Huygens–Fresnel principle

Huygens' principle and Huygens principle, also called huygens - Fresnel principle (after Augustin Jean Fresnel and Christiaan Huygens ), states that each point of a wavefront as a starting point for a new wave of so-called elementary wave can be considered. The new position of the wavefront is obtained by superposition ( superposition ) of all elementary waves. Since the elementary wave has a spherical shape or circular shape, also a returning wave forms. From the Huygens principle many special cases, such as diffraction phenomena in the far field (Fraunhofer diffraction ) or Nahfeldbeugung follow ( Fresnel diffraction ).

Huygens principle in physics

The further propagating wave front is obtained as the outer envelope of the wavelets, as the elementary waves in the same medium and at the same rate spread as the original wave. In different media, the propagation velocity and thus the direction of wave propagation, which is manifested as refractive changes. At the edge of an obstacle (eg gap) Huygens' principle leads to diffraction.

Huygens was able to explain with his theory of the diffraction and refraction of light, what with the corpuscular theory of Newton, however, was possible. Only when the speed of light could be determined in two different media, it was possible to decide which of the two theories applicable. As a medium of propagation of the light waves Huygens postulated the airwaves. This is no longer needed as a physical concept since the general acceptance of the theory of special relativity published in 1905 by Albert Einstein.

Huygens principle in mathematics

In mathematics, the Huygens principle is applied in the theory of partial differential equations. It states that wave equations have a rear wavefront in the room for. One speaks of the existence of a rear wavefront when a disturbance of the output data in a neighborhood of a point does not affect the solution of the wave equation for sufficiently large times t.

Statement by the Huygens principle to the simple wave equation

As initial data ( t = 0) the following applies:

With time as a variable and as a local variable.

The case n = 1

According to d'Alembert's solution formula is valid for u = u ( x, t):

We disturb the start date in the interval [a, b], then it can be seen from the above formula that for the point the fault at the time has no influence, because the initial data and were not disturbed. → To apply the Huygens principle.

Be and to disturb the start date in [a, b]. Then it will be seen that for any time t, the fault is still affects the solutions u (x, T), for integrating over the " Störintervall "

Conclusion: In the one-dimensional Huygens principle generally does not apply, but it only applies to the start date.

The case n = 2

The general solution formula for the two-dimensional case (after the descent method) is:

B (x, t) referred to the (filled ) circular disc with a center radius and x t.

Using this formula, one sees immediately that the Huygens principle does not apply. For disturbs to the initial data or in a rectangle R = [ A, B ] includes x [ c, d] be the interference effect from even to each point in time t = T for all the points, for the disc B (x, t) for these points the rectangle R. So is again integrated over disturbed data.

The case n = 3

According to Kirchhoff's formula is the solution for the wave equation:

S ( x, t ) is the spherical surface of the ball with the center and radius of x t. denotes the surface element of the sphere.

Using this formula one can easily see that in the 3 -d case applies the Huygens principle. If the initial data or on a cube Q = [a, b] × [c, d] × [e, f] disturbed, this disturbance does not affect the solution for the points x0 ∈ Q for large t> T. You just have to choose t so large that the spherical surface completely surrounds the square, and is thus no longer integrated over the perturbed data Q. Obviously must

. apply