the eikonal equation
/ eye-KON-al /
Drop a pebble in a pond and watch the ring of ripples expand. Or imagine a grass fire spreading outward from a match, or light fanning out from a bulb. In each case a front moves outward, and there is a natural question: at each point, how long until the front arrives? That arrival-time function, and the fronts as its level sets, is governed by the eikonal equation — the master equation of front propagation and the short-wavelength limit of all wave phenomena.
The equation is |grad u| = 1, or more generally |grad u(x)| = 1/c(x), where u(x) is the arrival time of the front at the point x and c(x) is the local speed of propagation. Read it geometrically: the gradient of u points in the direction the front travels, and its magnitude being 1/c says the front advances at speed c. The level sets {u = constant} are the successive positions of the front (the wavefronts), and the curves orthogonal to them — along which u increases fastest — are the rays. So solving the eikonal equation is exactly finding travel times and ray paths. The clean solution of |grad u| = 1 with u = 0 on a set S is u(x) = distance from x to S: the arrival time is just the (speed-weighted) shortest distance, and you build it by sending out characteristics (rays) straight from S. This is a nonlinear, fully nonlinear first-order PDE, and it is the canonical example for HUYGENS-style front construction and for viscosity solutions.
Why does it matter? The eikonal equation is the bridge from wave optics to ray optics — the high-frequency limit of the wave or Helmholtz equation, the reason light 'travels in straight rays' and how lenses are designed. It governs seismic travel-time computation, robot path planning, and the fast 'distance transform' used in image processing (fast marching and fast sweeping methods solve it efficiently). The honest subtlety, and a recurring theme of this whole field: smooth solutions break down where rays cross or fronts collide (think of the sharp ridge where two ripples meet, or a shock in the arrival time), so the right global solution is the VISCOSITY solution, which selects the correct first-arrival time and the physically meaningful kinks.
Solve |grad u| = 1 with u = 0 on a single point (the source). The solution is u(x) = |x|, the distance to the source, whose level sets are expanding circles (the wavefronts) and whose rays are the radial lines straight out from the point — exactly the picture of ripples spreading from a dropped pebble. Where wavefronts from two sources meet, u has a ridge, and the viscosity solution keeps the earlier arrival.
Arrival-time of a front: its level sets are the wavefronts, gradients are rays.
Classical (smooth) solutions of the eikonal equation generically fail where rays cross or fronts collide — the arrival-time function develops kinks. The correct global solution is the viscosity solution, which picks the genuine first-arrival time; an everywhere-smooth solution usually does not exist.