= Vanishing two-to-one forcing coefficient for Stokes convection
{title2=$\beta=0$}
In the free-slip <Stokes flow> <temperature> model, let the critical <temperature> be $g=\sin\pi z$ and vertical <velocity> $f=sg$, $s=k_c^2+\pi^2$. A forced positive second harmonic $(W,\Theta)e^{2ik_cx}$ couples to the negative critical harmonic. Its <temperature> <solvability condition> has integrand
$$
g(2f'\Theta+f\Theta'+W'g/2+Wg')=(sg^2\Theta+g^2W/2)'.
$$
The identity follows by substituting $f=sg$ and differentiating. Its integral is zero because $g$ vanishes on both plates, regardless of the forced boundary value of $\Theta$. Thus the first resonant forcing coefficient vanishes. A generic nonzero conjugate-amplitude term cannot be inferred from wave-number matching alone.
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