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

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- Solution b

Solution

Without self-gravity and for

k > 0

,

ω^{2} = Ω^{2} + c_{s}^{2} k^{2} .

The phase velocity and group velocity are

c_{p} = \frac{ω}{k}, c_{g} = \frac{d ω}{d k} = \frac{c _{s}^{2} k}{ω},

so

c_{p} c_{g} = c_{s}^{2} .

With self-gravity restored,

c_{g} = \frac{c _{s}^{2} k - π G Σ}{ω}, c_{p} c_{g} = c_{s}^{2} - \frac{π G Σ}{k} .

The product changes sign at

k_{crit} = \frac{π G Σ}{c _{s}^{2}} = \frac{1}{Q H} .

For

k > k_{crit}

, crests and a localized wave packet travel in the same radial direction. For

k < k_{crit}

, the group velocity and phase velocity have opposite signs: the envelope and its wave energy propagate opposite to the individual crests. At

k_{crit}

the group velocity vanishes and neighboring wave components cause the packet to spread.

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