Variables
$\rho =$ density
$v^i =$ velocity
$P =$ pressure
$\gamma =$ heat capacity ratio

Euler Fluid Equations
$\rho_{,t} + (\rho v^j)_{,j} = 0$
$(\rho v^i)_{,t} + (\rho v^i v^j + \delta^{ij} P)_{,j} = 0$
$(\frac{1}{2} \rho v^j v_j + \frac{1}{\gamma-1} P)_{,t} + (v^j (\frac{1}{2} \rho v^k v_k + \frac{\gamma}{\gamma-1} P))_{,j} = 0$

Expanded:
$\rho_{,t} + \rho_{,j} v^j + \rho {v^j}_{,j} = 0$
$\rho_{,t} v^i + \rho {v^i}_{,t} + \rho_{,j} v^i v^j + \rho {v^i}_{,j} v^j + \rho v^i {v^j}_{,j} + \delta^{ij} P_{,j} = 0$
$\frac{1}{2} \rho_{,t} v^j v_j + \rho {v^j}_{,t} v_j + \frac{1}{\gamma-1} P_{,t} + {v^j}_{,j} (\frac{1}{2} \rho v^k v_k + \frac{\gamma}{\gamma-1} P) + v^j (\frac{1}{2} \rho_{,j} v^k v_k + \rho v^k v_{k,j} + \frac{\gamma}{\gamma-1} P_{,j}) = 0$

Let $q = \hat{q} exp(i \tilde{q}_\mu x^\mu)$ for $x^\mu \in \{ t,x,y,z \}$.
Therefore $q_{,\mu} = \tilde{q}_\mu q$.

Let $v^t = 1$ for simplicity sake.
Use $\eta_{\mu\nu}$ as the background metric, so $v_t = -1$.

$\rho_{,t} + \rho_{,j} v^j + \rho {v^j}_{,j} = 0$
$\tilde{\rho}_t \rho + \tilde{\rho}_j \rho v^j + \rho \tilde{v}_j v^j = 0$
$\tilde{\rho}_t + \tilde{\rho}_j v^j + \tilde{v}_j v^j = 0$
$\tilde{\rho}_t = - (\tilde{\rho}_j + \tilde{v}_j) v^j$
$\tilde{\rho}_\mu v^\mu = -\tilde{v}_j v^j$

$\rho_{,t} v^i + \rho {v^i}_{,t} + \rho_{,j} v^i v^j + \rho {v^i}_{,j} v^j + \rho v^i {v^j}_{,j} + \delta^{ij} P_{,j} = 0$
$\tilde{\rho}_t \rho v^i + \rho \tilde{v}_t v^i + \tilde{\rho}_j \rho v^i v^j + \rho \tilde{v}_j v^i v^j + \rho v^i \tilde{v}_j v^j + \tilde{P}^i P = 0$
substitute $\tilde{\rho}_t + \tilde{\rho}_j v^j + \tilde{v}_j v^j = 0$
$\rho \tilde{v}_t v^i + \rho \tilde{v}_j v^i v^j + \tilde{P}^i P = 0$
$\rho v^i (\tilde{v}_t + \tilde{v}_j v^j) + \tilde{P}^i P = 0$
$\tilde{v}_\mu v^\mu v^i = -\frac{1}{\rho} P \tilde{P}^i$

$\frac{1}{2} \rho_{,t} v^j v_j + \rho {v^j}_{,t} v_j + \frac{1}{\gamma-1} P_{,t} {v^j}_{,j} (\frac{1}{2} \rho v^k v_k + \frac{\gamma}{\gamma-1} P) + v^j (\frac{1}{2} \rho_{,j} v^k v_k + \rho v^k v_{k,j} + \frac{\gamma}{\gamma-1} P_{,j}) = 0$
$\frac{1}{2} \tilde{\rho}_t \rho v^j v_j + \rho \tilde{v}_t v^j v_j + \frac{1}{\gamma-1} \tilde{P}_t P + \frac{1}{2} \rho \tilde{v}_j v^j v^k v_k + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \frac{1}{2} \tilde{\rho}_j v^j \rho v^k v_k + \rho v^k \tilde{v}_j v^j v_k + \frac{\gamma}{\gamma-1} \tilde{P}_j v^j P = 0$
substitute $\tilde{\rho}_t + \tilde{\rho}_j v^j + \tilde{v}_j v^j = 0$
$\rho \tilde{v}_t v^j v_j + \frac{1}{\gamma-1} \tilde{P}_t P + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \rho v^k \tilde{v}_j v^j v_k + \frac{\gamma}{\gamma-1} \tilde{P}_j v^j P = 0$
substitute $\rho \tilde{v}_t v^i + \rho \tilde{v}_j v^i v^j + \tilde{P}^i P = 0$
$\frac{1}{\gamma-1} \tilde{P}_t P + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \frac{\gamma}{\gamma-1} \tilde{P}_j v^j P - \tilde{P}^k P v_k = 0$
$\tilde{P}_t P + \gamma \tilde{v}_j v^j P + \tilde{P}_j P v^j = 0$
Assume $P > 0$
$\tilde{P}_t + \tilde{P}_j v^j + \gamma \tilde{v}_j v^j = 0$
$\tilde{P}_t + (\tilde{P}_j + \gamma \tilde{v}_j) v^j = 0$
$\tilde{P}_\mu v^\mu = -\gamma \tilde{v}_j v^j$

Summary:
$\tilde{\rho}_\mu v^\mu = -\tilde{v}_j v^j$
$\tilde{v}_\mu v^\mu v^i = -\frac{1}{\rho} P \tilde{P}^i$
$\tilde{P}_\mu v^\mu = -\gamma \tilde{v}_j v^j$