= Positivity condition for a phase-locked standing wave
{title2=$z=|A|^2>0$}
Writing $U=\operatorname{Re}(\beta+\gamma)$ and $V=\operatorname{Im}(\beta+\gamma)$, the squared amplitude is $z_\pm=[\mu U\pm\sqrt{(U^2+V^2)\nu^2-\mu^2V^2}]/(U^2+V^2)$. Reality of the square root does not guarantee positivity. For example, $U=1,V=0,\mu=-1,\nu=1/2$ gives two negative roots. Under the usual supercritical assumptions $\mu>0,U>0$, the plus root is positive whenever the <discriminant> is nonnegative.
// Destination: mathematical-biology.bigb
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