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Ray tracing

For the zero-frequency dispersion relation
$$
\Omega=\alpha|\mathbf k|^{1/2}-Vk_1,
$$
write $\mathbf k=\kappa(\cos\phi,\sin\phi)$. Then
$$
\kappa=\frac{\alpha^2}{V^2\cos^2\phi},
\qquad
\tan\psi=-\frac{\tan\phi}{1+2\tan^2\phi},
$$
where $\psi$ is the direction of the <group velocity>. Maximizing the angular slope gives a downstream wake wedge of semi-angle
$$
\tan^{-1}(2^{-3/2}).
$$

Stationary square-root-dispersion wake

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 38A / c / Solution

Stationary square-root-dispersion wake

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Stationary square-root-dispersion wake

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