Translations:The eikonal equation - book/15/en

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Suppose now that a ray points away from the wavefront. The wave wants to take the least time to travel to the new wavefront. By isotropy, the wave’s velocity is the same in all directions. Because the traveltime is velocity multiplied by distance, the wave wants to take the raypath that goes the shortest distance. The shortest distance is along the path that has no component along the wavefront; that is, the shortest distance is along the normal to the wavefront. In other words, the raypath must be orthogonal (i.e., at right angles) to the wavefront. Thus, the ray’s unit tangent vector u must be orthogonal to the wavefront. By definition, the gradient is a vector that points in the direction orthogonal to the wavefront. Thus the ray’s unit tangent vector u and the gradient grad t of the wavefront must point in the same direction.