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Let $K^2=|\mathbf{k}{|}^2=k_x^2+k_z^2+{\Omega }^2{t}^2{k}_x^2$. For the complex wave before taking its real part,since ${\dot{k}}_y=-\Omega k_x$. The asserted velocity equation therefore follows directly from $\dot{\mathbf{v}}+\Omega v_y\hat{\mathbf{x}}=-i\mathbf{k}q$. To verify the pressure amplitude, solenoidality is $\mathbf{k}\cdot \mathbf{v}=0$. Its derivative requiresand hence $q=2i\Omega k_x{v}_y/K^2$, as claimed.

The transverse amplitude equations areBoth coefficients are real. Differentiating $v_y{v}_z^{\ast }-v_z{v}_y^{\ast }$ makes the terms proportional to $b|v_y{|}^2$ cancel, while the remaining terms give $a(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$. Therefore the helicity invariant of an inviscid shearing wave obeysHere $H$ is real because the difference in its definition is purely imaginary. Its sign is preserved, and shear reduces its magnitude away from $t=0$.

For background shear $\Omega y\hat{\mathbf{x}}$ and uniform field $B_0\hat{\mathbf{x}}$, first-order smoothing gives the magnetic shearing-wave amplitude equationIf magnetic relaxation is fast compared with shear, $c_y,c_z\simeq i{k}_x{B}_0(v_y,v_z)/(\eta K^2)$ after free transients decay. Thus the doubled Fourier-mode electromotive quantity $\mathcal{E}=\mathrm{Re}(v_y{c}_z^{\ast }-v_z{c}_y^{\ast })$ is $-B_{0}H/(\eta K^2)$, where $H=i{k}_x(v_y{v}_z^{\ast }-v_z{v}_y^{\ast })$. 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 $\Omega k_x\ne 0$, givesIndeed $K^2=A+{\Omega }^2{k}_x^2{t}^2$, $A=k_x^2+k_z^2$, and substitution $z=|\Omega k_x|t/\sqrt{A}$ uses ${\int }_{-\infty }^{\infty }(1+z^2{)}^{-2}dz=\pi /2$.