Finite-depth Eady dispersion relation (${\left(c-\Lambda D/2\right)}^2=(\Lambda /m{)}^2[(mD{)}^2/4-(mD)\coth (mD)+1]$)

For constant shear $U=\Lambda z$ between impermeable horizontal walls at $0,D$, with constant buoyancy frequency and no interior potential-vorticity gradient, define $m=NK/|f_0|$. The boundary conditions yieldThe bracket is negative for $0<mD<2.399\dots$, giving baroclinic instability for nonzero zonal wavenumber. For large separation the boundary waves decouple; the lower-wave speed tends to $\Lambda /m$, matching the dispersion relation of a semi-infinite Eady edge wave.