Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / ii

Table of contents
- Solution ii

Solution

The boundary condition gives complex amplitude

w_{0} = U k h_{0}

. Substitution of the internal gravity wave form

w = w_{0} e^{i (k x + m z - ω t)}

into the internal-wave equation gives

ω^{2} = \frac{N ^{2} k ^{2}}{k ^{2} + m ^{2}} .

Since

∣ ω ∣ = U k

,

m^{2} = \frac{N ^{2}}{U ^{2}} - k^{2} .

For

k < N / U

, choose the sign of

m

that gives upward energy propagation; with

ω = - U k

, this is

m > 0

.

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