[ \left [ \begin{array}{c} \dot{\rho} \\ \dot{v} \\ \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 \\ \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 \\ \end{array} \right ] \] \[ + \left [ \begin{array}{c} -\partial_x \left ( \rho_1 v_0 \right ) - \partial_x \left ( \rho_0 v_1 \right ) \\ -v_0 \partial_x v_0-c_s^2\partial_x \rho_0 -\partial_x \phi_0 \\ \end{array} \right ] \]