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the backscattering direction. The overall angular width of the peak in the backscattering direction is of the order of Kelk = 3.18 X 10- 3 = 0.18 . In Fig. 8.3.4 the results are illustrated for a smaller w = 0.4. Multiple scattering plays a lesser role. Compared to the case w= 0.95, the backscattering peak here is wider. The second-order solution is also closer to the full multiple scattering solution than that of w= 0.95. In Fig. 8.3.5 we plot the bistatic scattering coefficients with Td = 4 at W = 0.85. The case of Td = 4 in Fig. 8.3.5 approximates well the half-space solution. Comparison of the results in the two figures indicates that larger optical thickness gives a larger bistatic coefficient and a sharper backscattering peak.



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In previous sections we studied the backscattering enhancement of isotropic scatterers. The approach was based on the second-order theory and also on a summation of all the ladder and cyclical terms. For isotropic scatterers the angular width of backscattering enhancement is of the order of the coherent





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wave-attenuation rate divided by the wavenumber. In this section we study the backscattering enhancement of waves by scatterers that do not scatterer isotropically. Nonisotropic scattering is characteristic of particles with sizes comparable with or larger than a wavelength. By restricting the observation angle to the vicinity of the backscattering direction, it is shown that both the ladder- and cyclical-term summations can be related to the unidirectional point-source Green's function of the transport equation. The Green's function is then solved by two methods. For the case of a small albedo or a small optical thickness, the second-order theory is applied. The angular width of enhancement is of the order of the coherent wave-attenuation rate divided by the wavenumber. For the case of a large albedo and a large optical thickness, diffusion approximation is used to calculate the unidirectional point-source Green's function. The angular width in this case is of the order of the transport rate divided by the wavenumber. The transport rate is equal to the product of the extinction rate and one minus the mean cosine of the scattering angle Ji. For strong forward scattering, the mean cosine Ii is close to unity. Hence the angular width is substantially smaller for particles with strong forward scattering, a result that is shown to be in good agreement with experimental observations.

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Consider a plane wave incident upon a slab of discrete scatterers lying between z = 0 and z = -d (Fig. 8.4.1). The incident wave is in the direction kid, and the scattered wave is measured in the direction ks . The subscript d is used to denote downward motion. We assume that the particles are uncorrelated in positions. Let

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In August 1994, with Mr. B.J. Pinckney of the church, George Fore and I opened the sidewall balcony framing for observation by removing some boards. We were able to observe the condition of the bearing seat at the upper end of the balcony framing beams. The principal framing of the balconies consisted of sloped 8-inch by 11-inch timbers that ran from the interior columns to the exterior wall where they rested on a 10 - by 10 -inch sill plate supported and enveloped by the exterior wall. The exterior sidewalls had deformed outward over time. The maximum movement was at the balcony sill beam level. George Fore s report showed that the maximum horizontal deformation of the wall was 2 5 8 inches on the north side and 3 9 16 inches on the south side. A horizontal crack had formed that ran along the full length of the sidewall, except at the ends where the masonry was tied back into the corners of the building. The great concern with regard to the adequacy of the balconies was based on the deformation shown in George Fore s report. Two disturbing possibilities were put forward by others regarding this movement. First, the underlying soils might be overstressed, allowing the sidewalls to settle and rotate about the toe of the footing. This appeared to be a possibility because of the settlement observed in the oor joists and the door openings in endwalls, which had deformed into a

(8.4.1)

(8.4.2)

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