n}|_e&i\omega h&0\\
0&i\omega&-ikn&-ik\\
-ik\kappa_V \frac{\partial \alpha}{\partial e}|_n&-ik\kappa_V \frac{\partial \alpha}{\partial n}|_e&0&1+i \omega \tau_V
\end{pmatrix} , \nonumber \\\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}^L_{xy} = \mathcal{M}^L_{xz} =
\begin{pmatrix}
i\omega h &0 \\
0&1+i\omega \tau_V 
\end{pmatrix} ,\end{aligned}$$ using the enthalpy density $h = e+P$. They have non-trivial solutions when the matrices have vanishing determinants. The longitudinal equations $\det(\mathcal{M}^L_{xx}) = 0$ lead to $$\begin{aligned}
\omega^2 - c_s^2 k^2 = \frac{i \kappa_V (c_2 \omega^2 - c_4 k^2) k^2 }{\omega (1+i \tau_V \omega)} , \label{eq:MLxx}\end{aligned}$$ where the sound velocity is defined as $$\begin{aligned}
c_s^2 = \frac{\partial P}{\partial e}\bigg|_n + \frac{n}{h} \frac{\partial P}{\partial n}\bigg|_e , \label{eq:cs2}\end{aligned}$$ and the thermodynamic coefficients as $$\begin{aligned}
c_2 &=& \frac{\partial \alpha}{\partial n} \bigg|_e ,\\
c_4 &=& \frac{\partial \alpha}{\partial n} \bigg|_e \frac{\partial P}{\partial e}\bigg|_n  - \frac{\partial \alpha}{\partial e}\bigg|_n \frac{\partial P}{\partial n}\bigg|_e .\end{aligned}$$ The Routh-Hurwitz stability criteria [@Routh; @Hurwitz] indicate that the $\mathrm{Im} (\omega)$ stays semi-positive when $c_2 \geq 0$ and $c_s^2 c_2 - c_4 \geq 0$. Those conditions are satisfied in thermodynamic systems since the former follows from the thermodynamic requirement that the fugacity should increase as the number density increases at a fixed energy density and the latter from $$\begin{aligned}
c_s^2 c_2 - c_4 &=& \frac{\beta}{h} 
\bigg( \frac{\partial P}{\partial \alpha}\bigg|_\beta \frac{\partial \alpha}{\partial n} \bigg|_e - 