| 16 | |

| 17 | When |

| 18 | |

| 19 | [[latex($\omega^2 < 0$)]] |

| 20 | |

| 21 | or |

| 22 | |

| 23 | [[latex($4 \pi G \rho_0 > C_s^2 k^2 $)]] |

| 24 | |

| 25 | we see that [[latex($\omega$)]] is imaginary. This implies that there is an exponential growth or decay factor multiplying the oscillatory [[latex($e^{i(\vec{k}\cdot \vec{x})}$)]] in the solution for [[latex($\rho_1$)]]. That is, |

| 26 | |

| 27 | [[latex($\rho_1 = e^{i(\vec{k} \cdot \vec{x})} e^{-i(^+_- i \omega t)} =$)]] |

| 28 | |

| 29 | [[latex($ e^{i(\vec{k} \cdot \vec{x})} e^{^+_- \omega t}$)]] |

| 30 | |

| 31 | Taking the real part, we have: |

| 32 | |

| 33 | [[latex($\rho_1 = \cos(kx)e^{\omega t}$)]] |

| 34 | |

| 35 | Now, we can rewrite [[latex($\omega$)]] into a more intuitive form. |

| 36 | |

| 37 | |

| 38 | |