# Dictionary:Eikonal equation

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

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

where ${\displaystyle n}$ is an index of refraction and is the wave function. the quantity ${\displaystyle \phi }$ is identified as wave propagation travel time. Valid only where the variation of properties is small within a wavelength, sometimes called the ‘‘high-frequency condition.’’

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

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

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 ${\displaystyle \mathbf {p} =(p_{1},p_{2},p_{3})}$ where ${\displaystyle p_{i}={\frac {\partial \phi }{\partial x_{i}}}}$ for ${\displaystyle i=1,2,3}$

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

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