This browser-side companion to the Alfvén-wave lecture scans the uniform ideal-MHD branches from low-β to high-β using the exact dispersion relation for the shear, fast, and slow modes. The polar plot displays phase speed as a function of propagation angle relative to the magnetic field, while the smaller panels track how the parallel and perpendicular limits change as the pressure becomes more or less important compared with magnetic tension.
This is the cleanest knob for the lecture’s polar plot: small values are low-β, order-unity values are mixed, and large values are pressure-dominated.
Used to mark one specific propagation direction on the polar plot and read off the three branch speeds there.
Shows the component structure of δv and δB for the selected branch at the highlighted angle.
The large panel is the polar phase-speed diagram. The radius is the phase speed normalized to vA, and the polar angle is the propagation angle relative to the background magnetic field. The two smaller panels show the directional cuts that matter most in the lecture: along the field and across the field.
The marked direction shows how the three phase speeds separate for one chosen propagation angle.
The lecture’s key limits are recovered directly: the parallel field-aligned branches and the across-field cutoff of the shear branch.
These bars show the signed component structure of δv and δB for the selected branch, up to an overall normalization and common phase.
The compressive branches follow the quadratic for the normalized phase-speed squared
u = ω²/(k² v_A²),
u² - (1 + c_s²/v_A²)u + (c_s²/v_A²) cos²θ = 0.
The two roots are the slow and fast branches, while the shear-Alfvén branch is
u = cos²θ.
So the low-β limit gives a fast branch near v_A, a slow branch mostly along the field,
and the shear branch exactly controlled by field-line tension.