We can combine jump conditions for mass flux

• \rho (v_x - \lambda_f) = \rho^*(v_x^* - \lambda_f)
• \rho^* (v_x^* - \lambda_s) = \rho^{**}(v_x^{**} - \lambda_s)

along with the symmetry requirements (v_x^{**} = v_y^{**} = 0) to get

• \rho(v_x - \lambda_f) = \rho^*(\lambda_s - \lambda_f)-\rho^{**} \lambda_s

as well as jump conditions for y-momentum

• \rho(v_x - \lambda_f) v_y - B_xB_y = \rho^*(v_x^* - \lambda_f) v_y^* - B_x B_y^*
• \rho^*(v_x^* - \lambda_s) v_y^* - B_xB_y^* = -B_x B_y^{**}

to get

• \rho(v_x - \lambda_f) v_y - B_xB_y = \rho^*v_y^*(\lambda_s - \lambda_f)-B_x B_y^{**}

and the jump conditions for magnetic flux

• B_y (v_x - \lambda_f) - B_x v_y = B_y^* (v_x^* - \lambda_f) - B_x v_y*
• B_y^*(v_x^* - \lambda_s) - B_x v_y* = -B_y^{**} \lambda_s

to get

• B_y (v_x - \lambda_f) - B_x v_y = B_y^*(\lambda_s - \lambda_f) - B_y^{**} \lambda_s

This gives us

• \rho(v_x - \lambda_f) = \rho^*(\lambda_s - \lambda_f)-\rho^{**} \lambda_s
• \rho(v_x - \lambda_f) v_y - B_xB_y = \rho^*v_y^*(\lambda_s - \lambda_f)-B_x B_y^{**}
• B_y (v_x - \lambda_f) - B_x v_y = B_y^*(\lambda_s - \lambda_f) - B_y^{**} \lambda_s

Now in our setup, the flow and the field are parallel, so we have B_y v_x = B_x v_y. Combining this with the third equation gives us

• B_y^{**} \lambda_s + B_y^* ( \lambda_f - \lambda_s ) = B_y \lambda_f

Which is evident from the geometry of the problem.

So we have

• \rho(v_x - \lambda_f) = \rho^*(\lambda_s - \lambda_f)-\rho^{**} \lambda_s
• \rho(v_x - \lambda_f) v_y - B_xB_y = \rho^*v_y^*(\lambda_s - \lambda_f)-B_x B_y^{**}
• B_y^{**} \lambda_s + B_y^* ( \lambda_f - \lambda_s ) = B_y \lambda_f
• -B_y \lambda_f = B_y^* (v_x^* - \lambda_f) - B_x v_y*

Now we can combine the 1st and 3rd equations and the 2nd and 4th equations to get

• \rho^*(v_x^* - \lambda_f) (v_y - v_y^*) = B_x(B_y - B_y^*)
• - \rho^{**}\lambda_s v_y^* = B_x ( B_y^* -B_y^{**})

So we have the following six equations for 8 unknowns (including the 2 wave speeds)

• \rho(v_x - \lambda_f) = \rho^*(\lambda_s - \lambda_f)-\rho^{**} \lambda_s
• \rho(v_x - \lambda_f) v_y - B_xB_y = \rho^*v_y^*(\lambda_s - \lambda_f)-B_x B_y^{**}
• B_y^{**} \lambda_s + B_y^* ( \lambda_f - \lambda_s ) = B_y \lambda_f
• -B_y \lambda_f = B_y^* (v_x^* - \lambda_f) - B_x v_y*
• \rho^*(v_x^* - \lambda_f) (v_y - v_y^*) = B_x(B_y - B_y^*)
• - \rho^{**}\lambda_s v_y^* = B_x ( B_y^* -B_y^{**})

Because of symmetry, the streamline coming in from the right, must remained pinned to the point along the contact. And given that Bx is a constant, we have

• B_y^{**} = B_x \tan \theta^{**}
• B_y^{*} = B_x \tan \theta^{*}
• B_y = B_x \tan \theta

and that

\tan \theta^{**} \lambda_s + \tan \theta^{*} \left ( \lambda_f - \lambda_s \right) = \lambda_f \tan \theta

We also have

• v_y^{**} = v_x^{**}=0
• v_y^* = v_x^* \tan \theta_v^* = v_x^* \tan \theta^* + \lambda_s \tan \theta^{**} + \lambda_s \tan \theta^*

v_y^* = v_x^*\tan \theta^* + \lambda_f \left ( \tan \theta - \tan \theta^* \right ) + 2 \lambda_s \tan \theta^*

• v_y = v_x \tan \theta
• v_x = c_s M \cos \theta
• B_x = \sqrt{\rho} v_A \tan \theta

Ideally, our givens would be

• M
• \mu = \frac{1}{\beta}
• \theta

And we can set (without loss of generality)

• c_s = 1
• \rho = 1

and we would need to solve for

• \rho^*
• \rho^{**}
• P^{*}
• P^{**}
• v_x^{*}
• \lambda_f
• \lambda_s
• \theta^*

Conservation of mass gives

\rho (v_x - \lambda_f) = \rho^* \left ( v_x^* - \lambda_f \right )

and conservation of momentum gives

• \rho (v_x - \lambda_f)^2 + P+B_x^2 \left ( \tan^2 \theta - 1 \right ) = \rho^* (v_x^* - \lambda_f)^2 + P^* + B_x^2 \left ( \tan^2 \theta^* - 1 \right )

• \rho (v_x - \lambda_f) v_x \tan \theta - B_x^2 \tan \theta= \rho^*(v_x^* - \lambda_f) \left (v_x^* \tan \theta + \lambda_f \left ( \tan \theta - \tan \theta^* \right ) + 2 \lambda_s \tan \theta^* \right )

and conservation of magnetic flux gives

• B_x \tan \theta \left (v_x - \lambda_f \right ) - B_x v_x \tan \theta = B_x \tan \theta^* \left (v_x^* - \lambda_f \right ) - B_x \left ( v_x^*\tan \theta^* + \lambda_f \left ( \tan \theta - \tan \theta^* \right ) + 2 \lambda_s \tan \theta^* \right )