# Difference between revisions of "Dictionary:Eikonal equation"

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(ī kōn’ ∂l) A form of the wave equation for harmonic waves in which the local velocity $V$ is compared to a reference velocity $V_{R}$ (analogous to comparing a velocity to the speed of light in vacuum):

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

where n is an index of refraction and is the wave function. 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 $V(\mathbf {x} )$ where $\mathbf {x} =(x_{1},x_{2},x_{3})$ , as

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