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Not all patterns are as perfect as the ones we have seen so far. Let s look at some other historical examples to help train your eyes to nd these patterns in your own analysis. Figure 10.8 shows an example of a somewhat rougher cup with handle. But even though there is a lot of noise in the chart, the rules we established early in the chapter still hold here: We have

(5.3.92)

(5.3.93)

i p(P1,152) Cp(P1,152)

drjCjE(G)

(5.3.94)

dpJp(151, 15:I)Gc(153)H(153 - 152)C p(P3, 152)

(5.3.95)



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The oldest covered bridge in the United States, built in 1812, is a Burr located near Lewisburg, Pennsylvania. Nearly 300 Burr arch-trusses are still standing, with 175 located in Pennsylvania, more than in any other state.

(5.3.96)

3.6 Low-Frequency Solution!;

FIGURE 10.8 The stock ACMR consolidated between $2 and $4.70 over a period of several years while creating this pattern.





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Hence the mass operator under QCA-CP is no.f dr/J j A system of nonlinear equations were developed for the case of dielectric spheres under QCACP [Tsang and Kong, 1980]. 3.6 Low-Frequency Solutions In this section the effective propagation constants under EFA, QCA, EFA-CP and QCA-CP will be calculated for spherical particles in the lowfrequency limit. Both the real part of the effective propagation constant J(T and the imaginary part of the effective propagation constant Ie will be studied. We consider cases where J(i J(,. with J( = J(T + iJ(i. In solving the integral equations of the various approximations, the leading term of the real part and the leading term of the imaginary part will be kept while the other terms can be safely discarded. The introduction of the coherent potential into the effective field approximation and the quasi-crystalline approximation results in a nonlinear equation for the mean Green's function. This is analogous to the relation between nonlinear approximation and the bilocal approximation of random medium, where it was found that the difference between the results of the two approximations is insignificant if the real part of the effective propagation constant J(,. is approximately equal to k m . For the case of random discrete scatterers, the permittivity of the particles can be appreciably different from that of the background medium. When the particles are densely distributed, the real part of the effective wavenumber J(" is appreciably different from k. Because of the difference between J(T and k, the introduction of any nonlinear approximation can significantly change the effective propagation constant. Thus the results of EFA-CP differ significantly from EFA, and the results of QCA--CP differ significantly from QCA.

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Effective Field Approximation (EFA)

The Lippmann-Schwinger equation for the transition operator is (5.3.97) In space representation, (rlU Ir') = U (r) 7 (rlr') where

(5.3.98)

and a is the radius of the particle of permittivity Es . To solve (5.3.97), a mixed representation (rITIP) == T m (r, 15) can be used. Applying to (5.3.97)

gives (5.3.99) Since the particles are spherical, the singularity of the dyadic Green's function will be separated out in the following manner (5.3.100)

1980 Of 893 historic covered bridges remaining in the United States, 231 are in Pennsylvania and 6 in North Carolina. 1989 Surviving Covered Bridges in Northampton County near Bethlehem, Pennsylvania include:

r = - r'.

where P S stands for principal value with a spherical exclusion volume at At low frequencies, p can be set equal to zero in (5.3.99). We further let T m(r)

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