Anelastic reduced-pressure Kelvin wave (${\omega }^2=N^2{k}^2/[k^2+m^2+1/(4H^2)]$)

For an isothermal anelastic approximation with vertical momentum $w_t-b+{\Phi }_z=0$, buoyancy $b_t+N^{2}w=0$, and zero meridional velocity, eliminating pressure yields ${\partial }_t^2(w_{xx}+w_{zz}-w_z/H)+N^2{w}_{xx}=0$. A factor $e^{z/(2H)+imz}$ gives the displayed dispersion relation. Meridional balance produces $e^{-y^2/(2L_D^2)}$, with $L_D^2=(\omega /k)/\beta$ and an eastward intrinsic speed. The limit $L_D^2=2NH/\beta$ requires both $|m|H\ll 1$ and $|k|H\ll 1$.