 Gauge transformation for conducting-square Darcy onset

At zero growth rate put $P=\psi +iq\theta$, $q=\sqrt{R}$. The coupled Darcy equations become $\Delta P+iq{P}_x=0$. Substituting $P=e^{-iq(x-1/2)/2}F$ gives $\Delta F+(q^2/4)F=0$ with homogeneous Dirichlet boundary conditions. The square Dirichlet Laplacian eigenvalues give $q^2=4{\pi }^2(m^2+n^2)$, $m,n\ge 1$, whose minimum is $8{\pi }^2$. For $S=\sin \pi x\sin \pi z$, $a=q_c/2$, the complex multiples $F=S,iS$ give real pairs $(S\cos (a\xi ),-S\sin (a\xi )/q_c)$ and $(S\sin (a\xi ),S\cos (a\xi )/q_c)$, $\xi =x-1/2$. Their reflection parities are opposite, yielding two independent physical modes.