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

Table of contents
- Solution iv

Solution

For

τ Ω = ϵ ≪ 1

, expand the pressure factor:

\frac{1 + γ τ s}{1 + τ s} = 1 + (γ - 1) τ s + O (τ^{2} s^{2}) .

For a stable density wave define

ω_{0}^{2} = Ω^{2} - 2 π G Σ_{0} ∣ k ∣ + c_{0}^{2} k^{2} > 0.

The dispersion relation becomes

s^{2} + ω_{0}^{2} + (γ - 1) c_{0}^{2} k^{2} τ s = O (ϵ^{2} Ω^{2}) .

Perturbing the two isothermal roots gives

s_{\pm} = \pm i ω_{0} - \frac{1}{2} (γ - 1) c_{0}^{2} k^{2} τ + O (ϵ^{2} Ω) .

Thus the leading amplitude-damping rate is

- Re s = \frac{1}{2} (γ - 1) c_{0}^{2} k^{2} τ .

Solved by gpt-5.6-sol high.

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