Variables
$\rho =$ density
$v^i =$ velocity
$B^i =$ magnetic field
$P =$ pressure

Ideal MHD Equations
$\rho_{,t} + (\rho v^j)_{,j} = 0$
$(\rho v^i)_{,t} + (\rho v^i v^j + \delta^{ij} (P + \frac{1}{2 \mu_0} B^k B_k) - \frac{1}{\mu_0} B^i B^j)_{,j} = 0$
${B^i}_{,t} + (B^i v^j - B^j v^i)_{,j} = 0$
$(\frac{1}{\gamma-1} P + \frac{1}{2} \rho v^k v_k + \frac{1}{2 \mu_0} B^k B_k)_{,t} + (( \frac{\gamma}{\gamma-1} P + \frac{1}{2} \rho v^k v_k + \frac{1}{\mu_0} B^k B_k ) v^j - \frac{1}{\mu_0} B^k v_k B^j)_{,j} = 0$

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} + P^{,i} + \frac{1}{\mu_0} \delta^{ij} B^k B_{k,j} - \frac{1}{\mu_0} {B^i}_{,j} B^j - \frac{1}{\mu_0} B^i {B^j}_{,j} = 0$
$ {B^i}_{,t} + {B^i}_{,j} v^j + B^i {v^j}_{,j} - {B^j}_{,j} v^i - B^j {v^i}_{,j} = 0$
$ \frac{1}{\gamma-1} P_{,t} + \frac{1}{2} \rho_{,t} v^k v_k + \rho v^k v_{k,t} + \frac{1}{\mu_0} B^k B_{k,t} + \frac{\gamma}{\gamma-1} P_{,j} v^j + \frac{\gamma}{\gamma-1} P {v^j}_{,j} + \frac{1}{2} \rho_{,j} v^k v_k v^j + \rho v^k v_{k,j} v^j + \frac{1}{2} \rho v^k v_k {v^j}_{,j} + \frac{2}{\mu_0} B^k B_{k,j} v^j + \frac{1}{\mu_0} B^k B_k {v^j}_{,j} - \frac{1}{\mu_0} {B^k}_{,j} v_k B^j - \frac{1}{\mu_0} B^k v_{k,j} B^j - \frac{1}{\mu_0} B^k v_k {B^j}_{,j} = 0$

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

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

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

$\rho_{,t} v^i + \rho {v^i}_{,t} + \rho_{,j} v^i v^j + \rho {v^i}_{,j} v^j + \rho v^i {v^j}_{,j} + P^{,i} + \frac{1}{\mu_0} \delta^{ij} B^k B_{k,j} - \frac{1}{\mu_0} {B^i}_{,j} B^j - \frac{1}{\mu_0} B^i {B^j}_{,j} = 0$
$\tilde{\rho}_t \rho v^i + \tilde{v}_t \rho v^i + \tilde{\rho}_j \rho v^i v^j + 2 \tilde{v}_j \rho v^i v^j + \tilde{P}^i P + \frac{1}{\mu_0} \tilde{B}^i B^k B_k - \frac{2}{\mu_0} \tilde{B}_j B^i B^j = 0$
Substitute $\tilde{\rho}_t + \tilde{\rho}_j v^j + \tilde{v}_j v^j = 0$.
$\tilde{v}_t \rho v^i + \tilde{v}_j \rho v^i v^j + \tilde{P}^i P + \frac{1}{\mu_0} \tilde{B}^i B^k B_k - \frac{2}{\mu_0} \tilde{B}_j B^i B^j = 0$
$\tilde{v}_\mu v^\mu \rho v^i + \tilde{P}^i P + \frac{1}{\mu_0} \tilde{B}^i B^k B_k - \frac{2}{\mu_0} \tilde{B}_j B^i B^j = 0$

