Vanishing two-to-one forcing coefficient for Stokes convection ($\beta =0$)

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 $(W,\Theta )e^{2i{k}_{c}x}$ couples to the negative critical harmonic. Its temperature solvability condition has integrandThe 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.