Dictionary:Eikonal equation

(ī kōn’ ∂l) (from Greek $\iota \kappa o\nu$ (ikon) meaning image. An equation derived from the wave equation through the substitution of a harmonic wave trial solution into the wave equation. In one form of the eikonal equation seen in physics literature, the local velocity $V$ is compared to a reference velocity $V_{R}$(analogous to comparing a velocity to the speed of light in vacuum):

$\left(\nabla \phi \right)^{2}=\left({\frac {V}{V_{R}}}\right)^{2}=n^{2}$,

where $n$ is an index of refraction and the quantity $\phi$ is identified as wave propagation phase advance function, which is the travel time of a point on a wave front. The use of index of refraction reflects the physicists' desire to work in dimensionless coordinates.

More commonly in geophysical literature, the eikonal equation (for scalar waves) is written in terms of medium velocity only $V(\mathbf {x} )$ where $\mathbf {x} =(x_{1},x_{2},x_{3})$, as

$\left(\nabla \phi (\mathbf {x} )\right)^{2}={\frac {1}{V^{2}(\mathbf {x} )}}.$

Solutions to the eikonal equation yield a high-frequency or large-wavenumber asymptotic representation of the wave field as a family of rays, represented by ray position and ray direction---the so-called kinematic aspect of wave propagation.

Another form of the eikonal equation is written in terms of the ray direction vector $\mathbf {p} =(p_{1},p_{2},p_{3})$ where the gradient of traveltime (or slowness) vector $p_{i}={\frac {\partial \phi }{\partial x_{i}}}$ for $i=1,2,3$

$p^{2}=\mathbf {p} \cdot \mathbf {p} =p_{1}^{2}+p_{2}^{2}+p_{3}^{2}={\frac {1}{V(\mathbf {x} )}}$

thus $\mathbf {x} =(x_{1},x_{2},x_{3})$ are the generalized coordinates and $\mathbf {p} =(p_{1},p_{2},p_{3})$ are the generalized momenta from Hamiltonian mechanics, and the eikonal equation corresponds to the Hamiltonian function or the Hamilton-Jacobi equation of analytical mechanics.

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