\( \renewcommand{\div}{{\vec{\nabla} \cdot}} \newcommand{\del}{{\dfrac{\partial}{\partial t}}} \)


For more details see for example Goedbloed, Keppens & Poedts, 2010 Advanced Magnetohydrodynamics.
Notations and definitions
The velocity is decomposed \begin{equation} \label{eq:ms2} \vec{u} = u_{n}\vec{e}_{n} + u_{t} \vec{e}_{t} \end{equation} in a component \(u_{n}\) parallel to the vector normal to the shock (the normal vector \(\vec{n}\)), and a tangential component \(\vec{v}_{t}\). Note, that there are two tangential directions perpendicular to each other. The indices \(n,t\) are then the normal, respective tangenial components of the vectors, \(\rho,P, \vec{u},\vec{B}, \gamma\) are the denisty, thermal pressure, plasma (bulk) velocity, magnetic field and polytropic index. The indes \(i\in\{1,2\}\) denote the upstream, repsective the downstream direction. Further, we will use \(\vec{B} = \vec{B}' / \sqrt{\mu}\), \(\vec{B}'\) is the magnetic induction in physical units (SI or cgs). The following notations will be used (\(i \in \{u,d\}\), where "u" are the parameter in upwind direction and "d" those downwind:
\(v_{A,n,i}\)\(\equiv\) \(\frac{B_{n,i}}{\sqrt{\rho}}\) normal Alvfén speed
\(v_{c}\) \(\equiv\) \( \sqrt{\frac{\gamma P}{\rho}}\) sound speed
\(M_{A,n,i}\)\(\equiv\) \(\frac{u_{n,i}}{V_{A,n,i}}\) normal Alvfénic Mach number
\(M\) \(\equiv\)\(M_{A,n,u}\) shorthand notation
\(M_{A,t,i}\)\(\equiv\) \(\frac{B_{t,i}}{u_{n,i}\sqrt{\rho_{i}}}\) ``tangential'' Alfv\'enic Mach number
\(M_{s,n,i}\)\(\equiv\)\(\frac{u_{n}}{v_{c}}\) normal sonic Mach number
\(s\)\(\equiv\) \(\frac{\rho_{d}}{\rho_{u}}=\frac{u_{n,u}}{u_{n,d}}\) the compression ratio
\(\beta_{t,i} \)\(\equiv\)\(2P_{i} /B^{2}_{t,i}\) tangential plasma beta
\(=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{t,i}}\) \(=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,t,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,t,i}}{M_{s,n,i}}\) alternative represnetation
\(\beta_{n,i} \)\(\equiv\)\(2P_{i} /B^{2}_{n,i}\) normal plasma beta
\(=\frac{2}{\gamma}\frac{\gamma \rho_{i}u_{n,i}^{2}P_{i}}{\rho_{i}u_{n,i}^{2}B^{2}_{n,i}}\) \(=\frac{2}{\gamma}\, \frac{v_{c,i}^{2}}{v_{A,n,i}^{2}} =\frac{2}{\gamma}\, \frac{M_{A,n,i}}{M_{s,n,i}}\) alternative represnetation
\(\beta_{i}\)\(\equiv\)\(\beta_{n,i}\) shorthand notation

We define the following angles: \begin{align} \vartheta_{i} &= \measuredangle \vec{B}_{i},\vec{n} \to \tan\vartheta_{i}=\frac{B_{t,i}}{B_{n}}\\ \varphi_{i} &= \measuredangle \vec{u}_{i},\vec{n} \to \tan\varphi_{i}=\frac{u_{t,i}}{u_{n}}\\ \alpha_{i} &= \measuredangle \vec{B}_{i},\vec{u}_{i}\to \cos\alpha_{i} =\frac{\vec{u}_{i}\cdot \vec{B}_{i}}{u_{i}B_{i}} \end{align} To calculate the Rankine Hugoniot equations, we replace \(\partial d/\partial t \to -U\) where \(U\) is the shock speed and \(\vec{\nabla}\to \vec{n}\) and switch to the shock rest frame \(\vec{u}' = \vec{u} - \vec{U}\). To save writings we negfelct the prime at \(\vec{u}' \) in the following.
Further, we will use \(\vec{B} = \vec{B}' /\sqrt{\mu}\), \(\vec{B}'\) is the magnetic induction in physical units (SI or cgs).

The Rankine Hugoniot equations

Then we obtain for the Rankine Hugoniot equations in the shock rest frame: \begin{align} \left\{ \rho u_{n}\right\} &=0 &\text{(continuity)}&\\ \left\{\rho u_{n}^{2} +P + \frac{1}{2} B^{2}_{t}\right\} &=0 &\text{(normal momentum)} &\\ \rho u_{n}\left\{\vec{u}_{t}\right\}- B_{n}\left\{\vec{B}_{t}\right\} &=0 &\text{(tangential momentum)} &\\ \rho u_{n}\left\{\frac{1}{2} (u^{2}_{t}+u_{n}^{2}) + \frac{1}{\rho}\left(\frac{\gamma}{\gamma-1} P+\frac{B^{2}_{t}}{2}\right)\right\} -B_{n}\left\{(\vec{u}_{t}\cdot\vec{B}_{t})\right\} &=0 &\text{(energy)}&\\ \left\{B_{n}\right\} &=0 &\text{(normal B-flux)} &\\ \rho u_{n} \left\{\frac{\vec{B}_{t}}{\rho}\right\}-B_{n}\left\{\vec{u}_{t}\right\}&= 0 &\text{(tangential B-flux)} &\\ \end{align} where the curly brackets are the shorthand notation for \(\{a\}=a_u-a_d\) where the indices \(u,d\) are the upstream respctivly the downstream direction.

The above system are 8 equations for eight unkowns \(\rho, u_n, P, B_n\) and two vector components for \(\vec{u}_t\) and \(\vec{B}_t\). The shocks and discontinuities are characterized as follows:
To caluclate the donwstrema values we have to solve the above equations system with respect t \(s\). The general case is a tedious algebrac excercise, where one has to handle the perpendicular shocks separatly, because, in general, because in the general case a divison by \(B_n\) is required. its solutions are found here.