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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 becomesTake $k\ne 0$ 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 $\hat{q}=0$; regularity then extends that condition through the isolated level. HenceDecay at infinity excludes the growing exponential, giving the decaying vertical structure of a semi-infinite Eady edge waveThe 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.

A lower-boundary Eady edge wave in the semi-infinite Eady model has a smooth exponentially decaying vertical structure and a real frequency. It is supported by the horizontal boundary buoyancy gradient while its interior potential vorticity anomaly vanishes. Its speed equals the basic velocity at one penetration depth.