I have three equations as follows (for a polytropic gas):
1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$
2) $\displaystyle\quad\rho \left( \frac{\partial \mathbf{u}}{\partial t}+\mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p$
3) $\displaystyle\quad \frac{\partial}{\partial t} \left(\rho \varepsilon + \frac{\rho u^2}{2}\right) = \nabla\cdot \left[\rho \mathbf{u} \left(\varepsilon\frac {u^2}{2}+p \mathbf{u}\right)\right]$
where $u$, $\rho$, and $p$ are the velocity, density, and pressure, respectively.
From these equations,
I would like to derive:
$$ \left( \frac{\partial}{\partial t} + \mathbf{u}\cdot \nabla \right) p-c_s^2 \left( \frac{\partial} { \partial t}+\mathbf{u}\cdot \nabla \right) \rho = 0 $$
I am not sure how to approach this problem and so I have come here for some help. Can anyone see a way forward for deriving this final equation from the ones provided?
I know that:
1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\mathbf{u}\cdot\nabla p = -\gamma p\nabla \cdot \mathbf{u}$
$\frac{\nabla P}{\rho U^2/L}$ ~ $\frac{P}{L}/\frac{\rho u^2}{L} = \frac{P}{\rho u^2} $ which is proportional to $\frac {C_s^2}{u^2} $
Note that: $\displaystyle\quad\varepsilon = \frac{p}{[\rho(\gamma -1)]}$