= Solution
Work with the explicitly corrected reduced-pressure system in <5/b>. Write $d=1/(2H)$ and use the complex wave amplitude, taking real parts at the end. The vertical factor satisfies $(\partial_z^2-H^{-1}\partial_z)w=-(m^2+d^2)w$, so the wave equation gives
$$
\boxed{\omega^2=\frac{N^2k^2}{k^2+m^2+d^2},\qquad
c=\frac\omega k=\frac{N}{\sqrt{k^2+m^2+d^2}}>0.}
$$
For a <Equatorial Kelvin wave>[Kelvin wave], continuity, zonal momentum and the <buoyancy> equation then determine all polarizations:
$$
\boxed{u=-\frac{m+id}{k}\,w,\qquad
\Phi=\frac\omega k u=-\frac{\omega(m+id)}{k^2}w,
\qquad p'=\rho_0\Phi,\qquad b=-i\frac{N^2}{\omega}w,\qquad v=0.}
$$
In particular $u,w,b,\Phi$ have amplitude proportional to $e^{z/(2H)}$, whereas <pressure> $p'$ has amplitude proportional to $e^{-z/(2H)}$. Directly substituting these expressions into vertical momentum gives $\Phi_z=-i\omega(m^2+d^2)w/k^2=i(\omega-N^2/\omega)w$, verifying both the sign and the <dispersion relation>.
Meridional momentum is $\beta yu+\Phi_y=0$. Since $\Phi=cu$, it integrates to a <Gaussian distribution>[Gaussian] $\exp[-\beta y^2/(2c)]$. Thus
$$
\boxed{L_D^2=\frac c\beta=\frac{N}{\beta\sqrt{k^2+m^2+1/(4H^2)}}.}
$$
The eastward intrinsic branch $c>0$ is selected by decay away from the equator; choosing $c<0$ would make the meridional amplitude grow without bound. Although the squared <dispersion relation> has both <frequency> signs, a meridionally trapped <Equatorial Kelvin wave>[Kelvin wave] has this one <Phase velocity>[phase-propagation] direction.
For \b[short vertical <Wavelength>[wavelengths]], $m^2\gg k^2+d^2$, so $\omega^2\simeq N^2k^2/m^2$, $c\simeq N/|m|$ and $L_D^2\simeq N/(\beta|m|)$. The wave is a slowly oscillating, <Hydrostatic equilibrium>[hydrostatic] internal <gravity wave> with strong vertical variation and relatively narrow trapping. Here $u\simeq-(m/k)w$ and $p'/\rho_0\simeq-(\omega m/k^2)w$; the <buoyancy> expression above remains valid.
For \b[long vertical <Wavelength>[wavelengths]], $|m|\ll\sqrt{k^2+d^2}$, the exact limiting speed is $c=N/\sqrt{k^2+d^2}$. To obtain the stated height-controlled limit also requires long horizontal <wavelength>, $|k|H\ll1$. With both $|m|H\ll1$ and $|k|H\ll1$,
$$
\boxed{\omega\simeq2NHk,\qquad L_D^2\simeq\frac{2NH}{\beta}.}
$$
The finite <Scale height>[density scale height] prevents the <Hydrostatic equilibrium>[hydrostatic] $1/|m|$ <Phase velocity>[phase speed] from diverging as $m\to0$. The polarizations tend to $u\simeq-iw/(2Hk)$, $\Phi\simeq-iNw/k$ and $b\simeq-iNw/(2Hk)$. Long vertical <wavelength> alone at fixed large $kH$ would not justify $c=2NH$. In the opposite horizontally short limit the <frequency> approaches $N$ instead. The growth $e^{z/(2H)}$ compensates decreasing reference <density> in wave-energy <density>; a radiating wave train is a local modal description, not a finite-total-energy eigenfunction over an infinite vertical domain.
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