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The theory of multiple scattering can be treated from several viewpoints. In this chapter, we take the viewpoint of cascade of layers. The basic idea is to first use a thin layer that can only have single scattering. The precise nature of "thinness" is discussed. We then find the bistatic scattering results of such a thin layer. To simplify the problem, we use point scatterers. The scattering of a thin layer is important because a thick layer can be considered as a cascading of many thin layers together with single scattering occurring in each thin layer. We first study the propagation of coherent wave with this cascading approach. The results give the effective propagation constant that agrees with Foldy's approximation. Then we study the secondorder solution of a thin layer. The radiative transfer equations can then be developed from the point of view of a result of ca.'5cading of thin layers. The cascading approach illustrates, in a more intuitive way, the role of multiple scattering in a thick layer. It also gives results such as Foldy's approximation and radiative transfer theory that are usually derived from more formal multiple scattering theory. We also study the effects of clustering taking into account the effects of pair distribution function between clusters and within a cluster. When compared with independent scattering, the pair distribution between different clusters and the pair distribution function within a cluster can have opposite effects on the extinction.



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ascending trendline early in April 2000. It tried to recover, but once the trend was broken, the damage was done.

Consider a plane wave impinging upon a slab of point scatterers of thickness D. The background permittivities of regions 0, 1, and 2 are all equal to E which is not absorptive so that the boundaries at z = 0 and z = - Dare nonreflective (Fig. 3.1.1). The incident wave is





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The most appropriate stabilization for a tabby wall should be the same as that employed for any historic brick masonry wall suffering damage. Traditional repairs are the most direct, using similar technology to that available to the original builder. Walls function best when they are braced by oors and roofs. All masonry walls provide better service when they are part of an occupied building with an intact roof system. Methods for preserving tabby should be similar to those of any other masonry system, with repairs determined on a case-by-case basis. Protecting a tabby structure from moisture should be the primary goal.

01).

(3.1.1)

(3.1.2) (3.1.3) (3.1.4)

1jJ(s)(l)(r) =

L 1jJi )(l\r)

(3.1.5)

to:.. ::

:". to .:.:

FIGURE 7.4 Following the violation of its trendline, Yahoo collapsed, losing almost its entire value.

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Structural stabilization of tabby ruins is required if close inspection by the public is to be allowed. Over a long period of time, tall, unbraced walls and chimney structures tend to collapse due to high wind events. If subjected to a suf cient number of storms of suf cient magnitude, what remains are walls that are stable until deterioration reduces their capacity further. Bracing may be required if analysis indicates that a wall exceeds a certain maximum unbraced

8 ....... '.

.. . . . ,'..

_, = 0

0 ,

to':

. ' "

to:,;,

: .

FIGURE 7.5 Here is the SPY over a period close to a decade. Its ascending trendline was intact from 1994 through 2000, but once it was violated, the fall that followed was severe.

where 'lj)~S)(l) (r) is the first-order scattered field from the point scatterer located at T = Tt = PI + ZZt:

length, which has been calculated, or if calculations indicate that suf cient wind can be applied to overstress the free-standing wall. Because of the open nature of ruins, a free-standing wall is subject to wind forces from two directions normal to its face. Analysis is similar to a sign or solid fence, using building-code wind-load factors for such structures.

'!p )(l)(r) =

ikJr-r, 1_ _I eik,p:p,-ikiZz/

(3.1.6)

where kip kixx + kiyY, and f is an isotropic scattering amplitude. The plane wave representation of spherical waves is, from (2.1.14) of Volume I, 2.

___ = _ Ttl 27f ,

Ir -

__ _

(3.1.7)

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