= Solution
Put $S=M^2/f_0$. The <thermal wind> is $\bar U(z)=-Sz$, and the basic <potential vorticity> is spatially constant: $\bar q=2f_0$ when the full reference buoyancy is included, or $f_0$ when that reference contribution has been subtracted. Either way $\nabla_h\bar q=0$.
For a smooth <normal mode>, the linear <quasi-geostrophic potential-vorticity equation> becomes
$$
ik(\bar U-c)\widehat q=0,\qquad \widehat q=-K^2\widehat\psi+\frac{f_0^2}{N^2}\widehat\psi'',\qquad K=\sqrt{k^2+l^2}.
$$
Take $k\ne0$ and $K>0$, as required for a nontrivial wave with the printed phase-speed parameterization. Except possibly at an isolated <critical level of a shear-flow wave>, this implies $\widehat q=0$; regularity then extends that condition through the isolated level. Hence
$$
\widehat\psi''-m^2\widehat\psi=0,\qquad m=\frac{NK}{|f_0|}.
$$
Decay at infinity excludes the growing exponential, giving the <decaying vertical structure of a semi-infinite Eady edge wave>
$$
\boxed{\widehat\psi(z)=C e^{-mz},\qquad m=\frac{N\sqrt{k^2+l^2}}{|f_0|}.}
$$
The amplitude $C$ is arbitrary. The <semi-infinite Eady model> supports a wave trapped at its lower boundary with penetration depth $m^{-1}$. The zero-horizontal-wavenumber case has no nonzero decaying solution of this homogeneous vertical equation. Singular neutral interior <potential vorticity> sheets belong to a different continuous-spectrum class; they are not the smooth decaying edge-wave eigenfunction requested here.
The PDF contains the request for this vertical structure and uses the phase factor $e^{i(kx+ly-kct)}$. The TeX aid omits that request and also inserts a duplicate factor without $c$. The solution uses the original PDF form throughout.
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