ChapterZero

Eikonal equation

The eikonal equation is a non-linear PDE related to the least action principle and the Hamilton-Jacobi equations of mechanics:

$|\nabla u(x)| = \frac{1}{F(x)}.$

One application of the equation is describing the propagation of a wavefront through a medium where the local speed is a function of location F(x) . In this case, the level sets of correspond to the wavefronts at different times; that is, the set $\{x : u(x)=t}$ is the wavefront at time . The derivation of this equation is straightforward: let $\sigma(x)$ represent the differential path of the light ray emitted from point at time , which reaches the new wavefront at time t + dt . This path is perpendicular to the wavefront it starts from, and has length |F(x) dt| , so

$dt = \left|\frac{du}{d\sigma}(x)\right| = \left| \langle \nabla u(x) , \sigma \rangle\right| = |\nabla u(x)|\cdot |\sigma(x)| = |\nabla u(x)| \cdot |F(x)| \cdot dt \Rightarrow |\nabla u(x)| = \frac{1}{|F(x)|}$

yields the eikonal equation.

Note that if F=1 identically and u=0 on the boundary of a region $\Omega$ , then for outside $\Omega$ , the solution u(x) to the eikonal equation measures the distance from to $\Omega$ . At least, I think it should

Possibly relevant posts:

Exactness of differential forms (7/28/2007)
Notes from the underbelly (11/26/2007)
Differential of the determinant (6/29/2007)

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Eikonal equation

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