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aU'vI)(I)(k z ) =AMM(kop, k z, k pp, k z, a l"n )c(M)(I)(kz ) n n + A~N (k op , k z , k pp , k z , al)c~'V)(I)(kz)
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202 First- and Second-Order Solutions
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[e~~)(j) (k~)Rg A~M (k op , k z , k~p, k~, aj)
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+ c~r:)(j) (k~)Rg A~N (k op , k z , k~p, k~, aj)] + (U(k z ) X
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dk;sinc ( (k;
,,2ik, ("
In-n,(kop!Pj - Pll)e-i(n-n') PjPI-ikz(ZJ-ZI) ROl(k z )
[c~~)(j) (k~)Rg A~M (k op , -k z , k~p, k~, aj)
(N)(j)(k')R 9 AMN(kop, - k z, k' k'z, aj )] + cn' z n' pp' (N)(l)(k z ) - n -ANM(kop, k z, k pp, k z, al ) cn (M)(l)(k z ) an + A;;N (k op , k z , kpp , k z , al)e~N)(I) (k z )
= _ iRiorl ~Vi (_1)ne-in ,~imrj28(kz + kiz )
- ikiOrl SOl (kiz)e2ikiZd ~V1 (-1 )ne-in</>,-imrj28(kz - k iz )
f; N (
L 2;
0) n'~oo
1 dk~sinc (k~
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L ) - kz)-;f
1-0 _-!)e-i(n-n')r/JPjPI-ikz(zJ-ZI) op PJ PI NM(k op, k z, k' k'z,aj ) c(M)(j)(k/) pp' z [R 9 A n' n'
n-n' (k
NN(k (N)(j)(k/)] + R 9 A n' op, k z, k'pp' k'z,aj ) en' z
+ (U(k z ) +
t, (-[;:) n~= 1:
U(k z - k))
((k; + k.) ~i) e2ik ,(.,+d)
In-nl (koplPj - (11)e -i(n-n')<'&PjPl-ikz(zj-Zl) SOl (k z ) [Rg A~M (k op , -k z , k~p,
k~, aj)c~~1)(j) (k~)
+ Rg A;;,N (k op , -kz , k~p, k~, aj)c~~)(j)(k~)]
where U is Heaviside step function. Equations (12.2.53a) and (12.2.53b) contain 2N integral equations for the 2N unknowns dn,M)(l) (k z ) and c~N)(I) (k z ), l = 1,2, ... ,N. Once they are solved numerically, these will give the "final" internal field coefficients c~M)(l) (k z ) and dn,N)(I) (k z ). These can then be substituted back into (12.2.48) to find the "final" scattered field from each cylinder j. Summation over all E~j) then gives the final scattered field E s (which can be taken in the far field) in the scattered direction of
(Os, s),
((k~ -
ksz ) L
(M)(j)(k') n pp' z,aj ) cn z { -2 hA s [R 9 AMM(ksp, k sz, k' k'
+ Rg A~fN (k sp , k sz , k~p, k~, aj )c~N)(j) (k~)]
(M)(j)(k') - vs [R9 ANM(ksp, k sz, k' k'z' aj ) cn n pp' z
+ Rg A;;N (k sp , k sz , k~p, k~, aj)cW)(j) (k~)] }
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j=l n=-(X)
dk z'e2ikSZ(ZJ+d)sinc ((k:
+ k sz ) L ) d
. { -iltsROl (k sz ) [c~M)(j) (k~)Rg A~M (k sp , -k sz , k~p, k~, aj)
2.2 First- and Second-Order Solutions
+ c~N)(j) (k~)Rg Ar:!N (k SP1 -k sz , k~p, k~, aj)]
- vsS01(k sz ) [c~M)(j)(k~)Rg A~M (k sp , -kSZ1 k~Pl k~, aj)
+ c~N)(j) (k~)Rg A~N (k sp , -k SZ1 k~Pl k~, aj)] }
where k sp = k sin Os and k sz = k cos Os' A convenient way of solving the Foldy-Lax multiple scattering equations of (12.2.53a) and (12.2.53b) is by iteration. The first-order solution can also be derived analytically. Let c(M)(j) (k z ) = c(M)(j)d(_ktZ )<5(k z + k tZ )iKi '-Tj n n + c~M)(j)U(kiz)<5(kz - kiz)ik"TJe2ik,zd + c~N)(j)H (k z ) (12.2.55)
c(N)(j)(k Z ) = c(N)(j)d(_k 'lZ )8(kZ 'n n
)iKi 'TJ
+ c~N)(j)U(kiz)<5(kz - kiz)ik;'Tie2ikiZd + c~N)(j)H (k z ) (12.2.56)
where superscripts 7L and d denote upward- and downward-going first-order solutions, respectively, and superscript H denotes higher-order solution. On substituting (12.2.55)-(12.2.56) into (12.2.53) and balancing the coefficients of the delta functions, the first-order solution of the internal field obeys the equation MM(k tp, -k k -k aJ')c(M)(j)d(_k' ) An tZ, n
~z, ~Ppl ~z
MN(k + A n tp, -k tZ, k
-k 1 aJ')c(N)(j)d(_k'tz ) tz n (12.2.57a)
= _ ~hi (_lte- in i-imr/2
NM(k tpl -k k tpp, - k aJ )c(M)(j)d(_k ) An tZ, n NN(k "n + A n tp, -k k tpp, -k tz, aJ')c(N)(j)d(_k'tZ ) = _ E Vi (_1)n e -in</Ji-imr/2 kip MM(k tp, k tZ, k tpp, k tz 1 aJ )c(M)(j)U(k ) An n tz