Changes between Version 1 and Version 2 of u/johannjc/scratchpad

Timestamp:

07/03/13 18:18:35 (12 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad

@@ -3 +3 @@
 \[ \left [ 
 \begin{array}{c}
-\dot{\rho} \\
-\dot{v} \\
+\partial_t{\rho_1} \\
+\partial_t{v_1} \\
+\partial_{xx}{\phi_1} \\
 \end{array}
 \right ] 
 =\left [
 \begin{array}{c}
--\partial_x \left ( \rho v \right ) \\
--v \partial_x v-c_s^2\partial_x \rho -\partial_x \phi \\
+-\rho_0\partial_x v_1 \\
+-\frac{c_s^2}{\rho_0}\partial_x \rho_1 -\partial_x \phi_1 \\
+4 \pi G \rho_1
 \end{array}
 \right ] = \left [
+\begin{array}{ccc}
+0 & -\rho_0\partial_x & 0 \\
+-\frac{c_s^2}{\rho_0}\partial_x & 0 & -\partial_x \\
+4 \pi G & 0 & 0
+\end{array}
+\right ]
+\left [
 \begin{array}{c}
--\partial_x \left ( \rho_0 v_0 \right ) \\
--v_0 \partial_x v_0-c_s^2\partial_x \rho_0 -\partial_x \phi_0 \\
+\rho_1\\
+v_1 \\
+\phi_1 \\
+\end{array}
+\right ] 
+\]
+\[
+\partial_t
+\left [ 
+\begin{array}{c}
+\rho_1 \\
+v_1 \\
+\end{array}
+\right ] 
+=
+\left [
+\begin{array}{cc}
+0 & -ik\rho_0 \\
+-ik\frac{c_s^2}{\rho_0} + \frac{4 \pi i G}{k} &  0 \\
+\end{array}
+\right ] 
+\left [
+\begin{array}{c}
+\rho_1\\
+v_1 \\
+\end{array}
+\right ] 
+\]
+
+which gives a characteristic equation
+
+}}}
+[[latex($\lambda^2 + k^2 c_s^2 -4 \pi G \rho_0 = 0$)]]
+
+and we have 
+
+[[latex($\lambda = \pm \sqrt{4 \pi G \rho_0-k^2 c_s^2}$)]]
+
+with eigen vectors
+
+
+{{{
+#!latex
+\[
+\left [
+\begin{array}{c}
+k \rho_0 \\
+i \lambda
 \end{array}
 \right ]
 \]
+}}}
+
+So for stable waves we have [[latex($\lambda = i\omega$)]] where [[latex($\omega$)]] is real.  And there are two solutions...
+
+{{{
+#!latex
 \[
-+
 \left [
 \begin{array}{c}
--\partial_x \left ( \rho_1 v_0 \right ) - \partial_x \left ( \rho_0 v_1 \right ) \\
--v_0 \partial_x v_0-c_s^2\partial_x \rho_0 -\partial_x \phi_0 \\
+\rho_1 \\
+v_1 \\
 \end{array}
-\right ] 
+\right ]
+=
+\left [
+\begin{array}{c}
+d\rho e^{\pm i \omega t} e^{i k x} \\
+dv e^{\pm i \omega t} e^{i k x} \\
+\end{array}
+\right ]
 \]
 }}}
+where 
+
+{{{
+#!latex
+$dv = -\frac{\omega}{k}\frac{d\rho}{\rho}$
+}}}
+
+So if [[latex($\lambda=\omega$)]] where [[latex($\omega$)]] is real -  then we have two solutions as well.
+
+{{{
+#!latex
+\[
+\left [
+\begin{array}{c}
+\rho_1 \\
+v_1 \\
+\end{array}
+\right ]
+=
+\left [
+\begin{array}{c}
+d\rho e^{\pm \omega t} e^{i k x} \\
+dv e^{\pm \omega t} e^{i k x + i \pi} \\
+\end{array}
+\right ]
+\]
+}}}
+