1D Alfvén Wave in MHD

What you are seeing: a driver at the coronal base shakes a bundle of magnetic field lines; the transverse kink runs outward along $B_0\hat x$ at the Alfvén speed $v_A = B_0 / \sqrt{\mu_0 \rho}$ , with magnetic tension as the restoring force. Plasma parcels are frozen into the lines (ideal MHD) and ride them transversely with $v_y$ , illustrating the Walén relation $v_y = \mp b_y / \sqrt{\mu_0 \rho}$ (cyan $v_y$ and orange $b_y$ arrows are antiphase). This is how Alfvén waves carry energy up coronal field lines into the solar wind. The lower strip plots $b_y(x,t)$ and $v_y(x,t)$ . Raising $B_0$ or lowering the density speeds the wave up.

Figure 1. Transverse perturbations of magnetic field and velocity propagating at

v_A

B_0 (nT)5.0

n (amu/cm³)5.0

WHAT TO TRY

Raise B_0: the Alfven speed v_A = B_0/sqrt(mu0 rho) climbs, so the transverse kink runs out along the field faster. The crest races to the right edge sooner.
Raise the density n: heavier plasma slows the wave, since v_A falls as 1/sqrt(rho). The field lines feel the same tension but more inertia.
Watch the cyan plasma velocity v_y against the orange field perturbation b_y: they stay antiphase, the Walen relation v_y = -/+ b_y/sqrt(mu0 rho) that marks a shear Alfven wave riding magnetic tension.