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we can replace K z in the argument of hand e by K~. In deriving the dense media radiative transfer equation, we distinguish two distance scales. The long-distance scale is characterized by the mean free path lmfp, with lmfp = 1/(2K"), K" = Im(K). The short distance scale is characterized by the wavelength scale Ae = 21T / K' with K' = Re(K). Transport-type equations are derived based on the assumption that lmfp Ae, lmfp a Sj ' That is, the mean free path is much larger than the wavelength and the particle size. The condition is generally met for many problems involving multiple scattering except in the case of strong localization. Thus the envelope amplitude " u and d in (7.3.11) and (7.3.12) are varying on the lmfp scale. For z and z' separated on the Ae scale, we assume the trial solution

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(7.3.13a)

FIGURE 7-19 All exterior surfaces were renewed, including the steeple. (See color insert.)

where m, n = u, d. The assumption of (7.3.13a) is that incoherent fields with different transverse wave vectors kJ.. of propagation are uncorrelated. For the same transverse wave vector, there is nonzero correlation. The upward and downward propagation waves with the same transverse kJ.. wave vector are correlated because of the effective reflecting interface at z = O. This correlation is represented by A . Using (7.3.13a) and (7.3.10), we have

FIGURE 3.13 You can have easy access to the chart of any major U.S. index with the Indices module. FIGURE 3.14 There are nearly 200

( (1') *(1"))

dkJ..eikl-.(rl--r~) {A1!lL(z,z"kJ..)eiK~(Z-Z') +Add(z,z',kJ..)e-iK~(Z-Z')

(z, z', kJ..)e-iK~(Z+Z')}

(7.3.13b)

For l' i= 1", the mean Green's function of (7.2.2), (7.2.3), and (7.2.18) can be written in the form

(G(1',1")) =

dkJ..e ikdr l--r'l) {9p (kJ..' z, z')

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ot only did Charleston s 1841 Market Hall survive war, tornados, hurricanes, re, and the 1886 earthquake, but also it suffered underutilization and neglect forced by Reconstruction and the Depression that hit South Carolina s economy especially hard. During the bombardment of the city from Morris Island by federal forces, an artillery shell burst through the roof, grazing the bottom chord of one kingpost roof truss. Damage by historic events such as the Civil War and the 1886 earthquake, combined with the low country s harsh climate of sun, rain, and humidity, were abundantly evident when Phillips & Oppermann, PA obtained the architectural design contract for its restoration in 1992. This chapter will relate the design team s struggle to apply conservation principles by integrating high-tech solutions of analysis and materials testing with simple traditional approaches of timber framing and masonry restoration. Once the strategy for structural rehabilitation was developed, it had to be defended on several occasions prior to the start of construction and through a construction period that lasted more than three years. Major items of discussion were shoring and bracing, masonry rehabilitation, and timber framing. Although Charleston, being a port city, is hardly isolated, much of the expertise for the rehabilitation was obtained at a distance from the project. The Winston-Salem based architect Joseph K. Oppermann assembled a team consisting of a structural engineer, materials conservator, and a mechanical engineer from such diverse locations as Raleigh, North Carolina; Philadelphia, Pennsylvania; and East Norwalk, Connecticut.

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+ 9R (kJ..' z, z')}

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for z > z' for z < z'

(7.3.14)

(7.3.15) (7.3.16)

[e (Kz) e(Kz) + h (Kz) h(K z)]

(7.3.17) (7.3.18)

a< (Kz) = 87[;K z [e (- K z) e(- K z) + h(- K z) h (- K z) ]

[R e (k-L) e(-K z ) e (K z )

Structural Investigation of Historic Buildings: A Case Study Guide to Preservation Technology for Buildings, Bridges, Towers, and Mills. David. C. Fischetti 2009 John Wiley & Sons, Inc.

(7.3.19)

+ R h (7~-l) h(-K z ) h(K z )]

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In (7.3.8b) the integrand includes the correlation of the total field which is a sum of the correlation of the incoherent field. From (7.2.26) on the coherent field, we have the correlation of the coherent field (7.3.20) with

~(KiZ) = [EihTihh(-KiZ) + Eie Tie e (-KiZ )]

. [E,j,Ti'hh (- K iz ) + E:eTi:e (- K iZ )] In (7.3.20), K:~ = Im(Kiz ). Thus (7.3.22) Note that we have assumed the wave dependence of ( (rI) *(rI)) in (7.3.13b). Substituting (7.3.20), (7.3.22), (7.3.14), and (7.3.13) into (7.3.8b), the objective is to balance the wave dependences based on the phase dependence of the various terms so that equations can be obtained for the =mn covariance A of the incoherent field. Without loss generality, let z > z' in (7.3.8). Then (7.3.21)

A (z, z', k-l) eiJ<~(z-z') + Add (z, z', k-l) e-iJ<~(z-z')

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