 0$. They satisfy the causality condition $$\begin{aligned}
\bigg | \frac{\partial \mathrm{Re} (\omega)}{\partial k} \bigg | \leq 1.\end{aligned}$$ The stability condition $$\begin{aligned}
\mathrm{Im} (\omega) \geq 0,\end{aligned}$$ is satisfied for $c_2 c_s^2 - c_4\geq 0$, which is consistent with the Routh-Hurwitz stability conditions.

The solutions to the transverse equations $$\begin{aligned}
\det(\mathcal{M}^L_{xy}) &=& \det(\mathcal{M}^L_{xz}) \nonumber \\
&=& i \omega h (1+i\omega \tau_V) = 0, \label{eq:MLxy}\end{aligned}$$ are the non-propagating modes $\omega = 0$ and 