Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / c / iv

Table of contents
- Solution iv

Solution

When

k_{y} = 0

,

k_{x}

is constant and the system becomes

\dot{u}_{x} = 2 (Ω + \frac{c _{s}^{2} k _{x}^{2}}{Ω}) u_{y}, \dot{u}_{y} = - \frac{Ω}{2} u_{x} .

Thus

\ddot{u}_{x} + (Ω^{2} + c_{s}^{2} k_{x}^{2}) u_{x} = 0.

For time dependence

e^{- iω t}

, the dispersion relation is

ω^{2} = Ω^{2} + c_{s}^{2} k_{x}^{2} .

Explicitly,

u_{x} = C_{1} cos (ω t) + C_{2} sin (ω t), u_{y} = \frac{u ˙ _{x}}{2 ( Ω + c _{s}^{2} k _{x}^{2} /Ω )},

and

δ = 2 i k_{x} u_{y} /Ω

. This is an axisymmetric inertial-acoustic wave: pressure supplies the

c_{s}^{2} k_{x}^{2}

restoring term and epicyclic motion supplies the

Ω^{2}

term.

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