= Isothermal coplanar magnetohydrodynamic characteristic matrix
{title2=$v=u_x\pm c_f,\ u_x\pm c_{\rm slow}$}
For one-dimensional flow obeying an <isothermal equation of state> with $u_z=B_z=0$, the primitive variables $(\rho,u_x,u_y,B_y)$ form a four-variable hyperbolic system and $B_x$ is constant. The speeds relative to $u_x$ satisfy $c^4-(c_s^2+v_a^2)c^2+c_s^2v_{ax}^2=0$. These are the fast and slow <magnetosonic waves>. The independent out-of-plane <Alfvén wave> polarization and a freely advected entropy mode are absent from this reduced isothermal system.
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