= Quasistatic magnetic response of a helical shearing wave
{title2=$\mathcal E\simeq-B_0H/(\eta|\mathbf k|^2)$}
For background shear $\Omega y\widehat{\mathbf x}$ and uniform field $B_0\widehat{\mathbf x}$, <first-order smoothing> gives the magnetic shearing-wave amplitude equation
$$
\dot{\mathbf c}=\Omega c_y\widehat{\mathbf x}+ik_xB_0\mathbf v-\eta K^2\mathbf c.
$$
If magnetic relaxation is fast compared with shear, $c_y,c_z\simeq ik_xB_0(v_y,v_z)/(\eta K^2)$ after free transients decay. Thus the doubled Fourier-mode electromotive quantity $\mathcal E=\operatorname{Re}(v_yc_z^*-v_zc_y^*)$ is $-B_0H/(\eta K^2)$, where $H=ik_x(v_yv_z^*-v_zv_y^*)$. The spatially averaged <mean-field electromotive force> of real waves has an extra factor $1/2$. Using the <helicity invariant of an inviscid shearing wave>, and assuming the approximation holds uniformly in time with $\Omega k_x\ne0$, gives
$$
\int_{-\infty}^{\infty}\mathcal E\,dt\simeq-\frac{\pi B_0H(0)}{2\eta|\Omega k_x|\sqrt{k_x^2+k_z^2}}.
$$
Indeed $K^2=A+\Omega^2k_x^2t^2$, $A=k_x^2+k_z^2$, and substitution $z=|\Omega k_x|t/\sqrt A$ uses $\int_{-\infty}^{\infty}(1+z^2)^{-2}dz=\pi/2$.
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