= Exact magnetic shearing wave
{title2=$\dot k_x=2Ak_y,\quad\dot{\bar b}_y=-2A\bar b_x,\quad\mathbf k\cdot\bar{\mathbf b}=\mathrm{constant}$}
In an incompressible <shearing sheet>, a single real-wavevector velocity and magnetic Fourier mode transverse to that wavevector has identically zero quadratic self-advection and cross-advection. Advecting its phase by background shear and stretching its spatially uniform field give the displayed evolution laws. Their opposite contributions keep the <Alfvén frequency> constant. The resulting viscous and resistive wave-amplitude equations are exact finite-amplitude solutions; physical pressure adjusts to the quadratic magnetic-pressure terms.
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