Linearization about $U\widehat{\mathbf x}$ changes the frequency to the intrinsic value $\omega-Uk$. Thus
$$
\boxed{
\omega=Uk-\frac{\beta k}{K^2+f_0^2m^2/N^2}.}
$$
A mountain-fixed disturbance has $\omega=0$, so its vertical wavenumber satisfies
$$
\boxed{
m^2=\frac{N^2}{f_0^2}\left(\frac\beta U-K^2\right).}
$$
The <quasi-geostrophic mountain wave> propagates vertically only when $m^2>0$. For $U>0$, this requires
$$
\boxed{0<U<\frac\beta{k^2+l^2}.}
$$
Otherwise the disturbance is vertically evanescent.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 3 / iii / Solution

{title2=QG mountain wave}

A quasi-geostrophic mountain wave is a stationary <Rossby wave> forced when a geostrophic basic flow crosses topography. Its vertical wavenumber is real only when the intrinsic Rossby-wave phase speed permits vertical propagation.

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