= Linear Alfvén wave equations
{c}
{title2=$\omega=\pm k_\parallel v_A$}
Take a static uniform <fluid> of <mass density> $\rho>0$, constant gas <pressure>, and background <magnetic field> $\mathbf B_0=B_0\mathbf e_z$ with $B_0>0$. Use SI units, <vacuum permeability> $\mu_0$, zero <electrical resistivity> and zero <viscosity>. Write $\mathbf B=\mathbf B_0+\mathbf b_\perp(z,t)$ and $\mathbf u=\mathbf u_\perp(z,t)$, with both disturbances perpendicular to $\mathbf e_z$. Retain only <linear order> in their <amplitudes>. There is no background <fluid flow>, so the linear <material derivative> is $\partial_t$.
The <continuity equation> has $\nabla\cdot\mathbf u_\perp=0$, hence $\partial_t\delta\rho=0$; choosing no initial <mass density> fluctuation gives $\delta\rho=0$. For this shear mode the gas <pressure> fluctuation is also zero. In the <ideal magnetohydrodynamic momentum equation>, the <Lorentz force density> can be written
$$
\frac{(\nabla\times\mathbf B)\times\mathbf B}{\mu_0}
=\frac{(\mathbf B\cdot\nabla)\mathbf B}{\mu_0}
-\nabla\frac{B^2}{2\mu_0}.
$$
Because $\mathbf B_0\cdot\mathbf b_\perp=0$, the perturbed <magnetic pressure> starts at quadratic order. The surviving transverse <magnetic tension> gives the first equation below. Expanding the <ideal magnetohydrodynamic induction equation> gives the second:
$$
\rho\,\partial_t\mathbf u_\perp=\frac{B_0}{\mu_0}\partial_z\mathbf b_\perp,
\qquad
\partial_t\mathbf b_\perp=B_0\partial_z\mathbf u_\perp.
$$
Differentiate the first in time and substitute the second. Both <velocity> and <magnetic field> disturbances obey the <one-dimensional wave equation>:
$$
\partial_t^2\mathbf u_\perp=v_A^2\partial_z^2\mathbf u_\perp,
\qquad v_A=\frac{B_0}{\sqrt{\mu_0\rho}}.
$$
For a <plane wave> proportional to $e^{i(k_\parallel z-\omega t)}$, this gives $\omega^2=k_\parallel^2v_A^2$, or $\omega=\pm k_\parallel v_A$. The magnetic and <velocity> <amplitudes> satisfy $\mathbf b_\perp=-(B_0k_\parallel/\omega)\mathbf u_\perp$: they have opposite signs for propagation along $+\mathbf e_z$ and equal signs for propagation along $-\mathbf e_z$. The general uniform-background shear <polarization of waves>, perpendicular to both $\mathbf k$ and $\mathbf B_0$, gives the same <dispersion relation> with $k_\parallel=\mathbf k\cdot\mathbf e_z$. Its <group velocity> is $\pm v_A\mathbf e_z$, whereas its <phase velocity> normal to an oblique wavefront has magnitude $v_A|k_\parallel|/|\mathbf k|$.
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