Simple wave in magnetohydrodynamics ($A_x{\mathbf{U}}_q=v(q){\mathbf{U}}_q$)

An ideal-MHD simple wave traces a one-parameter state-space curve whose tangent is a right eigenvector of the characteristic matrix. Its parameter obeys $q_t+v(q)q_x=0$ and its characteristic speed obeys the Inviscid Burgers equation. Nonlinear Alfvén waves are a linearly degenerate example.

For a one-dimensional velocity perpendicular to a transverse magnetic field, $c_f^2=\gamma p/\rho +B^2/({\mu }_0\rho )$. The primitive-variable characteristic matrix has two advective eigenvalues and the two speeds $u\pm c_f$. Integrating the fast eigenvectors keeps $p/{\rho }^{\gamma }$ and $B/\rho$ constant. The state parameter obeys ${\rho }_t+(u\pm c_f){\rho }_x=0$, so the characteristic speed satisfies the Inviscid Burgers equation.

With $B_x=0$, a fast simple wave in magnetohydrodynamics has constant $u_y$ and $B_y/\rho =K$. Its longitudinal velocity satisfies $d{u}_x/d\rho =\pm c_f/\rho$, and ${\rho }_t+(u_x\pm c_f){\rho }_x=0$. Along this state curve the speed derivative is $\pm [c_f/\rho +K^2/(2{\mu }_0{c}_f)]$, which is nonzero. Compressive profiles therefore exhibit shock formation by characteristic intersection, unlike a linearly degenerate characteristic field.