# Dictionary:Eikonal equation

(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.