Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1 / c

Solution

Write the perturbations as

(σ, u, v) e^{i k_{x} x + λ t}

k = ∣ k_{x} ∣

, and evaluate all unmarked background quantities at

Σ_{0}

v_{s}^{2} = \frac{d P}{d Σ}_{Σ_{0}}, β = \frac{d l n ( ν Σ )}{d l n Σ}_{Σ_{0}} .

λσ + i k_{x} Σ_{0} u = 0.

The radial and azimuthal momentum equations are

[λ + (ν_{b} + \frac{4}{3} ν) k^{2}] u - 2Ω v = - i k_{x} (v_{s}^{2} \frac{σ}{Σ _{0}} + ϕ),

(λ + ν k^{2}) v + (2Ω - S) u = - i k_{x} Sβ ν \frac{σ}{Σ _{0}},

ϕ = - 2 π G σ / k

. The term proportional to

β

comes from perturbing the density-dependent background shear stress

T_{x y} = - ν Σ S

Eliminating

σ, u, v

and using

S = 3Ω/2

κ_{r}^{2} = 2Ω (2Ω - S) = Ω^{2}

(λ + ν k^{2}) {λ [λ + (ν_{b} + \frac{4}{3} ν) k^{2}] - 2 π G Σ_{0} k + v_{s}^{2} k^{2}} + λ Ω^{2} + 3 β Ω^{2} ν k^{2} = 0 .

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