 Isothermal perpendicular magnetosonic simple wave

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.