Past exam of the topics in applied mathematics course of the University of Oxford / 2008 / A / 4 / e

Here $L_1=1$ and $L_2=1/2$, so a perturbation varying in both coordinates has $n,m\ge 1$ andWith the given unit coefficients the determinant becomes $D(s)=(1+s{)}^2-\beta s$. The trace remains negative, so instability occurs precisely when $D(s)<0$, orThe right-hand side is strictly increasing for $s>1$, since its derivative is $1-s^{-2}>0$. Its smallest allowed value is attained by the mode $(n,m)=(1,1)$, giving the onset thresholdAt this value the first mixed mode has a zero eigenvalue; it becomes unstable for $\beta >{\beta }_{\mathrm{crit}}$. Thus the critical value is the infimum of the strictly unstable range, rather than a strictly unstable point itself. Its spatial factor is $\cos (\pi x)\cos (2\pi y)$.