Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 39C / b / i

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- Solution i

Solution

For the given internal gravity wave,

Ω (k, m; x, z) = \frac{N k}{k ^{2} + m ^{2}} + γ (k x - m z) .

The wavevector part of the Hamiltonian ray-tracing equations is

\dot{k} = - \frac{\partial Ω}{\partial x} = - γk, \overset{m}{˙} = - \frac{\partial Ω}{\partial z} = γm .

Applying the initial data gives

k (t) = k_{0} e^{- γ t}, m (t) = m_{0} e^{γ t} .

In particular,

k (t) m (t) = k_{0} m_{0}

is conserved.

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