Huygens' Principle

The Huygens' Principle,named after Christiaan Huygens, is an interesting propertry of the solutions of the wave equation:
You define the influence area of a point x at time t in Rⁿ as the set
I(x,t) = { y in Rⁿ | u(y,t) is not constant if the inital conditions are canged at x }. That's the set of points which depend on x at time t. If you solve the wave equation you would notice that for n = 1 or n even I(x,t) is a closed ball. But for n > 1 and n odd, it's a sphere.
This fact makes communication in the real world much easier: The signals (light or sound, anything transmitted by waves in space) reaching you from a point x at time t are the only signals you receive from that point. That's because we live in a world with 3 space dimensions (odd, >1 !). With 4 space dimensions this would be problematic: all signals from the past would interfere with new signals and after some time it might be troublesome to spot the new information in this mess.

Huygens' Principle, simply stated, is the following:

Every point on a wave can be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself. The new wave front is the envelop of all the wavelets -- that is, the tangent to all of them. (Douglas A. Giancoli, Physics)

Huygen's principle, developed by Dutch physicist Christiaan Huygens (1629-1695) is used to determine the future position of a wave front when its past position is known.

For example, consider a wave front from A to B, traveling away from point S at velocity v. This wave is assumed to be isotropic, meaning that the wave is moving away from S at the same speed in all directions. To find the position of the wave after t seconds, we can draw tiny circles, or wavelets from every point (in reality, as many as possible) with radius vt. The tangent to these wavelets is the position of the wave after t seconds. Huygen's principle is very important for determining the position of a given wave, especially after the wave diffracts after hitting an obstacle.