# Difference between revisions of "Dictionary:Eikonal equation"

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(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, 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): | (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, 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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<center><math>\left(\nabla \phi \right)^2 =\left(\frac{V}{V_R}\right)^2=n^2 </math>,</center> | <center><math>\left(\nabla \phi \right)^2 =\left(\frac{V}{V_R}\right)^2=n^2 </math>,</center> | ||

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where <math>n</math> is an index of refraction and the quantity <math> \phi</math> is identified as wave | where <math>n</math> is an index of refraction and the quantity <math> \phi</math> 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 | 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. | the physicists' desire to work in dimensionless coordinates. | ||

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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> | ||

where <math> \mathbf{x} = (x_1,x_2,x_3) </math>, as | where <math> \mathbf{x} = (x_1,x_2,x_3) </math>, as | ||

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<center> <math> \left(\nabla \phi(\mathbf{x}) \right)^2 = \frac{1}{V^2(\mathbf{x})} . </math> </center> | <center> <math> \left(\nabla \phi(\mathbf{x}) \right)^2 = \frac{1}{V^2(\mathbf{x})} . </math> </center> | ||

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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. | ||

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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 the gradient of traveltime (or ''slowness'') vector | 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 the gradient of traveltime (or ''slowness'') vector | ||

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

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<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> | <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> | ||

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

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## Latest revision as of 19:12, 8 April 2019

(ī kōn’ ∂l) (from Greek **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V }**
is compared to a reference velocity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_R }**
(analogous to comparing a velocity to the speed of light in vacuum):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left(\nabla \phi \right)^2 =\left(\frac{V}{V_R}\right)^2=n^2 }**,

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n}**
is an index of refraction and the quantity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V(\mathbf{x} ) }**
where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = (x_1,x_2,x_3) }**
, as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} = (p_1,p_2, p_3) }**
where the gradient of traveltime (or *slowness*) vector
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_i = \frac{\partial \phi}{\partial x_i} }**
for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i = 1, 2, 3 }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p^2 = \mathbf{p} \cdot \mathbf{p} = p_1^2 + p_2^2 + p_3^2 = \frac{1}{V(\mathbf{x})} }**

thus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = (x_1,x_2,x_3) }**
are the *generalized coordinates* and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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.