Kirchhoff approximation

We can represent the total field as a superposition of the incident and scattered fields above the refiecting boundary B, while we identify it with the transmitted wave, p r,Lj), below the reflector  

The incident and scattered fields satisfy the following equations everywhere in the upper half-space above the reflector  

We assume also that all wavefields - the total, the incident, the scattered, and the transmitted fields - satisfy the Sommerfeld radiation conditions, which are 

Note that in most cases we prefer to use the normal direction pointing upwards from the surface of integration, n = —riB- By changing the direction of the normal in expression (14.62), we arrive at the following integral representation for the scattered field  

We may recall now that the surface B is a reflecting surface. This means that the incident field experiences a reflection at this boundary, which can be characterized approximately by the reflection coefficient A providing the relationship between the leading order (high frequency asymptotics) incident and scattered fields (Bleistein et ah, 2001)  

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The Kirchhoff approximation is based on expressing the scattered field p (r, oj) at some point r in the upper half-space, using its values at the reflecting boundary (see Figure 14-2). To solve this problem we apply the Kirchhoff integral formula (13.197) to the scattered field in the upper half-space ... 

Muinonen, K., 2003 Light scattering by tetrahedral particles in the kirchhoff approximation. In T. Wriedt, ed., Electromagnetic and Light Scattering — Theory and Applications VII, pp. 251-254. 

The GTD formalism can be applied conveniently to the calculation of the field diffracted by a slit of width 2a and infinity length. For simplicity, we assume a plane incident wave normal to the edges. As a first approximation we take the field on the aperture coincident with the incident field (Kirchhoff approximation). Then, following J. B. Keller, we can say that the field point P at a finite distance is reached by two different rays departing from the two edges and by a geometrical optics ray, if any (see Fig. 1 la). The contribution of the diffracted rays can be expressed in the form... 

A point that deserves comment is the use of the Kirchhoff approximation, which can be considered valid when the slit width is much larger than the field wavelength. This approximation can be improved, as shown by Keller, by taking into account the multiple diffraction undergone by the rays departing from each and diffracted from the opposite one (see dashed line in Fig. 1 lb). 

The exact values of E and 5E / 5n are in general unknown and the Kirchhoff or physical optics method consists in approximating the values of these two quantities on the surface and then evaluating the Helmholtz integral. We shall approximate the field at any point of the surface by the field that would be present on a tangent plane at the point. With this approximation, the field on the surface and its normal derivative are... 

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

In plate theory, the problem is reduced from the deformation of a solid body to the deformation of a surface by use of the Kirchhoff hypothesis (normals to the undeformed middle surface remain straight and normal after deformation, etc., as discussed in Chapter 4). Then, we attempt to apply boundary conditions to that surface which is usually the middle surface of the plate. There should be no surprise that the boundary conditions for the unapproximated solid body are not the same as those for the solid approximated with a surface. The problem arises when these boundary conditions are applied to an approximate set of equilibrium equations that result when force-strain and moment-curvature... 

This fourth-order partial differential equation can have only two boundary conditions on each edge for a total of eight boundary conditions. Thus, some step in the approximations leading to Equation (D.27) must limit the boundary conditions from those displayed in Equations (D.23) and (D.24) because there three boundary conditions occur for each edge for a total of twelve boundary conditions. This dilemma has been resolved historically by Kirchhoff who proved that the boundary conditions consistent with the approximate differential equation. Equation (D.27), are... 

The simplesf mefhod of solution of fhe Kirchhoff equations that correspond to the random network of conducfance elemenfs in three dimensions is in the single-bond effective medium approximation (SB-EMA), wherein a single effective bond between two pores is considered in an effective medium of surrounding bonds. The conductivify (7b, of fhe effective bond is obtained from the self-consistent solution of fhe equation ... 

