Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 345 / 1 / b

Solution

For a single phase

ϕ = k x + m z - ω t

, every disturbance field is a function of

ϕ

alone. Its velocity is tangent to the phase surfaces:

k \cdot u = k (- ψ_{z}) + m ψ_{x} = 0.

Consequently both nonlinear Jacobians vanish exactly, irrespective of amplitude. Substitution gives

iω (k^{2} + m^{2}) ψ = ik b, - iω b + i N^{2} k ψ = 0.

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

The wave is an exact nonlinear solution of the inviscid Boussinesq equations. The parameter

ϵ

merely sets its amplitude; it does not change its frequency or waveform. Very large

ϵ

can nevertheless invalidate the Boussinesq model or make the total stratification locally overturn.

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