Solution

Assume the medium varies on scales much longer than the wavelength and period, so the leading-order WKB method gives the local eikonal equation. For the phase

θ

in the stated ansatz, define

k = \nabla θ, ω = - \partial_{t} θ .

(375)

The local dispersion relation becomes the Hamilton--Jacobi equation

- \partial_{t} θ = Ω (\nabla θ; x, t) .

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Differentiating it with respect to

x_{i}

, using equality of mixed derivatives, and choosing a ray whose velocity is

\overset{x}{˙}_{i} = \frac{\partial Ω}{\partial k _{i}}

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gives

\dot{k}_{i} = \partial_{t} k_{i} + \overset{x}{˙}_{j} \partial_{x_{j}} k_{i} = - \frac{\partial Ω}{\partial x _{i}} .

(378)

Here the dot, or

d / d t

, is the total derivative along a ray,

\frac{d}{d t} = \frac{\partial}{\partial t} + \dot{x} \cdot \nabla_{x},

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and not merely a partial time derivative at fixed position.

Finally, differentiate

ω = Ω (k; x, t)

along the same trajectory:

\frac{d ω}{d t} = \frac{\partial Ω}{\partial t} + \frac{\partial Ω}{\partial x _{i}} \overset{x}{˙}_{i} + \frac{\partial Ω}{\partial k _{i}} \dot{k}_{i} = \frac{\partial Ω}{\partial t},

(380)

because the last two terms cancel. The Hamiltonian ray-tracing equations are therefore

\frac{d x _{i}}{d t} = \frac{\partial Ω}{\partial k _{i}}, \frac{d k _{i}}{d t} = - \frac{\partial Ω}{\partial x _{i}}, \frac{d ω}{d t} = \frac{\partial Ω}{\partial t} .

(381)

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 39C / a / Solution

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