Eikonal equation

The eikonal equation (from other Greek εἰκών - image) is a nonlinear partial differential equation that occurs in wave propagation problems when the wave equation is approximated using the semiclassical approximation . This equation is derived from Maxwell's equations , and connects wave optics with geometric optics .

Content

1 Formulation
2 Examples
3 References
4 Literature

Wording

The eikonal equation can be represented in the form:

|\nabla u(x)|=F(x),\ x\in \Omega

{\ displaystyle | \ nabla u (x) | = F (x), \ x \ in \ Omega}

$u|_{\partial \Omega }=0$ ${\ displaystyle u | _ {\ partial \ Omega} = 0}$ where

$\Omega$ ${\ displaystyle \ Omega}$ there is a subset in $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ . Here

$F(x)$ ${\ displaystyle F (x)}$ - a function with positive values associated with the speed of wave propagation in the medium.
$\nabla$ ${\ displaystyle \ nabla}$ - indicates the gradient ,
$|\dots |$ ${\ displaystyle | \ dots |}$ - Euclidean norm .

Examples

If $F\equiv 1$ ${\ displaystyle F \ equiv 1}$ then the distance function to $\partial \Omega$ ${\ displaystyle \ partial \ Omega}$ satisfies the eikonal equation.

Links

Short wavelength approximation. Eikonal equation

Literature

Born M., Wolf E. Fundamentals of Optics / Per. from English - M., 1973.