For $\tau\Omega=\epsilon\ll1$, expand the pressure factor:
$$
\frac{1+\gamma\tau s}{1+\tau s}
=1+(\gamma-1)\tau s+O(\tau^2s^2).
$$
For a stable density wave define
$$
\omega_0^2=\Omega^2-2\pi G\Sigma_0|k|+c_0^2k^2>0.
$$
The dispersion relation becomes
$$
s^2+\omega_0^2+(\gamma-1)c_0^2k^2\tau s
=O(\epsilon^2\Omega^2).
$$
Perturbing the two isothermal roots gives
$$
\boxed{s_\pm=\pm i\omega_0
-\frac12(\gamma-1)c_0^2k^2\tau+O(\epsilon^2\Omega)}.
$$
Thus the leading amplitude-damping rate is
$$
\boxed{-\operatorname{Re}s
=\frac12(\gamma-1)c_0^2k^2\tau}.
$$

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 321 / 3 / b / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-321/3/b/iv/solution