= Resistive damping of an Alfvén wave
{c}
{title2=$\omega^2+i\eta_m k_\parallel^2\omega-v_A^2k_\parallel^2=0$}
Finite <electrical conductivity> $\sigma$ allows <magnetic diffusion>, with <magnetic diffusivity> $\eta_m=1/(\mu_0\sigma)$. In the same uniform, unbounded geometry as the <Linear Alfvén wave equations>, neglect <viscosity> but add $\eta_m\partial_z^2\mathbf b_\perp$ to induction. For real $k_\parallel$ and time dependence $e^{-i\omega t}$, elimination gives
$$
\omega^2+i\eta_m k_\parallel^2\omega-v_A^2k_\parallel^2=0,
\qquad
\omega=-\frac{i\eta_m k_\parallel^2}{2}\pm\sqrt{v_A^2k_\parallel^2-\frac{\eta_m^2k_\parallel^4}{4}}.
$$
Thus weak <magnetic diffusion>, $\eta_m|k_\parallel|\ll v_A$, gives an <amplitude> decay rate $\eta_m k_\parallel^2/2$. For $\eta_m|k_\parallel|>2v_A$, both temporal roots are purely decaying; they no longer describe an oscillating traveling mode. This temporal calculation at fixed real <wavenumber> is different from a driven vessel at fixed real <frequency>, where a complex spatial <wavenumber> describes penetration and attenuation. <Viscosity> also dissipates <kinetic energy>, and walls introduce transverse gradients and additional losses.
For an axial length $L$, comparing the <magnetic diffusion> time $L^2/\eta_m$ with the transit time $L/v_A$ gives the ratio $Lv_A/\eta_m$. A large value favors propagation before diffusion, but a narrow vessel can introduce a shorter transverse diffusion scale. In the actual 1949 analysis, finite <electrical conductivity> and vessel dimensions restrict the useful <frequency> interval; above its upper limit the response becomes a skin wave.<Experimental Investigations of Magneto-Hydrodynamic Waves by Stig Lundquist>[Abstract: finite-conductivity frequency interval, upper limit and geometrical lower limit]{ref}
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