Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2019 / Section A / 2 / Solution

The exponential factor removes the first spatial derivative. Apply the chain rule explicitly:while ${\partial }_t^2\Delta \rho =e^{-z/(2H)}{\Psi }_{tt}$. Substituting into the supplied density equation, the terms proportional to ${\Psi }_z$ cancel, leavingThis is the atmospheric acoustic cutoff transformation. There is a sign error in the target equation in the original PDF, as well as in the transcription: the last term must be $+{\omega }_0^2\Psi$ for real positive ${\omega }_0$. If the displayed minus sign is retained literally, coefficient matching would require ${\omega }_0^2=-c^2/(4H^2)$, so it would not define a real characteristic oscillation period.

For the corrected equation, a plane wave $\Psi \propto e^{i(kz-\omega t)}$ has dispersion relationThus ${\omega }_0$ is the acoustic cutoff frequency: vertically propagating acoustic waves have $|\omega |\ge {\omega }_0$. The scale height and sound speed have units metres and metres per second, respectively, so ${\omega }_0$ has units ${\mathrm{s}}^{-1}$.

For an order-of-magnitude estimate in the lower terrestrial atmosphere, take $H\simeq 8.0\mathrm{km}$ and $c\simeq 340\mathrm{m}{\mathrm{s}}^{-1}$. These are representative atmospheric values, not additional data specified by the paper. They giveThe isothermal atmosphere approximation is local: a real atmospheric temperature profile and the thermodynamic response of the disturbance modify the precise cutoff and period.