Difference between revisions of "Dictionary:Eikonal equation"

From SEG Wiki
Jump to: navigation, search
Line 1: Line 1:
 
{{lowercase}}{{#category_index:E|eikonal equation}}
 
{{lowercase}}{{#category_index:E|eikonal equation}}
(&#x012B; k&#x014D;n&#x2019; &#x2202;l) A form of the wave equation for harmonic waves 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):  
+
(&#x012B; k&#x014D;n&#x2019; &#x2202;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):  
  
  
Line 6: Line 6:
  
  
where ''n'' is an index of refraction and [[File:Fgr.gif]] is the wave function. Valid only where the variation of properties is small within a wavelength, sometimes called the &#x2018;&#x2018;high-frequency condition.&#x2019;&#x2019;
+
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 &#x2018;&#x2018;high-frequency condition.&#x2019;&#x2019;
  
 
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>
Line 14: Line 15:
  
 
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 12:48, 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 Fgr.gif 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.

External links

find literature about
Eikonal equation
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png