Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 2 / d

Solution

For an incompressible perturbation,

Δ = 0

, so

Ψ = g ξ_{z}

. The vertical equation gives

\frac{d ξ _{z}}{d z} = \frac{ω ^{2}}{g} ξ_{z} .

Incompressibility together with the horizontal equation also gives

\frac{d ξ _{z}}{d z} = \frac{g k _{x}^{2}}{ω ^{2} - 4 Ω ^{2}} ξ_{z} .

Equating the two expressions yields

ω^{2} (ω^{2} - 4 Ω^{2}) = g^{2} k_{x}^{2} .

ω^{2} = 2 Ω^{2} + (g k_{x})^{2} + (2 Ω^{2})^{2} .

Its vertical displacement decays as

e^{ω^{2} z / g}

into the atmosphere

z < 0

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