= Spatial damping of a driven Alfvén wave
{title2=$k=\frac\omega{v_A}(1-i\epsilon)^{-1/2},\quad\epsilon=\eta_m\omega/v_A^2$}
For fixed real positive <angular frequency> $\omega$, a transverse <Alfvén wave> proportional to $e^{i(kz-\omega t)}$ in a uniform fluid has $k^2=\omega^2/(v_A^2-i\eta_m\omega)$ when <magnetic diffusivity> $\eta_m$ is retained and <viscosity> is neglected. Choose the branch with positive real and imaginary parts for propagation and attenuation along $+z$. For $\epsilon\ll1$, $k=(\omega/v_A)(1+i\epsilon/2+O(\epsilon^2))$. The amplitude attenuation length is $L_A=1/\operatorname{Im}k$, whereas the <energy> attenuation length is $L_E=1/(2\operatorname{Im}k)$. With wavelength $\lambda=2\pi/\operatorname{Re}k$, $L_A/\lambda\sim1/(\pi\epsilon)$ and $L_E/\lambda\sim1/(2\pi\epsilon)$. Fixed-real-<wavenumber> temporal <Resistive damping of an Alfvén wave> is a different boundary-value prescription.
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