$\frac{1}{\gamma-1} P_{,t} + \frac{1}{2} \rho_{,t} v^k v_k + \rho v^k v_{k,t} + \frac{1}{\mu_0} B^k B_{k,t} + \frac{\gamma}{\gamma-1} P_{,j} v^j + \frac{\gamma}{\gamma-1} P {v^j}_{,j} + \frac{1}{2} \rho_{,j} v^k v_k v^j + \rho v^k v_{k,j} v^j + \frac{1}{2} \rho v^k v_k {v^j}_{,j} + \frac{2}{\mu_0} B^k B_{k,j} v^j + \frac{1}{\mu_0} B^k B_k {v^j}_{,j} - \frac{1}{\mu_0} {B^k}_{,j} v_k B^j - \frac{1}{\mu_0} B^k v_{k,j} B^j - \frac{1}{\mu_0} B^k v_k {B^j}_{,j} = 0$
$\frac{1}{\gamma-1} \tilde{P}_t P + \frac{1}{2} \tilde{\rho}_t \rho v^k v_k + \rho v^k \tilde{v}_t v_k + \frac{1}{\mu_0} B^k \tilde{B}_t B_k + \frac{\gamma}{\gamma-1} \tilde{P}_j P v^j + \frac{\gamma}{\gamma-1} P \tilde{v}_j v^j + \frac{1}{2} \tilde{\rho}_j \rho v^k v_k v^j + \frac{3}{2} \rho \tilde{v}_j v^j v^k v_k + \frac{2}{\mu_0} B^k \tilde{B}_j B_k v^j + \frac{1}{\mu_0} B^k B_k \tilde{v}_j v^j - \frac{2}{\mu_0} \tilde{B}_j B^k v_k B^j - \frac{1}{\mu_0} B^k \tilde{v}_j v_k B^j = 0$
Substitute $\tilde{\rho}_t + \tilde{\rho}_j v^j + \tilde{v}_j v^j = 0$.
$\frac{1}{\gamma-1} \tilde{P}_t P + \rho v^k \tilde{v}_t v_k + \frac{1}{\mu_0} B^k \tilde{B}_t B_k + \frac{\gamma}{\gamma-1} \tilde{P}_j P v^j + \frac{\gamma}{\gamma-1} P \tilde{v}_j v^j + \rho \tilde{v}_j v^j v^k v_k + \frac{2}{\mu_0} B^k \tilde{B}_j B_k v^j + \frac{1}{\mu_0} B^k B_k \tilde{v}_j v^j - \frac{2}{\mu_0} \tilde{B}_j B^k v_k B^j - \frac{1}{\mu_0} B^k \tilde{v}_j v_k B^j = 0$
Substitute $\tilde{v}_t \rho v^i + \tilde{v}_j \rho v^i v^j + \tilde{P}^i P + \frac{1}{\mu_0} \tilde{B}^i B^k B_k - \frac{2}{\mu_0} \tilde{B}_j B^i B^j = 0$
$ - \tilde{P}^i P v_i + \frac{1}{\gamma-1} \tilde{P}_t P + \frac{1}{\mu_0} \tilde{B}_t B^k B_k + \frac{\gamma}{\gamma-1} \tilde{P}_j v^j P + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \frac{1}{\mu_0} \tilde{B}_j v^j B^k B_k + \frac{1}{\mu_0} \tilde{v}_j v^j B^k B_k - \frac{1}{\mu_0} \tilde{v}_j B^j B^k v_k = 0$
$ \frac{1}{\gamma-1} \tilde{P}_t P + \frac{1}{\gamma-1} \tilde{P}_j P v^j + \frac{1}{\mu_0} \tilde{B}_t B^k B_k + \frac{1}{\mu_0} \tilde{B}_j v^j B^k B_k + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \frac{1}{\mu_0} \tilde{v}_j v^j B^k B_k - \frac{1}{\mu_0} \tilde{v}_j B^j B^k v_k = 0$
Substitute $ \tilde{B}_t B^i + \tilde{B}_j B^i v^j + B^i \tilde{v}_j v^j - \tilde{B}_j B^j v^i - B^j \tilde{v}_j v^i = 0$
$\frac{1}{\gamma-1} \tilde{P}_t P + \frac{1}{\gamma-1} \tilde{P}_j P v^j + \frac{\gamma}{\gamma-1} \tilde{v}_j v^j P + \frac{1}{\mu_0} B_i \tilde{B}_j B^j v^i = 0$
Assume $P > 0$.
$\tilde{P}_t + \tilde{P}_j v^j + \gamma \tilde{v}_j v^j + \frac{\gamma-1}{\mu_0 P} B_i \tilde{B}_j B^j v^i = 0$
$\tilde{P}_\mu v^\mu = - \gamma \tilde{v}_j v^j - \frac{\gamma-1}{\mu_0 P} \tilde{B}_j B^j B_k v^k$