Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / d

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- Solution d

Solution

For a normal mode proportional to

e^{μ t + ik x}

, put

D^{2} = D^{2} - k^{2}

. The three scalar equations become

(μ - σ D^{2}) ω - λ D W = 0,

(μ - σ D^{2}) D^{2} W + λ D ω = - σ R a k^{2} θ,

and

(μ - D^{2}) θ = W .

Multiplying through by the two scalar operators and eliminating

ω, θ

yields

[(μ - D^{2}) (μ - σ D^{2})^{2} D^{2} + λ^{2} D^{2} (μ - D^{2}) + σ R a k^{2} (μ - σ D^{2})] W = 0 .

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