 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-77/1/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For

0 < ω < N

, put

α = ω / N^{2} - ω^{2}

k = \pm α m

. On the positive-frequency branch, for

k > 0

ω = \frac{N k}{k ^{2} + m ^{2}}, c_{gx} = \frac{N m ^{2}}{( k ^{2} + m ^{2} ) ^{3/2}} > 0, c_{g z} = - \frac{N km}{( k ^{2} + m ^{2} ) ^{3/2}} .

The radiation condition therefore selects

k = α m

, since energy must travel into the fluid. Substitution into the internal-wave polarization relations gives the complex fields

u = ϵ e^{i (α m x + m z - ω t)}, w = - α ϵ e^{i (α m x + m z - ω t)}, σ = \frac{i N ^{2} α ϵ}{ω} e^{i (α m x + m z - ω t)} .

The physical fields are their real parts;

v = 0

. The vertical component of group velocity is negative, so this positive-

m

plane internal gravity wave carries energy down and right, although its phase propagates in a different direction.

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