= Gauge transformation for conducting-square Darcy onset
{title2=$P=e^{-iq(x-1/2)/2}F$}
At zero growth rate put $P=\psi+iq\theta$, $q=\sqrt R$. The coupled Darcy equations become $\Delta P+iqP_x=0$. 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 $q^2=4\pi^2(m^2+n^2)$, $m,n\geq1$, 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.
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