Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3 / c

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- Solution c

Solution

For a mode

e^{i k \cdot x + s t}

, put

D = s + ν k^{2}

. The linear equations become

D u_{x} - 2Ω u_{y} = - i k_{x} P / ρ_{0} - N^{2} θ, D u_{y} + \frac{1}{2} Ω u_{x} = 0, D u_{z} = - i k_{z} P / ρ_{0},

with

s θ = u_{x}

and

k_{x} u_{x} + k_{z} u_{z} = 0

. Eliminating

u_{y}, u_{z}, P, θ

gives the dispersion relation

s (D^{2} + ϖ^{2}) + n^{2} ϖ^{2} D = 0, ϖ^{2} = \frac{k _{z}^{2}}{k ^{2}} Ω^{2}, n^{2} = \frac{N ^{2}}{Ω ^{2}} .

Expanding,

s^{3} + 2 k^{2} ν s^{2} + [ϖ^{2} (1 + n^{2}) + k^{4} ν^{2}] s + n^{2} k^{2} ν ϖ^{2} = 0 .

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