Without self-gravity and for $k>0$,
$$
\omega^2=\Omega^2+c_s^2k^2.
$$
The <phase velocity and group velocity> are
$$
c_p=\frac\omega k,
\qquad
c_g=\frac{d\omega}{dk}=\frac{c_s^2k}{\omega},
$$
so
$$
\boxed{c_pc_g=c_s^2}.
$$

With self-gravity restored,
$$
c_g=\frac{c_s^2k-\pi G\Sigma}{\omega},
\qquad
\boxed{c_pc_g=c_s^2-\frac{\pi G\Sigma}{k}}.
$$
The product changes sign at
$$
\boxed{k_{\rm crit}=\frac{\pi G\Sigma}{c_s^2}
=\frac1{QH}}.
$$
For $k>k_{\rm crit}$, crests and a localized <wave packet> travel in the same radial direction. For $k<k_{\rm crit}$, the <group velocity> and <phase velocity> have opposite signs: the envelope and its wave energy propagate opposite to the individual crests. At $k_{\rm crit}$ the group velocity vanishes and neighboring wave components cause the packet to spread.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-321/2/b/solution