Dictionary:Eikonal equation/es

(ī kōn’ ∂l)(del griego $\iota \kappa o\nu$ (ikon) significando "imagen". Una ecuación derivada de la ecuación de onda a traves de la sustitución de una solución de prueba de onda harmonica dentro de la ecuación de onda. En una forma la ecuación eikonal es vista en la literatura de la fisica, la velocidad local $V$ es comparada a la velocidad de referencia $V_{R}$(analoga a comparar una velocidad a la rapidez de la luz en el vacio):

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

where $n$ is an index of refraction and the quantity $\phi$ 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 $V(\mathbf {x} )$ 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.

Another form of the eikonal equation is written in terms of the ray direction vector $\mathbf {p} =(p_{1},p_{2},p_{3})$ where the gradient of traveltime (or slowness) vector $p_{i}={\frac {\partial \phi }{\partial x_{i}}}$ for $i=1,2,3$

$p^{2}=\mathbf {p} \cdot \mathbf {p} =p_{1}^{2}+p_{2}^{2}+p_{3}^{2}={\frac {1}{V(\mathbf {x} )}}$

por lo tanto $\mathbf {x} =(x_{1},x_{2},x_{3})$ son las "coordenadas generalizadas" y $\mathbf {p} =(p_{1},p_{2},p_{3})$ son el "momento generalizado" de la mecanica Hamiltoniana, y la ecuación eikonal corresponde a la función Hamiltoniana o la ecuación Hamilton-Jacobi de la mecanica analitica.

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