= Displacement equation for a parallel magnetic shear flow
{title2=$[(\rho S^2-k_x^2B^2)f']'-k^2(\rho S^2-k_x^2B^2)f=0$}
For nonresonant incompressible <normal modes> of a uniform-density <shear flow> with aligned uniform <magnetic field>, let $S=\sigma+k_xU(z)$, $k^2=k_x^2+k_y^2$ and $f=w/S$. The normal <displacement> is $f/i$. The linearized horizontal equations and incompressibility give $ik^2p_1=\rho(Sw'-S'w)=\rho S^2f'$. Substituting the <ideal magnetohydrodynamic induction equation> into normal <momentum> gives the displayed conservative equation. Across a zero-thickness <shear layer>, continuity of $f$ and of $(\rho S^2-k_x^2B^2)f'$ supplies the interface conditions. They express common normal <displacement> and continuous perturbed <magnetohydrodynamic total pressure>, avoiding undefined products of distributional shear gradients.
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