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In this section we consider a plane electromagnetic wave incident onto a half-space of identical dielectric spherical scatterers. The simpler case of normal incidence is first treated (Fig. 6.1.1). The quasi-crystalline approximation and T-matrix formalism will be used. The case of point dipoles were treated in Born and Wolf [1975]. It was shown that the original incident wave was extinguished giving the Ewald-Oseen extinction theorem. A transmitted wave is obtained that has an effective propagation constant governed by Lorentz-Lorenz law. For our case here, finite size spheres are used. Two sets of equations are obtained, namely, the generalized Ewald-Oseen extinction theorem and the generalized Lorentz-Lorenz law. The generalized LorentzLorenz law is a set of homogeneous equations. Setting the determinant of the coefficient matrix equal to zero gives the dispersion relation for the coherent effective propagation constant K. For this new case, unlike the point dipoles treated in Born and Wolf [1975]' K has an imaginary part that accounts for the decay due to scattering.



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Consider a plane wave impinging upon a half-space of N identical dielectric spherical scatterers. The incident field is

(6.1.1)

(6.1.2)

ki = k(sin ()i cos cPd; + sin ()i sin cPi'Y + cos ()iZ)

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Einc(r-) = " L [(AI) a rnn Rg -Mrnn(kT) (N) + amn Rg -N rnn(kT) ]

(6.1.3)

(6.1.4)





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where m, n go through the vector spherical wave coefficients n = 1, 2, ... , N max , m = 0, 1, 2, ...' , n. Then the field exciting the lth scatterer is

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American Association of State Highway and Transportation Of cials (AASHTO), Standard Speci cations for Highway Bridges, 15th ed. (Washington, D.C.: AASHTO, 1995). American Concrete Institute (ACI), Building Code Requirements for Reinforced Concrete (Detroit: ACI, 1956 and 1965). American Institute of Timber Construction (AITC), Timber Construction Manual, 3rd ed. (New York: John Wiley & Sons, 1985). American Society of Civil Engineers, American Wooden Bridges, ASCE Historical Publication no. 4, 1976. ASCE Standard: Minimum Design Loads for Buildings and Other Structures (ASCE 7-95). Public Ballot Draft Copy. ASCE Washington, D.C. Conrad P Roberge and Glenn R. Bell, Regulated Structural Peer Review, Civil Engineering Practice (Fall/Winter 1994). Gerry Gilliam, Brazil, Movie by Embassy International Pictures, starring Jonathan Pryce, Robert De Niro, Catherine Helmond, Terry Gilliam, MCA produced. 1985. Herman Haupt, Reminiscences of General Herman Haupt (July 1901). Wright, John W. ed. The Universal Almanac. Kansas City: Andrews & McMeel Publishing, 1995. pp. 566 567. National Forest Products Association (NFPA), National Design Speci cation for Wood Construction (Washington, D.C.: NFPA, 1991), pp. 103 104, and 125. National Forest Products Association, National Design Speci cations for Wood Construction, (Washington, D.C.: National Forest Products Association, 1991). Squire Whipple, A Work on Bridge Building: Consisting of Two Essays (Utica, NY: H. H. Curtiss, Printer, Devereux Block, 1847).

[w~!,)(l) Rg Mrnn(kTr-l) + w~~(l} Rg Nrnn(kTr- I )]

(6.1.5)

-(I)

(6.1.6)

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as the column matrix containing the exciting field coefficients of particle i. In 10 of Volume II, the Foldy-Lax multiple scattering equations in matrix form were derived. Using the same notations,

.;-. -

=(j)

( .)

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(6.1.7)

Equation (6.1.7) can be interpreted by saying that the field exciting the lth scatterer is the sum of incident wave and scattered wave from all scatterers j except j = l. The matrix a(kqrj) represents vector translation formula translating the center at 'Fj to spherical waves centered at 'Fl. The matrix elements of a are given in 10 of Volume II. In index notation, we have

2: [w~~)(l)

Rg Mmn(krq)

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