n}|_e&i \omega \kappa_W \beta &1+i \omega \tau_W
\end{pmatrix} , \nonumber \\\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^E_{xy} = \mathcal{M}^E_{xz} =
\begin{pmatrix}
i\omega h&i \omega \\
i \omega \kappa_W \beta &1+i\omega \tau_V 
\end{pmatrix} .\end{aligned}$$ The longitudinal equations $\det(\mathcal{M}^E_{xx}) = 0$ lead to $$\begin{aligned}
\omega^2 - c_s^2 k^2 = \frac{i \kappa_W (d_2 \omega^2 - d_4 k^2) k^2 }{\omega [1+i (\tau_W - \kappa_W \beta/h ) \omega]} , \label{eq:MExx}\end{aligned}$$ where $$\begin{aligned}
d_2 &=& \frac{n}{h} \bigg( \frac{\partial \beta}{\partial n} \bigg|_e + \frac{\beta}{h} \frac{\partial P}{\partial n}\bigg|_e \bigg),\\
d_4 &=& \frac{n}{h} \bigg( \frac{\partial \beta}{\partial n} \bigg|_e \frac{\partial P}{\partial e}\bigg|_n  - \frac{\partial \beta}{\partial e}\bigg|_n \frac{\partial P}{\partial n}\bigg|_e \bigg) .\end{aligned}$$ Here, $\mathrm{Im} (\omega)$ stays semi-positive when $d_2 \geq 0$, $c_s^2 d_2  - d_4 \geq 0$, and $$\begin{aligned}
\tau_W - \frac{\beta}{h} \kappa_W \geq 0, \label{eq:twkw}\end{aligned}$$ according to the Routh-Hurwitz stability criteria. The first two conditions are again satisfied in thermodynamic systems as $$\begin{aligned}
d_2 &=& \frac{n^2}{h^2} \frac{\partial \alpha}{\partial n}\bigg|_e \geq 0,\\
c_s^2 d_2  - d_4 &=& \frac{n^2 \beta}{h^3} 
\bigg( \frac{\partial P}{\partial \alpha}\bigg|_\beta \frac{\partial \alpha}{\partial n} \bigg|_e - \frac{\partial P}{\partial \beta}\bigg|_\beta \frac{\partial \alpha}{\partial e} \bigg|_n \bigg)^2 \geq 0,\nonumber \\\end{aligned}$$ using the relations (\[eq:dpda\]) and (\[eq:bnae\]). Note that $d_2 = c_2 n^2/h^2$ and $d_4 = c_4 n^2/h^2$. The third condition is also consistent with the 