= Zero-mean magnetic shearing wave
{title2=$\dot{\widetilde{\mathbf B}}=-2A\widetilde B_x\mathbf e_y-\eta k^2\widetilde{\mathbf B}$}
In an incompressible <shearing sheet> with background velocity $-2Ax\mathbf e_y$, take both the perturbation <velocity> and the entire <magnetic field> to be real single <Fourier modes> with a common real <wavevector> and transverse amplitudes. Then every quadratic directional derivative vanishes, including terms involving complex conjugates. There is no uniform background magnetic field and consequently no linear magnetic-tension coupling between the amplitudes. Advecting the phase requires $\dot k_x=2Ak_y$, $\dot k_y=\dot k_z=0$. The velocity amplitude obeys $\dot{\widetilde{\mathbf v}}-2A\widetilde v_x\mathbf e_y+2\boldsymbol\Omega\times\widetilde{\mathbf v}=-i\mathbf k\widetilde\psi-\nu k^2\widetilde{\mathbf v}$; the magnetic amplitude obeys the displayed equation. The modified pressure includes <magnetic pressure>; the physical pressure can contain a mean and second harmonic even when the modified pressure is a single mode.
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