Outside point source

Particular integrals associated with the two inhomogeneous terms in Eq. (2.5.14) are also required to complete the solution. Before a specific analytic solution can be found for the integral containing the Planck intensity, it is necessary to specify the explicit form of B(r). This is deferred to the appropriate sections where such 

The integral containing Fq can be dealt with immediately. Writing Eq. (2.5.14) without the Planck intensity term yields 

Summarizing this chapter, we have derived both analytic and numerical procedures for calculating the emerging radiation field provided we can specify the vertical distributions of the temperature as well as the gas and particle compositions. It is also necessary to know the absorption and scattering properties of atmospheric volume elements on a microscopic scale. In the next chapter we discuss these properties before proceeding with the task of computing the intensity of the outgoing radiation field. 

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Consider the two original individually homogeneous layers with thicknesses tq and n — To. Let an outside point source of intensity ttFo 8 pi — pio)8 4> — 4>o) irradiate the composite from above, and ignore all thermally emitted radiation. By analogy with Eqs. (2.1.43), (2.1.46), and (2.3.1), the intensity at tq in the direction pi, thermal emission, is given by... 

On the other hand, the solution to Eq. (2.2.3) for a thin layer in the presence of an outside point source of radiation is given by... 

Here m is the mass enclosed by the surface of integration, S. As follows from Equation (1.126), the field outside coincides with that caused by a point source with the same mass, M, located at the sphere center. This is a well-known result, which is hard to predict. In fact, this behavior occurs regardless of how close the observation point is to the sphere, and it results from the superposition of fields, caused by elementary masses. This is rather an exception, since in general the field differs from that generated by an elementary mass. 

Applying again the Gauss s formula and taking into account the spherical symmetry, we find that inside the shell, Rai, it behaves as a point source situated at the origin. Thus, we have... 

We see that a summation of fields of elementary masses outside of the shell produces the same result as a point source, placed at the center of the shell. This is the second example of such equivalence, and again it is an exception. Consider a spherical surface S with radius R — a + r[Pg.46]

This emissive power is assumed to be over the whole flame surface area, and is significantly less than the emissive powers that can be calculated from point source measurements. Increasing the pool diameter reduces the emissive power due to the increasing black smoke outside the flame and the resulting obscuration effect on the luminous flame. 

The point source model assumes that the fire can be represented as a point that is radiating to a target at a distance, R, from the point. The model is most appropriate for calculating incident heat fluxes to targets where fluxes are in the range from 0 to 5 kW/m (SFPE, 1999). The point source model has been shown to be accurate for calculating the incident heat flux from a jet flame to a target outside the flame (Beyler, 2002). The literature contains more refined line or cylinder models (Beyler, 2002 SINTEF, 1997). 

For operability, if the location of point sources exceeds several Member States or if several Member States or countries outside the EU are experiencing effects of the pollution, the emissions should be controlled through the authorisation process. 

To this point we have assumed that an atom, be it heavy or otherwise, scatters as a point source of scattering power fj having phase 0j. Although the detailed physical explanation is outside the scope of this book and involves quantum mechanical properties, it must be pointed out that this is not entirely true. An atom scatters X rays in a somewhat more complex fashion, in that its scattered radiation is composed of two components. The major component, which arises from normal Thompson scattering, and is by far the largest component, has phase 0 dependent on the atom s position as we have assumed. But there is also a minor component of the scattering that has phase 0 + jt/2. This is because the electrons of the atom also absorb a small amount of radiation due to electron resonance phenomena and re-emit it with a phase change. This second component is called the anomalous dispersion, and to be entirely correct, we should properly describe the radiation scattered by an atom as a complex number,... 

In a free space, the sound source can be considered as a point source (see Figure 13.1). In practical industrial applications, however, sound is either radiated from the source of definite size (e.g., loudspeaker membrane) or, more frequently, reflected from the point source by surfaces of different shapes such as horn, paraboloid, ellipsoid, etc. In both cases such sound radiation can be regarded as coming from a plane source. This results in a specific pattern of sound intensity in the zone near the sound source, the sound intensity is constant (Fresnel zone), whereas outside this zone (the Fraunhofer zone) the sound intensity decreases inversely with the square of the distance from the plane source, i.e., in the same way as for a point source (Figure 13.4). 

Again, the boundaries of the system are considered to lie outside of the diffusion zone. The solution is then obtained simply by superimposing the solutions of an infinite number of point sources lying in the region x < 0. The notation is simplified by introducing the so-called error function (erf) ... 

For a thin disc, approximate equations have been derived and tables of factors have been published which can be used to correct the activity of a distributed source to a point source equivalent, (e.g. Faires and Boswell, 1981 and Debertin and Hehner, 1988). For volume sources, the integrations are more comphcated and caimot be reduced to a simple expression for calculating a geometry correction factor. Abbas (2006) has reported a direct mathematical method of calculating sohd angle subtended by a well-type detector for point sources, circular disc and cylindrical sources. The procedure can cope with sources inside and outside of the well. Presumably, apphcation of the equations to a cylindrical, rather than a well-type detector, would be possible. 

Some of the instruments 1 classify as non-binding in Chap. 5 are taken to reflect customary international law. Amongst others, this is true for GA Resolution 1514 on the end of colonialism. While the content of this resolution may have been accepted as custom, the resolution itself remains outside the sources of art. 38 (1) ICJ-S. Therefore, the resolution is found in the category of non-binding instruments. It is no more than the point of departure one the main points along the way of this project is to argue that this strict categorization as 1 put forward in Chap. 5 is not adequate and needs to be reconsidered. 

Nearly all analyses of earthquake source mechanisms explicitly exclude net forces and torques. The equivalent forces given by Eq. 3, which arise from the imbalance between true physical stresses and those in the model, are consistent with these restrictions. The stress glut source region, so Gauss s theorem implies that the total force vanishes at each instant. 

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