Huygens’ principle

P.D. Wilcox, R.S.C. Monkhouse, M.J.S. Lowe and P. Cawley, The use of Huygens principle to model the acoustic field from interdigital Lamb wave transducers, Review of Progress in Quantitative NDE, Vol 17, Plenum Press, New York, 1998 (in press). 

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The wave function v /ex (r) of electrons at the exit face of the object can be considered as a planar source of spherical waves according to the Huygens principle. The amplitude of diffracted wave in the direction given by the reciprocal vector g is given by the Fourier transformation of the object function, i.e. 

The distribution of light intensity in Figure 13 can be computed by application of Huygens principle which allows us to calculate the shape of a propagating wavefront provided the wavefront at an earlier instant is known. According to this principle, every point of a wavefront may be considered as a source of secondary waves (often called a wavelet) which spread out in all directions, i.e., all points on a wavefront are point sources for the production of spherical secondary wavelets. The new wave front 2 is then found by constructing a surface tangent to all the secondary wavelets as shown in... 

Figure 14. (a) Schematic of Huygens Principle showing construction of a new wave front 2 from the preceeding wave front S 0>) plane wave front incident on a slit of width b (c) diffraction for case where b < (d)... 

Comparison of Flame Propagation m a Channel and Cellular Flames If one considers the disturbances of finite amplitude at the infinite thin flame surface from the point of view of Huygens principle, then with time there appear angular points, which diminish the flame front surface, i.e.,... 

Here, cq is the speed of light in vacuum. According to Huygens principle, the rays are refracted as described by Snellius law of refraction ... 

The ray model of light is of limited usefulness. If we are to understand the fundamental processes involved in the formation of an image by a lens, we must consider the wave nature of light. The simplest form of the wave theory of light is based on a geometrical construction known as Huygens principle, which is usually stated as follows ... 

Even though Huygens principle makes no assumptions about the nature of the wave, the principle is extremely useful, especially when stated in a precise mathematical form (see Section 1.3). 

We can use Huygens principle to explain qualitatively the essential features of diffraction, but a quantitative treatment involves casting Huygens principle into a precise mathematical form, the Fresnel-Kirchhoff formula ... 

Multiple Scattering Theory was first formulated by Lord Rayleigh [17] in a paper published in 1892 dealing with the propagation of heat or electricity through an inhomogeneous media. However, for most people the application to optics known as Huygens principle [36], is probably better known. It states that ... 

As we have noted in the introduction, experimental evidence for a kinematic description of excitable wave fronts is rich. Based on hyperbolic wave equations and the Huygens principle, Wiener and Rosenblueth [81] recommend the eikonal approach of geometric optics waves propagate at a constant normal speed... 

The very first attempt to construct a simplified kinematical description of a rotating spiral wave has been done in the classical paper of N. Wiener and A. Rosenblueth [33]. This description is based on the assumption that wave fronts propagate in a uniform and isotropic medium with equal velocity from any stimulated points into a region where the medium is in the rest state. Due to Huygens principle, successive wave fronts are perpendicular to a system of rays which represent the position which may be assumed by stretched cords starting from the stimulated point. The back of the wave is another curve of the same form, which follows the wave front at a fixed distance Ag measured along these rays. 

Vernadsky often writes that life is a planetary and a cosmic phenomenon. Sometimes he uses these terms synonymously . .Considering life phenomena as living matter shows that it is a planetary, i.e. cosmic phenomenon [Huygens principle]" (Vernadsky, 1965, p. 227). Sometimes the term planetary phenomenon is used to stress that some properties of the biosphere are defined by the properties of the Earth as a planet, for example, by the form and shape of the geoid. 

Consider a plane wave whose direction of propagation is perpendicular to a wall containing two small holes (Fig, 3,4). According to Huygens principle, these holes subsequently become sources of spherical waves which then interfere. At a distance which is large with respect to A and to d, the spherical wave in the direction of observation s can be treated as a plane wave (Fraunhofer s approximation). The path difference A between the two waves and 2 the direction s is the projection of d onto s ... 

They allow us to illustrate the refraction of light using Huygens principle (Fig. 4.26). 

Fig. 4.26. Refraction according to Huygens principle the wave fronts and the light rays in a uniaxial crystal...

A strange pattern appears on the screen a number of high concentrations of traces are separated by regions of low concentration. This resembles the interference of waves e.g., a stone thrown into water causes interference behind two slits an alternation of high and low amplitudes of water level. Well, but what does an electron have in common with a wave on the water surface The interference on water was possible because there were two sources of waves (the Huygens principle) — that is, two slits. 

Huygens construction (Huygens principle) Every point on a wavefront may itself be regarded as a source of secondary waves. Thus, If the position of a wavefront at any instant is known, a simple constmction enables its position to be drawn at any subsequent time. The constmction was first used by Christiaan Huygens. 

In some physical circumstances, we can derive familiar elements of the less fundamental theory from the more fundamental by straightforward limiting inferences. Thus if ordinary cases of refraction and reflection are what is being considered, we may simply derive a geometrical law of reflection or a Snell law of refraction from wave optics and Huygens principle, with, perhaps, applications of the method of going to the limit where wavelengths are small relative to macroscopic relevant features of the experimental setup. 

Secondly, from the Huygens principle Eq. (4.46) there is abstracted also the limiting delta Dirac-flmction for a propagator connecting two space events simultaneously ... 

Generally, the amplitude of a wave scattered by material point O and measured at any distant point P (R) corresponds to the Huygens principle ... 

According to the above equations every point on a wavefront serves as the source of spherical wavelets. The field amplitude at any point is the superposition of the complex amplitudes of all these wavelets. The representation of a field as a superposition of many elementary wavelets is known as the Huygens principle, since it was formulated in his Traits de la Lumi6re in 1690. Eresnel completed the description of this principle with the addition of the concept of interference. 

The application of these results to a larger aperture is found from Huygens principle. This states that each point on a wavefront sends out secondary waves that can be combined to form a new wavefront. Thus by knowing the distribution of amplitude and phase of the fields across the aperture under examination,... 

Electron diffraction is a collective elastic scattering phenomenon in which the electrons are scattered by atoms in a regular array or a crystal. Secondary waves are generated when an incoming plane electron wave interact with the atoms. Analogous to Huygens principle for diffraction of light waves, the secondary waves interfere with each other, and diffraction patterns thus can be obtained. 

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