 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/39d/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Horizontal momentum and incompressibility give

p = ρ_{0} \frac{ω}{k} u = - ρ_{0} \frac{ωm}{k ^{2}} w .

Therefore

⟨ \overline{I}_{z} ⟩_{0} = \frac{1}{2} Re (p w^{*}) = - \frac{ρ _{0} ωm}{2 k ^{2}} w_{0}^{2} .

For upward radiation take

ω = - U k

and

m = N^{2} / U^{2} - k^{2} > 0

, obtaining

⟨ \overline{I}_{z} ⟩_{0} = \frac{1}{2} ρ_{0} U^{2} k h_{0}^{2} N^{2} - U^{2} k^{2} > 0.

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