 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 321

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

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

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)