Alfvén and magnetosonic waves

Take a homogeneous static plasma with constant density $\rho_0$, scalar pressure $p_0$, and magnetic field $\mathbf B_0$. Small perturbations vary as $\exp[i(\mathbf k\cdot\mathbf x-\omega t)]$. Linear continuity links density perturbation to $\mathbf k\cdot\delta\mathbf u$, while the adiabatic closure links pressure to density. The induction equation links magnetic perturbation to fluid displacement. Substituting these relations into momentum balance produces an eigenvalue problem for three velocity polarizations.

Define the Alfvén speed from background field strength and mass density, and the sound speed from the selected adiabatic pressure response. Let $\theta$ be the angle between wavevector and background magnetic field. These definitions refer to a uniform, ideal, scalar-pressure medium. Equilibrium gradients, background flow, kinetic resonances, and collisions can change the spectrum. Calling a measured disturbance Alfvénic is therefore a comparison with these assumptions, not just a claim that a magnetic field is present.

For the shear Alfvén mode, perturbed velocity lies normal to the plane containing $\mathbf k$ and $\mathbf B_0$. Its divergence vanishes, so density and scalar pressure are unperturbed to first order. Magnetic tension supplies the restoring force. The frequency depends on the parallel wave number; when that component is zero this ideal branch has zero frequency, rather than a propagating shear oscillation at the perpendicular Alfvén crossing rate. Its group velocity points along the background field.

The other two polarizations lie in the wavevector–field plane and generally compress the fluid. The magnetosonic quadratic gives fast and slow phase-speed squares. The nonnegative discriminant and nonnegative roots can be checked from their sum and product. Fast and slow denote eigenbranches, not two arbitrary wave packets moving at user-selected speeds. Their velocity and magnetic perturbations are coupled, so identifying a branch requires polarization information as well as a frequency estimate.

For propagation parallel to the field, magnetosonic phase speeds reduce to the larger and smaller of $v_A$ and $c_s$. One transverse magnetic branch is degenerate with the shear Alfvén speed; special-angle eigenvectors should be treated using the linear equations rather than a division by a vanishing component. For perpendicular propagation, fast speed becomes $\sqrt{v_A^2+c_s^2}$ while both slow and shear frequencies vanish in this homogeneous ideal limit.

At fixed angle the frequency is proportional to wave-number magnitude, yet the medium remains directionally anisotropic. A phase front and a wave packet need not travel in the same direction. Zero-frequency limiting branches are also not evidence that all plasma disturbances stop: other geometries, entropy perturbations, and physics beyond the ordering remain possible. The useful habit is to test angle, pressure closure, and microscopic scale before extrapolating a familiar sound-wave picture to a magnetized plasma.