= Solution
Assuming the mean wind remains $10\,\mathrm{m\,s^{-1}}$, the weakly stratified upper layer has
$$
m_{\rm upper}^2=\left(\frac{10^{-3}}{10}\right)^2-k^2
=10^{-8}\,\mathrm{m^{-2}}-k^2.
$$
For the inverse-width estimate $|k|\sim10^{-4}\,\mathrm{m^{-1}}$, it is near the turning threshold $m=0$, with negligible upward group-<velocity> component. Components of larger $|k|$ are evanescent aloft and are substantially reflected; only horizontal wavelengths exceeding $2\pi U/N_{\rm upper}\simeq62.8\,\mathrm{km}$ can propagate vertically there. For the alternative $10\,\mathrm{km}$ sinusoidal wavelength, $m_{\rm upper}^2<0$ and the evanescent decay length is about $1.6\,\mathrm{km}$.
The sharp inversion and change of stability can therefore cap the upward-radiating disturbance. Reflection between the ground and cap may trap <energy> in the lower layer and allow downstream lee-wave trains or resonant guided modes, with permitted wavelengths depending on the interface and lower-layer structure. Reflection, leakage and possible resonance are distinct: a specified $5\,\mathrm{km}$ height does not by itself fix a unique mode amplitude or wavelength.
Wave <clouds> may become confined below the cap or organized into persistent lee bands if there is enough moisture. Inversion-limited vertical mixing can also retain low <cloud>, haze or pollutants; sufficiently strong trapped motions can promote rotors or <turbulence>. These are possible changes, not inevitable weather outcomes: humidity, wave amplitude, detailed stability and wind shear are unspecified. The new structure suppresses upward <energy> transmission for the dominant short horizontal scales even though the lower atmosphere can still support waves.
Back to article page
