Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / e

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- Solution e

Solution

The homogeneous equation can be written

v^{''} + [\frac{S ^{2} k _{y}^{2} x ^{2}}{v _{s}^{2}} - k_{y}^{2} - \frac{κ _{r}^{2}}{v _{s}^{2}}] v = 0.

Its solutions are locally oscillatory where

S^{2} k_{y}^{2} x^{2} > κ_{r}^{2} + v_{s}^{2} k_{y}^{2},

equivalently

∣ S x ∣ > v_{s}^{2} + \frac{κ _{r}^{2}}{k _{y}^{2}} > v_{s} .

The relative background orbital speed is

∣ u_{0} ∣ = ∣ S x ∣

, so propagating wave zones occur only where the relative motion of the satellite and disk material is supersonic.

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