Magnetic-tension response to an incompressible displacement ($\delta \mathbf{B}=ik{B}_0\mathbf{\xi },\delta \mathbf{f}=-k^2{B}_0^2\mathbf{\xi }/{\mu }_0$)

In ideal magnetohydrodynamics, a small displacement of a uniform magnetic field obeys $\delta \mathbf{B}=\nabla \times (\mathbf{\xi }\times {\mathbf{B}}_0)$. For ${\mathbf{B}}_0=B_0{\mathbf{e}}_z$, a horizontal Fourier mode proportional to $e^{ikz}$ with $\nabla \cdot \mathbf{\xi }=0$ gives the displayed field and force perturbations. The linear magnetic pressure perturbation vanishes because $\delta B_z=0$. The remaining magnetic tension acts as a restoring spring, with Alfvén frequency $k{v}_A$.