Eikonal equation

The eikonal equation is a nonlinear first-order partial differential equation that arises in wave propagation problems, geometric optics, and the theory of excitable media. In its canonical form, it describes how wavefronts propagate normal to themselves with a speed that depends on position: |∇T(x)| = 1/c(x), where T(x) is the arrival time of the wavefront at position x and c(x) is the local propagation speed.

In the context of excitable media, the eikonal equation emerges as a singular limit of reaction-diffusion equations when the thickness of the wavefront becomes negligible compared to the radius of curvature. The equation predicts that the normal velocity of a curved wavefront equals the planar velocity minus a term proportional to the curvature — a relation that explains why expanding circular waves slow down and why spiral waves in cardiac tissue or chemical systems maintain stable rotation.

The eikonal equation is the optical approximation of excitable dynamics: it throws away the internal structure of the wavefront and keeps only its geometry. This is not an error but a strategy — the same strategy that allows us to trace light rays without solving Maxwell's equations.