Wavevector ( $k$ )

A wavevector points in the direction of phase propagation; its magnitude is the wavenumber and its scalar product with position gives the spatial phase.

Table of contents
- Hamiltonian ray-tracing equations Wavevector
  - WKB method Hamiltonian ray-tracing equations
    - Eikonal equation WKB method
  - Total derivative along a ray Hamiltonian ray-tracing equations

Hamiltonian ray-tracing equations

 0  0

For a slowly varying local dispersion relation

ω = Ω (k; x, t)

, wave-packet rays obey

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

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WKB method

 0  0

The WKB method seeks a rapidly oscillating field in the form

ϕ (x, t) = A (x, t; ε) e^{i θ (x, t) / ε}, 0 < ε ≪ 1,

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and determines the phase and slowly varying amplitude order by order in

ε

Eikonal equation

 1  0

The leading WKB phase obeys an eikonal equation. For a local dispersion relation

ω = Ω (k; x, t)

and

k = \nabla θ

, it is

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

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Total derivative along a ray

 0  0

Along a ray

x (t)

, the derivative of a field

f (x, t)

\frac{df}{d t} = \frac{\partial f}{\partial t} + \dot{x} \cdot \nabla f .

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