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A linear force-free magnetic field independent of $y$ can be written $\mathbf{B}=(-A_z,\lambda A,A_x)$. Choosing $A=(B_0/k)\sin (kx)e^{-qz}$ gives a decaying arcade with $k=n\pi /a$, $n\ge 1$, and $q=\sqrt{k^2-{\lambda }^2}>0$. Both horizontal components vanish at $x=0,a$. A fixed horizontal mode requires $|\lambda |<k$; if the mode number is unrestricted, some sufficiently high mode exists for every finite real $\lambda$. Its magnetic field line projections satisfy $A=\text{constant}$ and $dy/dx=\lambda /q$ within each cell.

A linear force-free magnetic field has a spatially constant proportionality factor, $\nabla \times \mathbf{B}=\lambda \mathbf{B}$. Taking its curl and using $\nabla \cdot \mathbf{B}=0$ givesHenceThe constant $\lambda$ has dimensions of inverse length in this convention. If it varied in space, the extra term $\nabla \lambda \times \mathbf{B}$ would remain, so the same Helmholtz equation would not follow merely from a nonlinear force-free condition.