= Decaying linear force-free magnetic arcade
{title2=$q^2=k^2-\lambda^2$}
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 $k=n\pi/a$, $n\ge1$, 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.
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