somewhere near the beginning.

Eikonal equation

Filed under: Mathematics — Alex @ 8:06 pm 11/14/2006

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 u correspond to the wavefronts at different times; that is, the set \{x : u(x)=t} is the wavefront at time t. The derivation of this equation is straightforward: let \sigma(x) represent the differential path of the light ray emitted from point x at time t, 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 x outside \Omega, the solution u(x) to the eikonal equation measures the distance from x to \Omega. At least, I think it should :)

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