# 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) | + | (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, in which the 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 | + | where <math>n</math> is an index of refraction and [[File:Fgr.gif]] is the wave function. 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.’’ | ||

More commonly in geophysical literature, the eikonal equation (for scalar waves) is written in terms of medium velocity only <math> V(\mathbf{x} ) </math> | More commonly in geophysical literature, the eikonal equation (for scalar waves) is written in terms of medium velocity only <math> V(\mathbf{x} ) </math> | ||

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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. | 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 <math> \mathbf{p} = (p_1,p_2, p_3) </math> where | ||

+ | <math> p_i = \frac{\partial \phi}{\partial x_i} </math> for <math> i = 1, 2, 3 </math> | ||

+ | |||

+ | <center> <math> p^2 = \mathbf{p} \cdot \mathbf{p} = p_1^2 + p_2^2 + p_3^2 = \frac{1}{V(\mathbf{x})} </math> </center> | ||

+ | |||

+ | thus <math> \mathbf{x} = (x_1,x_2,x_3) </math> are the ''generalized coordinates'' and <math> \mathbf{p} = (p_1,p_2, p_3) </math> are | ||

+ | the ''generalized momenta'' from Hamiltonian mechanics, and the eikonal equation corresponds to the Hamiltonian function or the | ||

+ | Hamilton-Jacobi equation of analytical mechanics. | ||

==External links== | ==External links== | ||

{{search}} | {{search}} |

## Revision as of 11:48, 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, in which the 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 is the wave function. 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.