 Exceptional root of the mixed Rossby-gravity mode

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 $v=C{e}^{-y^2/2}$, the original equations require ${\delta }^{\prime }-y\delta =i(2k-1/k)v$, 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.