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The interior potential-vorticity equation alone does not determine the frequency. Use the rigid-boundary buoyancy condition for quasi-geostrophic waves. Linear adiabatic buoyancy conservation about the basic state givesAt $z=0$, the impermeability condition sets $w^{\prime }=0$ and the basic velocity is zero. ThereforeSubstitution of ${\hat{\psi }}^{\prime }(0)=-m\hat{\psi }(0)$ yields the dispersion relation of a semi-infinite Eady edge waveFor $f_0>0$, this is $c=-M^2/[N\sqrt{k^2+l^2}]$. The sign is fixed by both the thermal-wind relation and the lower-boundary buoyancy budget.

The steering height of a semi-infinite Eady edge wave is obtained by cancelling the phase-speed term against the basic advection in the wave-following frame:Thus the phase velocity equals the basic thermal-wind velocity at one penetration depth. For nonzero $M$ they have the same sign in the laboratory frame. If “exactly opposes the thermal wind” is read literally as $c=-\bar{U}(z)$, it instead gives $z=-1/m$, outside the fluid: there is no positive physical height with opposite laboratory velocities. The positive height above is the physically meaningful critical level, where the $-c$ and $\bar{U}$ contributions to the wave-frame material derivative oppose and cancel. This distinction resolves the ambiguous printed wording without reversing the derived wave speed. If $M=0$, the shear and $c$ vanish, so there is no distinguished steering height.