= Solution
An <Alfvén wave> is a transverse disturbance restored by <magnetic tension>. In a uniform conducting <fluid> with background $\mathbf B_0=B_0\mathbf e_z$, a small transverse displacement $\boldsymbol\xi_\perp(z,t)$ bends the <magnetic field lines>. <Magnetic flux freezing> gives $\mathbf b_\perp=B_0\partial_z\boldsymbol\xi_\perp$, and the linear transverse <Lorentz force density> is $(B_0/\mu_0)\partial_z\mathbf b_\perp$. Thus the <Linear Alfvén wave equations> reduce to
$$
\rho\,\partial_t^2\boldsymbol\xi_\perp
=\frac{B_0^2}{\mu_0}\partial_z^2\boldsymbol\xi_\perp,\qquad
\boxed{v_A=\frac{B_0}{\sqrt{\mu_0\rho}},\quad\omega^2=k_\parallel^2v_A^2.}
$$
For this shear <polarization>, $\delta\rho=\delta p=0$ at linear order. The disturbance transports <energy> along the background <magnetic field>; a single traveling wave has equal perturbation <magnetic energy> and <kinetic energy>, with $\mathbf b_\perp=\mp\sqrt{\mu_0\rho}\,\mathbf u_\perp$ for propagation along $\pm\mathbf B_0$. These properties distinguish it from compressive <magnetosonic waves>.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2013/paper-1-alfven-wave.png]
{title=Transverse displacement of field lines in a traveling Alfvén wave}
The <ideal magnetohydrodynamics> approximation requires strong electrical coupling and wavelengths large compared with microscopic scales. Finite <electrical resistivity> and <viscosity> dissipate the wave's <energy>. In <stellar winds> and other astrophysical <plasmas>, <Alfvén waves> carry disturbances and can contribute to heating when dissipated. The <Lundquist liquid-metal Alfvén wave experiment> provides a laboratory counterpart: a conducting liquid in a strong applied <magnetic field> transmits a transverse magnetic disturbance, making the field's mechanical restoring action measurable.
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