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 $P>0$.
${\tilde{P}}_t+{\tilde{P}}_j{v}^j+\gamma {\tilde{v}}_j{v}^j+\frac{\gamma -1}{{\mu }_{0}P}{B}_i{\tilde{B}}_j{B}^j{v}^i=0$
${\tilde{P}}_{\mu }{v}^{\mu }=-\gamma {\tilde{v}}_j{v}^j-\frac{\gamma -1}{{\mu }_{0}P}{\tilde{B}}_j{B}^j{B}_k{v}^k$