It is a remarkable fact that the contemporary history of absorption and emission spectroscopy began simultaneously, from the simultaneous discoveries that Bunsen and Kirchhoff made in the middle of the 19th century. They observed atomic emission and absorption lines whose wavelengths exactly coincided. Stokes and Kirchhoff applied this discovery to the explanation of the Fraunhofer spectra. Nearly at the same time approximately 150 years ago, Stokes explained the conversion of absorbed ultraviolet light into emitted blue light and introduced the term fluorescence. Apparently, the discovery of the Stokes shift marked the birth of luminescence as a science. 

An alternative polymerization mechanism and polymer architecture has been proposed by Kirchhoff [1, 2, 3], Tan and Arnold [77], By this mechanism, polybenzocyclobutenes which do not contain reactive sites of unsaturation are proposed to polymerize by the 1,4 addition of the o-quinodimethane intermediates to give a substantially linear poly(o-xylylene) structure. Since the monomers all contain at least two benzocyclobutene units the net result of this reaction will to a first approximation be a ladder type polymer as shown in Fig. 17. The formation of a true ladder polymer however would require that all... 

When A Ha°SS0C), and A5a°SS0C)(. are dependent on temperature, plots of In k t versus 1/T do not follow linear dependencies. According to Kirchhoff s law, when temperature-dependent heat capacity conditions prevail, i.e., when A Cp i = 0, as observed for example with the heterothermic binding scenarios,29,30,39,62 256 258 respectively, then the dependency of In k) on T can be approximated by a polynomial expression as represented by... 

The Planck-Kirchhoff law allows a good approximation of the spectral radiance of any thermal radiator, the sources as well as the samples and detectors. Thermal radiators are characterized by a definite temperature as well as by their absorption coefficients f(i>) or a(i>), which describe the characteristic spectrum of the radiator ... 

In principle, the diffraction patterns can be quantitatively understood within the Fraunhofer approximation of Kirchhoff s diffraction theory as described in any optics textbook (e.g., [Hecht 1994]). However, Fraunhofer s optical diffraction theory misses an important point of our experiments with matter waves and material gratings the attractive interaction between the molecule and the wall results in an additional phase of the molecular wavefunction [Grisenti 1999], Although the details of the calculations are somewhat involved2, the qualitative effect of this attractive force on far-field diffraction can be understood as a narrowing of the real slit width to an effective slit width [Briihl 2002], For our fullerene molecules the reduction can be as big as 20 nm for the unselected molecular beam and almost 30 nm for the slower, velocity selected beam. The stronger effect on slower molecules is due to the longer and therefore more influential interaction between the molecules and the wall. 

Property Approximations. Because of the lack of accurate radiative properties in many situations, it is common practice to invoke certain approximations for the property behavior. The most common assumptions are that the surface properties are independent of wavelength (a gray surface), independent of direction (a diffuse surface), that the surface behaves as an ideal mirror (a specular surface), or that the surface is black. The assumption of a gray-diffuse surface is the most commonly invoked. For a surface that is truly both gray and diffuse, Kirchhoff s law applies for all of the property sets that is, a = e, and the computation of radia-... 

Electromagnetic radiation in thermal equilibrium within a cavity is often approximately referred to as the black-body radiation. A classical black hole is an ideal black body. Our own star, the Sun, is pretty black A perfect black body absorbs all radiation that falls onto it. By Kirchhoff s law, which states that a body must emit at the same rate as it absorbs radiation if equilibrium is to be maintained , the emissivity of a black body is highest. As shown below, the use of classical statistical mechanics leads to an infinite emissivity from a black body. Planck quantized the standing wave modes of the electromagnetic radiation within a black-body cavity and solved this anomaly. He considered the distribution of energy U among A oscillators of frequency... 

The simplest approach to solve the Kirchhoff equations that correspond to the random network of conductance elements is obtained in single-bond effective medium approximation (SB-EMA), wherein a single effective bond... 

We must take account of the temperature dependence of in first order, via the Kirchhoff relation, dLf/dT)p = Cp)i — (C )a = Cp, where A.Cp represents the difference in molar heat capacity of pure liquid A and pure solid A. If the temperature dependence of this difference is negligible we obtain in first approximation the linear dependence... 

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