= Alfvén wave
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= Shear Alfvén wave
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An <Alfvén wave> is a <magnetohydrodynamic wave> in which <magnetic tension> restores a sideways disturbance of a conducting <fluid>. In the small-<amplitude>, uniform-background <ideal magnetohydrodynamics> model, its shear <polarization of waves> has <displacement> perpendicular to both the background <magnetic field> and the <wavevector>. It is an <incompressible flow> disturbance: the <mass density> and gas <pressure> have no first-order fluctuation. Its <dispersion relation> is $\omega^2=(\mathbf k\mathbin\cdot\mathbf v_A)^2$, where $\mathbf v_A$ is the background <Alfvén velocity>.
The mechanism resembles a plucked taut string: <magnetic flux freezing> makes sideways <fluid flow> bend <magnetic field lines>, and their <magnetic tension> accelerates neighboring fluid. The fluid supplies inertia and the bent <magnetic field> supplies the restoring stress. This permits <energy> transport without a first-order compression, unlike an <acoustic wave>. <Hannes Alfvén>'s 1942 paper, Existence of Electromagnetic-Hydrodynamic Waves, considered precisely a uniform axial <magnetic field>, infinite <electrical conductivity> and sideways <velocity> depending on axial position and time; it proposed a solar application. This was a theoretical prediction, followed by the distinct <Lundquist liquid-metal Alfvén wave experiment>.<Existence of Electromagnetic-Hydrodynamic Waves by Hannes Olof Gösta Alfvén>[1942, pp. 405–406; ideal-conductor plane-wave argument, reproduced on slide 4 of Kinetic Alfvén waves in space plasmas]{ref}
\Image[/alfven-wave-magnetic-tension.png]
{title=Magnetic tension restores a transverse Alfvén-wave displacement}
{description=An original schematic of a small sideways bend in a uniform background <magnetic field>. The <displacement> and restoring <Lorentz force density> point in opposite directions at the crest; the transverse <magnetic field> follows the slope, so it vanishes at that crest. With <fluid Lagrangian displacement> $\boldsymbol\xi$, <magnetic flux freezing> gives $\mathbf b_\perp=B_0\partial_z\boldsymbol\xi_\perp$. The drawing exaggerates the bend for readability.}
For oblique propagation, the shear <Alfvén wave> has <velocity> normal to the plane containing the <wavevector> and background <magnetic field>. The <fast magnetosonic wave> and <slow magnetosonic wave> instead have <velocity> in that plane and generally involve compression, with gas <pressure> and <magnetic pressure> contributing to their restoring forces. For exactly parallel propagation the plane is not unique: either transverse <polarization of waves> is allowed, and some modal speeds coincide. For $k_\parallel=0$ the shear branch has $\omega=0$, while the <fast magnetosonic wave> can still propagate perpendicular to the <magnetic field>. One should identify the <polarization of waves> as well as the <phase velocity>. The statement that the shear mode is incompressible concerns this linear mode; a finite linearly polarized disturbance can drive a quadratic compressive response, as in the <magnetic-pressure obstruction to a linearly polarized Alfvén wave>.
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