[ \left [ \begin{array}{c} \partial_t{\rho_1} \\ \partial_t{v_1} \\ \partial_{xx}{\phi_1} \\ \end{array} \right ] =\left [ \begin{array}{c} -\rho_0\partial_x v_1 \\ -\frac{c_s^2}{\rho_0}\partial_x \rho_1 -\partial_x \phi_1 \\ 4 \pi G \rho_1 \end{array} \right ] = \left [ \begin{array}{ccc} 0 & -\rho_0\partial_x & 0 \\ -\frac{c_s^2}{\rho_0}\partial_x & 0 & -\partial_x \\ 4 \pi G & 0 & 0 \end{array} \right ] \left [ \begin{array}{c} \rho_1\\ v_1 \\ \phi_1 \\ \end{array} \right ] \] \[ \partial_t \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = \left [ \begin{array}{cc} 0 & -ik\rho_0 \\ -ik\frac{c_s^2}{\rho_0} + \frac{4 \pi i G}{k} & 0 \\ \end{array} \right ] \left [ \begin{array}{c} \rho_1\\ v_1 \\ \end{array} \right ] \] which gives a characteristic equation

\lambda^2 + k^2 c_s^2 -4 \pi G \rho_0 = 0

and we have

\lambda = \pm \sqrt{4 \pi G \rho_0-k^2 c_s^2}

with eigen vectors

[ \left [ \begin{array}{c} k \rho_0 \\ i \lambda \end{array} \right ] \]

So for stable waves we have \lambda = i\omega where \omega is real. And there are two solutions…

[ \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = \left [ \begin{array}{c} d\rho e^{\pm i \omega t} e^{i k x} \\ dv e^{\pm i \omega t} e^{i k x} \\ \end{array} \right ] \]

where

dv = -\frac{\omega}{k}\frac{d\rho}{\rho}$

So if \lambda=\omega where \omega is real - then we have two solutions as well.

[ \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = \left [ \begin{array}{c} d\rho e^{\pm \omega t} e^{i k x} \\ dv e^{\pm \omega t} e^{i k x + i \pi} \\ \end{array} \right ] \]