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For a steady flow, mass conservation gives $\rho w=\text{constant}$ and ${\rho }^{\prime }/\rho =-w^{\prime }/w$. Define the Alfvén velocity ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$ and $C=w^2-v_{az}^2$. Combining the horizontal momentum and MHD induction equations without dividing by $C$ givesDot these identities with $B_x/({\mu }_0\rho )$ and $B_y/({\mu }_0\rho )$, respectively:The vertical momentum equation and the isothermal equation of state giveMultiply by $C$ and eliminate the magnetic derivative to obtainNo division by $w^2-v_{az}^2$ was needed in deriving this necessary relation.

The magnetosonic critical speeds in the $z$ direction areThe plus sign gives the fast magnetosonic wave speed and the minus sign the slow magnetosonic wave speed. Thus the differential coefficient is $(w^2-c_{\mathrm{f}}^2)(w^2-c_{\mathrm{s}}^2)$. A smooth outflow proceeding from below both speeds to above both must normally pass through both magnetosonic critical speeds. At each crossing, the right-hand side must also vanish; this is the regularity at a magnetosonic point condition. The derivative coefficient changes sign at each nondegenerate crossing. For $w>0$ and $w^{\prime }>0$, the driving term must be positive below the slow point, negative between the points and positive above the fast point.

The Alfvén speed component satisfies $c_{\mathrm{s}}^2\le v_{az}^2\le c_{\mathrm{f}}^2$. With downward gravity $g>0$, the gravitational term is consequently nonnegative at the slow point and nonpositive at the fast point. The magnetohydrodynamic shear work contribution must balance it at each point, and can provide the upward driving needed to pass the fast point. For $a=0$ and strictly positive $g$, generic separated slow and fast points cannot satisfy the required zero numerator; special vanishing-gravity or coincident-speed cases need separate treatment.

There is also Alfvén-point compatibility in a plane-parallel sheared flow. At $w^2=v_{az}^2$, the original transverse equations requireThese restrictions are not generally visible as a zero of the scalar differential coefficient, which there equals $-v_{az}^2(v_{ax}^2+v_{ay}^2)$. In particular, a strictly accelerating regular solution must have $B_x=0$ at that point. The scalar relation is therefore a necessary wind equation, not a substitute for regularity of all the original ideal magnetohydrodynamic equations. Degenerate cases such as a purely longitudinal magnetic field can merge characteristic speeds and reduce the number of distinct critical conditions.

Take the primitive state vector $\mathbf{U}=(\rho ,p,u_x,u_y,u_z,B_x,B_y,B_z{)}^T$, with $\rho >0$. In the absence of gravity, the ideal magnetohydrodynamic equations are first order: expanding the material derivatives, pressure gradient and Lorentz force makes every term linear in a spatial derivative of $\mathbf{U}$, with coefficients depending on $\mathbf{U}$. Dividing momentum balance by $\rho$ therefore gives a quasilinear partial differential equation system ${\partial }_t\mathbf{U}+A_i(\mathbf{U}){\partial }_i\mathbf{U}=0$ with eight-by-eight matrices. The solenoidal magnetic-field constraint is an additional constraint on initial data; the ideal magnetohydrodynamic induction equation preserves it because the divergence of a curl vanishes.

For a one-dimensional simple wave in magnetohydrodynamics, write $\mathbf{U}=\mathbf{U}(q(x,t))$ with a nonzero state-space tangent. Substitution gives ${\mathbf{U}}_q{q}_t+A_x{\mathbf{U}}_q{q}_x=0$. A nonconstant profile requires the tangent to be a right eigenvector of $A_x$:The chain rule then gives the wave-speed equation:This is the Inviscid Burgers equation, including the special case of constant $v$. Its characteristic curves are $x=a+v_0(a)t$, with $v=v_0(a)$. As long as the mapping remains invertible,Thus a region with $v_0^{\prime }<0$ steepens: faster characteristics catch slower ones, and the first gradient catastrophe occurs at $t_{\ast }=-1/{\min }_a{v}_0^{\prime }(a)$ when the minimum is negative. A genuinely nonlinear compressive simple wave therefore forms a shock wave. Rarefactive profiles can spread instead; steepening is not inevitable for every initial profile.

To continue past this time, use weak solutions of the conservative mass, momentum, fluid total-energy equation and ideal magnetohydrodynamic induction equation. The Rankine-Hugoniot conditions fix the jumps and shock velocity, while physical entropy production selects admissible magnetohydrodynamic shocks. Diffusive shock wave layers are replaced by moving discontinuities, so their microscopic structure need not be explicitly resolved. In particular, the smooth adiabatic pressure equation must be replaced by total-energy conservation when imposing the ideal magnetohydrodynamic shock conditions.

For the Alfvén waves, impose the one-dimensional solenoidal magnetic-field constraint, so $B_x$ is constant and $\delta B_x=0$. Put ${\mathbf{B}}_{\perp }=(B_y,B_z)$ and $c=v-u_x$. The transverse components of the linearized ideal magnetohydrodynamic equations giveThe longitudinal momentum equation additionally gives $\delta p+{\mathbf{B}}_{\perp }\cdot \delta {\mathbf{B}}_{\perp }/{\mu }_0=0$. The Alfvén eigenvectors have $\delta \rho =\delta p=\delta u_x=0$, hence their transverse polarization is perpendicular to ${\mathbf{B}}_{\perp }$. For $s=\pm 1$, their speed and explicit right Alfvén characteristic eigenvectors areFor ${\mathbf{B}}_{\perp }\ne 0$ one may choose $(h_y,h_z)=(B_z,-B_y)$, up to a nonzero scalar factor. If ${\mathbf{B}}_{\perp }=0$, either transverse polarization is allowed and this characteristic speed is degenerate. These expressions describe the two propagating Alfvén branches for $B_x\ne 0$; if $B_x=0$, they coalesce with advected, nonpropagating transverse disturbances.

Integrating along an Alfvén simple wave leaves $\rho ,p,u_x,B_x$ unchanged and gives $d{\mathbf{u}}_{\perp }=-sd{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and ${\mathbf{B}}_{\perp }\cdot d{\mathbf{B}}_{\perp }=0$. Consequently the finite-amplitude nonlinear Alfvén wave relations areThe constant-magnitude condition is essential: otherwise a varying magnetic pressure would drive longitudinal compression. An arbitrary smooth phase profile gives an explicit family, $f_3=B_{\perp }\cos \Theta (x-v_{s}t)$ and $f_4=B_{\perp }\sin \Theta (x-v_{s}t)$, with $f_1,f_2$ as above. Direct substitution in the transverse momentum and ideal magnetohydrodynamic induction equation gives $(u_x-v_s){\mathbf{u}}_{\perp }^{\prime }=B_x{\mathbf{B}}_{\perp }^{\prime }/({\mu }_0\rho )$ and $(u_x-v_s){\mathbf{B}}_{\perp }^{\prime }=B_x{\mathbf{u}}_{\perp }^{\prime }$, confirming the solution without relying on the eigenvector argument. Longitudinal momentum holds because $p+B_{\perp }^2/(2{\mu }_0)$ is constant. The speed $v_s$ is independent of amplitude along this family: these linearly degenerate characteristic fields translate without steepening.