# Difference between revisions of "Dictionary:Eikonal equation"

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{{lowercase}}{{#category_index:E|eikonal equation}} | {{lowercase}}{{#category_index:E|eikonal equation}} | ||

− | (ī kōn’ ∂l) (from Greek <math> \iota \kappa o \nu </math> (ikon) meaning ''image''. An equation derived from the wave equation through the substitution of a harmonic wave trial solution, | + | (ī kōn’ ∂l) (from Greek <math> \iota \kappa o \nu </math> (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, ithe local velocity <math> V </math> is compared to a reference velocity <math> V_R </math>(analogous to comparing a velocity to the speed of light in vacuum): |

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− | where <math>n</math> is an index of refraction and | + | where <math>n</math> is an index of refraction and the quantity <math> \phi</math> 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.’’ | propagation travel time. Valid only where the variation of properties is small within a wavelength, sometimes called the ‘‘high-frequency condition.’’ | ||

## Revision as of 12:52, 10 June 2015

(ī kōn’ ∂l) (from Greek (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, ithe local velocity is compared to a reference velocity (analogous to comparing a velocity to the speed of light in vacuum):

where is an index of refraction and the quantity 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 where , as

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 where for

thus are the *generalized coordinates* and are
the *generalized momenta* from Hamiltonian mechanics, and the eikonal equation corresponds to the Hamiltonian function or the
Hamilton-Jacobi equation of analytical mechanics.