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(Epi - 1) sk2- - '" = . Pj --; Ei L Aij
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(EPi) sk 2 - - 1
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(Epi) sk 2L - - 1 j=1
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3.3 Other Shapes
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Multiply by ~V(Epi - Ei), and noting that
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1- C~ - 1) sk
3E~V
~V(Epi
- E)
(2.3.28)
Putting (2.3.23) in (2.3.28) gives
--------'--"'--~~,_____-__;;_~
C:i - 1)
+2 -
2 (~)(~~ ~ - 1 -2o:C
.~~) + ~-3-
2 o:C [k a ik a ] 1---~- - - + - -
(2.3.29)
Using a
= (3/47f)1/3d and
= d3 in (2.3.29) gives
----;=-_ _---'t'-----
_____=_
i2k 1- -47f-E-d- [(47f) 1/3 k 2d 2 + - 3d- ] -3 3
(2.3.30)
The term with imaginary part in the denominator of (2.3.30) is known as radiative correction, which arises for the same reasoning as when scattering by Rayleigh spheres was discussed in 2, Section 8.2 of Volume 1.
Other Shapes
Let the medium be discretized into rectangular parallelepipeds Vs of sizes dx x d y x d z . We let dx = ad, d y = bd, d z = cd where a, band c are dimensionless quantities and their ratios denote the relative sizes of the three sides of the rectangular parallelepiped. The exclusion volume Vb for the dyadic Green's function in this case will be an infinitesimal parallelepiped with dimensions Ox = aO, Oy = M and Oz = cO and the L is as given in (2.1.63). Note that the ratios of the sides of the finite small rectangular parallelepiped Vs is the same as that of the infinitesimal rectangular parallelepiped Vb. Then
2 INTEGRAL EQUATION FORMULATIONS AND NUMERICAL METHODS
it is useful to write a low frequency approximation of Go(R). We note that for R =I- 0
2- - " E - eikR { 2 (1 - ,;kR) 2 - - - } G(R) = -k 2A (R) = - 41rk 2R3 k (-R I+RR)+ R2 (R I-3RR)
(2.3.31) When expanding G(R) of (2.3.31), it is important to see that there is a singular part of 0(1/(k 2R 3)) that is non-integrable over the origin. We also have to expand to the leading term in the imaginary part because that accounts for radiative correction. The Green's function G(R), on expansion will give 0(1/(k 2R 3)) + 0(1/ R) + iO(k). Thus in (2.3.31), we write exp(ikR) '::::' 1 + ikR - k 2R 2/2 - ik 3R 3/6. We have to include -ik 3R 3/6 because this gives a term of order iO(k) when multiplied with the second term inside the curly bracket of (2.3.31). Thus for R =I- 0 and kR 1
2 2 G(R) '::::' - 41r:2R3 {k (_R ] +
+ RR)(l + ikR)
(1 - ikR) 2= -k 2R 2 k3 R 3 } R2 (R 1- 3RR)(1 +ikR- -2- - i --)
= - 41rk12 R5 (R 2] - 3R R) +
8~ {~3 (R 2] + R R) + 4~k]}
(2.3.32)
Note that the imaginary part term of (2.3.32) is just the product of a constant and a unit dyad. Following (2.3.17), let
= = - -E S 2
dr' = A(r r') - -L2 'k V,-V8
(2.3.33)
We use (2.3.32) to write A as a sum of a regular part A o that is integrable over the origin and a singular part As that is non-integrable over the origin. Thus (2.3.34) where
(2.3.35a) (2.3.35b)
Thus (2.3.36)
3.3 Other Shapes
Note that As is non-integrable over the origin. However, the origin is excluded in the integration over Vs - Vb. The second integral in (2.3.36) can be shown generally to be zero. Here we perform it for the case of rectangular parallelepipeds of Vs and Vb. The volume Vs - Vb can be formed from 8 octants. In integration, the cross terms vanish. The volume integration can also be combined into that of one octant. Thus
The integral over A o can be performed as follows. The dyad S is diagonal so that
S = Sxxx + S/iry + Szzz L = Lxxx + L/ir[; + Lzzz
and L
(2.3.38) (2.3.39)
S=D-2
(2.3.40) (2.3.41 )
D where
= Dxx + DyY + Dzz
(2.3.42) (2.3.43) (2.3.44)
2 INTEGRAL EQUATION FORMULATIONS AND NUMERICAL METHODS
(2.3.45)
(2.3.46)
+ [2 dz 2z tan- 1
(2.3.47)
Substituting (2.3.40) into (2.3.19), we get
_ _ _
---.l!..L = ~ . E inc - ~
E~V E~V E~V
L (E~ V)A
i ..
-.!!.L
(2.3.48)
where (2.3.49) (2.3.50)
3.3 Other Shapes
(3iy
(2.3.51)
(Epi E
1) (L Y - DY k 2 )
(2.3.52)
and ~V = dxdyd z . In the case of cells of circular cylindrical shape of radius a and length l, the corresponding results of Lx, L y , L z and D x , D y and D z are L - L _ l x - y - 2(4a 2 + l2)1/2
(2.3.53a) (2.3.53b)
=1-----,--------