= Exceptional root of the mixed Rossby-gravity mode
{title2=$k^2=1/2,\quad\omega=-k$}
For dimensionless Hermite index zero, the <equatorial shallow-water dispersion relation> factorizes as $(\omega+k)(\omega^2-k\omega-1)=0$. The root $\omega=-k$ is generally spurious: for $v=Ce^{-y^2/2}$, the original equations require $\delta'-y\delta=i(2k-1/k)v$, where $\delta=u-\eta$. Multiplication by $e^{-y^2/2}$ and integration over the real line shows that decay at both ends requires $C=0$ unless $k^2=1/2$. At that exception the root already belongs to the physical quadratic; $u=\eta=i\omega yv$ is regular and no extra wave branch is created. The assertion concerns nonzero $k$ and <angular frequency>.
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